Chapter 4 Practical Geometry (Additional Questions)

The correct option is (C) 5.

To construct a unique quadrilateral, we need a minimum of five independent measurements.

A quadrilateral is defined by 8 parameters (4 sides and 4 angles). However, these parameters are not all independent. For instance, the sum of the interior angles of a quadrilateral is always $360^\circ$.

Providing fewer than 5 measurements usually allows for multiple possible quadrilaterals to be constructed with those given values. For example, if you are only given the four side lengths, you can form different quadrilaterals (e.g., a rectangle and a parallelogram with the same side lengths but different angles).

The five measurements provide sufficient constraints to fix the shape and size of the quadrilateral uniquely. Common sets of five measurements that allow for unique construction include:

1. Four sides and one diagonal.

2. Three sides and two included angles.

3. Two adjacent sides and three angles.

Therefore, the minimum number of measurements required to construct a unique quadrilateral is 5.

The correct option is (C) Four sides and one diagonal.

To construct a unique quadrilateral, we need a specific set of measurements that completely determines its shape and size. A unique quadrilateral can be constructed if we have a minimum of five independent measurements in certain configurations.

Let's examine the options provided:

(A) Four sides and one angle: This set is generally insufficient for a unique quadrilateral. For example, knowing the four side lengths and one angle at a vertex does not prevent the other vertices from being in different positions, leading to different quadrilaterals (unless it's a specific type like a parallelogram where opposite angles are equal, or a cyclic quadrilateral with a specific diagonal position relative to the angle).

(B) Three sides and two diagonals: This set of measurements can sometimes lead to two possible non-congruent quadrilaterals.

(C) Four sides and one diagonal: This set is sufficient for constructing a unique quadrilateral. Let the four sides be $a, b, c, d$ and the diagonal be $p$. A quadrilateral can be uniquely divided into two triangles by a diagonal. If we have the lengths of the four sides and one diagonal, say $AC=p$, then we can construct $\triangle ABC$ with sides $a, b, p$ and $\triangle ADC$ with sides $d, c, p$.

By the SSS (Side-Side-Side) congruence criterion, a triangle is uniquely determined by the lengths of its three sides (provided the triangle inequality holds). Thus, $\triangle ABC$ and $\triangle ADC$ are unique.

By constructing $\triangle ABC$ and $\triangle ADC$ on opposite sides of the common diagonal $AC$, we obtain a unique quadrilateral ABCD.

(D) Three angles and one diagonal: This set is insufficient. While angles determine the shape (up to similarity), knowing only one diagonal length along with three angles does not uniquely fix the size of the quadrilateral. You could have similar quadrilaterals with different side and diagonal lengths but the same angles.

Therefore, the set of measurements consisting of Four sides and one diagonal is sufficient to construct a unique quadrilateral.

The correct option is (B) Triangle inequality theorem.

When constructing a quadrilateral with four sides and one diagonal, we essentially construct two triangles that share the diagonal as a common side.

For example, if the quadrilateral is ABCD and the diagonal is AC, we construct $\triangle ABC$ using sides AB, BC, and AC, and $\triangle ADC$ using sides AD, DC, and AC.

The vertices are located by finding the intersection points of arcs drawn from the endpoints of the diagonal with radii equal to the lengths of the sides connected to those endpoints.

The Triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This property is fundamental because it ensures that the three given side lengths can actually form a triangle and allows us to determine the position of the third vertex relative to the other two.

If the lengths of the sides do not satisfy the triangle inequality with respect to the diagonal, then the triangles (and thus the quadrilateral) cannot be constructed, meaning the vertices cannot be located as planned.

Other options like the sum of angles in a triangle (which is $180^\circ$) or properties of parallel lines are not directly used for locating vertices using side lengths and a diagonal. Pythagoras theorem is specific to right-angled triangles and not generally applicable to all quadrilaterals constructed this way.

The correct option is (C) 5.

We are given:

- Three sides: This accounts for 3 measurements.

- Two included angles: This accounts for 2 measurements.

The total number of measurements given is the sum of these quantities.

Total measurements = (Number of sides) + (Number of angles)

Total measurements = $3 + 2 = 5$

Therefore, when you are given three sides and two included angles of a quadrilateral, you are given a total of 5 measurements.

The correct option is (D) Four angles.

To construct a unique quadrilateral, we typically need five independent measurements that fix both the shape and size of the figure.

Let's analyze the given options:

(A) Four sides and one diagonal: This is a standard case for unique construction. As discussed earlier, the diagonal divides the quadrilateral into two triangles, each uniquely determined by SSS criterion.

(B) Three sides and two diagonals: This set of measurements can sometimes lead to ambiguity, resulting in two possible non-congruent quadrilaterals. Therefore, it's not always considered a standard case for *unique* construction.

(C) Two adjacent sides and three angles: This is a standard case for unique construction. For example, knowing sides AB and BC, and angles A, B, and C allows us to draw AB, then an angle at B and side BC, then angles at A and C, whose rays will intersect to determine the fourth vertex D.

(D) Four angles: If only the four angles of a quadrilateral are given, we can determine the shape of the quadrilateral (e.g., it's a rectangle, a parallelogram, etc.), but not its size. For example, any rectangle has four $90^\circ$ angles, regardless of its side lengths. Any two rectangles with different side lengths will have the same four angles but are not congruent. Quadrilaterals with the same four angles are similar, not necessarily congruent or unique in size.

Since four angles alone are insufficient to determine a unique size for the quadrilateral, this is NOT a standard case for constructing a unique quadrilateral. While option (B) can also sometimes lead to non-uniqueness, option (D) provides the least information for determining a specific quadrilateral.

Therefore, the set of measurements that is NOT sufficient for unique construction is Four angles.

The correct option is (B) Two adjacent sides and the included angle.

A parallelogram has specific properties: opposite sides are equal and parallel, opposite angles are equal, and adjacent angles are supplementary (sum to $180^\circ$). These properties reduce the number of independent measurements needed for unique construction compared to a general quadrilateral.

Let's analyse the given options in the context of constructing a unique parallelogram:

(A) All four sides: If only the four side lengths are given, and we know it's a parallelogram (so opposite sides are equal), say sides $a, b, a, b$. While this confirms it's a parallelogram, it does not fix the angles. You can form different parallelograms by changing the angles (e.g., a rhombus can have varying angles while keeping side lengths constant). Therefore, this is not sufficient for unique construction.

(B) Two adjacent sides and the included angle: Let the adjacent sides be $a$ and $b$, and the included angle be $\theta$.

Step 1: Draw a line segment of length $a$.

Step 2: At one endpoint of the segment, draw a ray making an angle $\theta$. Mark a point on this ray at a distance $b$ from the vertex. This gives two adjacent vertices and defines one side and part of another.

Step 3: From the endpoint of side $a$ (not at angle $\theta$), draw a line parallel to the side of length $b$. From the endpoint of side $b$, draw a line parallel to the side of length $a$. The intersection of these two lines gives the fourth vertex.

Since opposite sides of a parallelogram are parallel and equal, this construction uniquely determines the parallelogram based on these three measurements (length of $a$, length of $b$, and angle $\theta$). This set of measurements (3 in total) is sufficient and is the minimum required using only side and angle information.

(C) All four angles: For a parallelogram, opposite angles are equal ($\alpha, \beta, \alpha, \beta$) and adjacent angles sum to $180^\circ$ ($\alpha + \beta = 180^\circ$). So, knowing one angle is enough to determine all four. However, knowing only the angles determines the shape but not the size. You can have infinitely many parallelograms with the same angles but different side lengths (similar parallelograms). Not sufficient for unique construction.

(D) One side and two diagonals: While this set (3 measurements) is also sufficient to construct a unique parallelogram (using the property that diagonals bisect each other), the question asks for information "about its sides and angles". Option (B) directly uses sides and angles, and is a standard construction method taught using these properties.

Considering the focus on "sides and angles" in the prompt and options, option (B) provides the minimum sufficient information (3 measurements) for unique construction among the choices primarily using sides and angles.

The correct option is (A) Opposite sides are parallel.

When constructing a parallelogram using two adjacent sides and the included angle, say sides AB and BC with included angle $\angle ABC$, the typical steps involve:

1. Draw the first side, say AB.

2. At vertex B, construct the given included angle and draw the second side, BC, along the ray.

3. To locate the fourth vertex D, we use the defining property of a parallelogram: opposite sides are parallel.

Specifically, we know that AD must be parallel to BC and CD must be parallel to AB.

One common construction method uses this directly:

- Draw a line through C parallel to AB.

- Draw a line through A parallel to BC.

The intersection of these two parallel lines is the vertex D.

Alternatively, knowing that opposite sides are equal ($AD = BC$ and $CD = AB$), we can draw an arc from A with radius equal to BC and an arc from C with radius equal to AB. Their intersection is D. The property that opposite sides are equal is a consequence of opposite sides being parallel.

While other properties like opposite angles being equal or adjacent angles being supplementary are true for parallelograms, the property of opposite sides being parallel is the most direct one used to locate the missing vertex by drawing parallel lines or using the equality of opposite sides derived from parallelism, after the initial adjacent sides and angle are drawn.

Diagonals bisecting each other (C) is a property used in a different construction method.

The correct option is (A) Length and breadth.

A rectangle is a special type of quadrilateral where all four angles are right angles ($90^\circ$) and opposite sides are equal in length.

To construct a unique rectangle, we need enough information to fix its dimensions (length and breadth). Let's examine the options:

(A) Length and breadth: If we know the length ($l$) and the breadth ($b$), we can construct the rectangle.

Step 1: Draw a line segment equal to the length ($l$).

Step 2: At one endpoint, draw a line segment perpendicular to the first one, equal to the breadth ($b$). This forms a $90^\circ$ angle.

Step 3: Complete the rectangle using the properties of a rectangle (opposite sides equal and parallel, all angles $90^\circ$). This uniquely determines the rectangle.

This set requires 2 measurements (length and breadth).

(B) One diagonal: Knowing only the length of one diagonal is not sufficient. Many different rectangles (with varying lengths and breadths) can have the same diagonal length. For example, a rectangle with sides 3 and 4 has a diagonal of $\sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5$. A rectangle with sides 1 and $\sqrt{24}$ also has a diagonal of $\sqrt{1^2 + (\sqrt{24})^2} = \sqrt{1+24} = \sqrt{25} = 5$. These are different rectangles.

(C) Perimeter: The perimeter of a rectangle is $P = 2(l+b)$. Knowing the perimeter only gives the sum of the length and breadth, not their individual values. For instance, rectangles with dimensions $l=5, b=3$ (perimeter 16) and $l=6, b=2$ (perimeter 16) have the same perimeter but are different rectangles.

(D) Area: The area of a rectangle is $A = l \times b$. Knowing the area only gives the product of the length and breadth. For instance, rectangles with dimensions $l=6, b=4$ (area 24) and $l=8, b=3$ (area 24) have the same area but are different rectangles.

To uniquely define a rectangle, we need two independent parameters. Knowing the length and breadth provides these two parameters directly.

While other pairs of measurements like 'one side and one diagonal' or 'one diagonal and the angle it makes with a side' are also sufficient (these also effectively provide 2 independent measurements), among the given options, 'Length and breadth' is the fundamental set of measurements defining the sides of a rectangle.

The correct option is (A) One side and one angle.

A rhombus is a special type of quadrilateral where all four sides are equal in length. It is also a parallelogram, meaning opposite sides are parallel and opposite angles are equal.

Let's examine the given options to see which combination is sufficient to construct a unique rhombus:

(A) One side and one angle: Let the side length be $s$ and one angle be $\alpha$. Since all sides are equal, knowing one side means all four sides are $s$. Knowing one angle, say $\alpha$, means the opposite angle is also $\alpha$. The other two angles are $180^\circ - \alpha$. With a side and an included angle, a parallelogram (and thus a rhombus with all sides equal) can be uniquely constructed.

Step 1: Draw a line segment of length $s$.

Step 2: At one endpoint, construct an angle $\alpha$ and draw a line segment of length $s$ along this ray.

Step 3: From the endpoints of these two segments, draw arcs of radius $s$. Their intersection point is the fourth vertex. This uniquely determines the rhombus.

This set requires 2 measurements (side length and angle).

(B) The lengths of the two diagonals: Let the diagonals be $d_1$ and $d_2$. The diagonals of a rhombus bisect each other at right angles.

Step 1: Draw one diagonal, say length $d_1$. Mark its midpoint.

Step 2: At the midpoint, draw a line perpendicular to the first diagonal. Mark points on this perpendicular line on either side of the midpoint, each at a distance of $d_2/2$.

Step 3: Connect the endpoints of the diagonals. This forms a unique rhombus.

This set also requires 2 measurements (lengths of the two diagonals). However, option (A) is given as "One side and one angle" which is a standard approach. Option (B) uses properties of diagonals, not directly sides and angles.

(C) The perimeter: The perimeter of a rhombus is $4s$. Knowing the perimeter only tells you the side length $s = P/4$. It does not give any information about the angles, so you can form different rhombuses with the same side length but different angles (e.g., a square is a rhombus with $90^\circ$ angles, but a rhombus with $60^\circ$ and $120^\circ$ angles has the same side length). Not sufficient for unique construction.

(D) Only one side: Knowing only one side is equivalent to knowing all four sides are equal. As discussed in (C), this is not sufficient for unique construction.

Both (A) and (B) provide sufficient information for unique construction (2 independent measurements). However, option (A) "One side and one angle" directly relates to sides and angles, which is a common way to define and construct many quadrilaterals. Option (B) uses properties of diagonals.

Considering standard geometric construction methods, providing a side length and an interior angle is a very direct way to begin constructing a parallelogram/rhombus. Therefore, (A) is a standard sufficient combination.

The correct option is (B) Diagonals bisect each other at right angles.

When constructing a rhombus using the lengths of its two diagonals, say $d_1$ and $d_2$, the standard construction method relies directly on the key property of a rhombus's diagonals.

The steps for construction are typically:

1. Draw one diagonal, say of length $d_1$. Let its endpoints be A and C.

2. Find the midpoint of this diagonal AC. Let this midpoint be O.

3. At the midpoint O, draw a line segment perpendicular to AC.

4. Along this perpendicular line, mark two points, B and D, on either side of O, such that $OB = OD = d_2/2$.

5. Connect the points A, B, C, and D.

This construction directly uses the fact that the diagonals bisect each other (step 2 and 4, as O is the midpoint of both diagonals when constructed this way) and that they are at right angles (step 3, drawing the perpendicular line).

While the other properties (all sides equal, opposite angles equal, adjacent angles supplementary) are true for a rhombus, they are consequences of the diagonals' properties or are used in different construction methods. The construction using diagonal lengths fundamentally depends on their intersection at the midpoint and perpendicularity.

The correct option is (D) Any of the above.

A square is a regular quadrilateral, meaning all its sides are equal in length and all its interior angles are equal ($90^\circ$). Due to its high symmetry, a square can be uniquely determined by a single measurement related to its size.

Let's consider each option:

(A) Side length: If the side length, say $s$, is given, we can construct a square.

Step 1: Draw a line segment of length $s$.

Step 2: At each endpoint, construct a perpendicular line segment of length $s$.

Step 3: Connect the endpoints of the perpendicular segments. This forms a unique square.

This set of information is sufficient.

(B) Diagonal length: If the diagonal length, say $d$, is given, we can construct a square. The diagonals of a square are equal, bisect each other at $90^\circ$, and the relationship between the side $s$ and diagonal $d$ is $d = s\sqrt{2}$.

Step 1: Draw a line segment equal to the diagonal length $d$. This will be one diagonal.

Step 2: Find the midpoint of this diagonal.

Step 3: Draw the perpendicular bisector of the diagonal.

Step 4: Along the perpendicular bisector, mark points on either side of the midpoint at a distance of $d/2$. This forms the second diagonal.

Step 5: Connect the endpoints of the two diagonals. This forms a unique square.

This set of information is sufficient.

(C) Perimeter: The perimeter of a square with side length $s$ is $P = 4s$. If the perimeter $P$ is given, the side length is uniquely determined as $s = P/4$. Since knowing the side length is sufficient to construct a unique square, knowing the perimeter is also sufficient.

This set of information is sufficient.

Since knowing the measure of the side length, the diagonal length, or the perimeter is sufficient to construct a unique square, the answer is any of the above.

The correct option is (A) Ruler and Compass.

In classical Euclidean geometry, geometric constructions are performed using only a straightedge (often called a ruler, but used only for drawing straight lines, not measuring length) and a compass (for drawing circles and arcs). These two tools are considered the fundamental instruments for geometric constructions.

The ruler allows us to draw a line through any two given points.

The compass allows us to draw a circle with a given center and radius (or passing through a given point).

While a protractor (for measuring and drawing angles) and set squares (for drawing parallel and perpendicular lines) are useful drawing tools, they are not typically included in the basic set for formal geometric constructions unless specified.

Therefore, the tools typically used for geometric constructions are the Ruler and Compass.

Let's match each figure with the minimum measurements required for its unique construction:

(i) Quadrilateral: A general quadrilateral requires a minimum of 5 independent measurements for unique construction. This matches with option (d).

(ii) Parallelogram: A parallelogram is uniquely determined by two adjacent sides and the included angle. This is a set of 3 measurements (side-angle-side, which with parallelogram properties is sufficient). This matches with option (c).

(iii) Rectangle: A rectangle is a special parallelogram with all angles $90^\circ$. It is uniquely determined by its length and breadth (its two adjacent sides). This matches with option (b).

(iv) Square: A square is a special rectangle with all sides equal. It is uniquely determined by a single measure related to its size, such as its side length or diagonal length. Option (a) provides the side length. This matches with option (a).

Putting these together, we get the pairing: (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a).

This corresponds to option (A).

The correct option is (D) $a + b \le c$.

The possibility of constructing a triangle with given side lengths $a, b,$ and $c$ is governed by the Triangle Inequality Theorem.

The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

For a triangle to be constructible with side lengths $a, b,$ and $c$, the following three conditions must all be true:

$a + b > c$

$a + c > b$

$b + c > a$

If any one of these conditions is not met, then it is impossible to form a triangle with the given side lengths. The negation of "$> $" (greater than) is "$ \le $" (less than or equal to).

Let's look at the options:

(A) $a + b > c$: This is one of the conditions required for a triangle to be possible.

(B) $a + c > b$: This is another condition required for a triangle to be possible.

(C) $b + c > a$: This is the third condition required for a triangle to be possible.

(D) $a + b \le c$: This is the negation of the condition $a + b > c$. If $a + b$ is less than or equal to $c$, the two shorter "sides" combined are not long enough to "reach" across the third side.

Therefore, the condition that makes it impossible to construct a triangle is when the sum of the lengths of two sides is less than or equal to the length of the third side.

This corresponds to option (D) $a + b \le c$ (or equivalently, $a + c \le b$ or $b + c \le a$).

The correct option is (B) Any one diagonal.

We are given the lengths of the four sides of the quadrilateral ABCD: AB, BC, CD, and DA.

As discussed in previous questions, providing only the four side lengths is generally not sufficient to construct a unique quadrilateral. Multiple quadrilaterals with different shapes (angles) can have the same four side lengths.

To make the construction unique, we need additional information that fixes the shape. A common way to do this is to provide a fifth measurement.

Let's evaluate the options:

(A) Any one angle: Providing one angle (e.g., $\angle ABC$) along with the four sides is still often insufficient for unique construction. For example, if you know the four side lengths and one angle, you might still be able to "flex" the quadrilateral into different shapes.

(B) Any one diagonal: Providing the length of one diagonal (e.g., AC or BD) along with the four sides makes the construction unique. If we are given the lengths of AB, BC, CD, DA, and the diagonal AC, we can construct $\triangle ABC$ using sides AB, BC, and AC (SSS criterion). Then, we can construct $\triangle ADC$ using sides AD, CD, and AC (SSS criterion) on the opposite side of AC. Since each triangle is uniquely determined by its three side lengths, the resulting quadrilateral formed by joining these two triangles along the common diagonal AC is unique.

(C) The sum of two opposite angles: Knowing the sum of opposite angles (e.g., $\angle A + \angle C$) is useful, particularly for cyclic quadrilaterals (where the sum is $180^\circ$), but it is not generally sufficient with just four side lengths to guarantee a unique quadrilateral.

(D) The perimeter: The perimeter is simply the sum of the four given side lengths ($AB + BC + CD + DA$). Knowing the perimeter provides no new information beyond what is already given by the individual side lengths. Thus, it does not help in achieving unique construction.

Therefore, to construct a unique quadrilateral when four sides are given, the essential additional piece of information is the length of Any one diagonal.

The correct option is (D) None of the above (need at least one more measurement).

To construct a unique quadrilateral, we generally need a minimum of five independent measurements. Giving only the lengths of the four sides provides only four measurements, which is insufficient for a unique construction in the general case (as seen in Question 1 and Question 15).

Let's consider the specific types listed:

(A) Square: A square has four equal sides, say of length $s$. While knowing the four side lengths (all equal to $s$) tells us the side length, it does not inherently provide the angle information ($90^\circ$). If you were told it's a square and given its side length $s$, you could construct it uniquely using the $90^\circ$ angle property. But if you are only given the four side lengths {s, s, s, s} without explicitly stating it's a square, you could construct various rhombuses with side $s$ and different angles. Thus, the square itself isn't uniquely determined by *only* the side lengths as numeric values without using its angle property.

(B) Rhombus: A rhombus also has four equal sides of length $s$. Knowing the four side lengths {s, s, s, s} tells you the side length is $s$. However, the angles of a rhombus can vary (except for a square, which is a special rhombus). You can construct many non-congruent rhombuses with the same side length by changing the angles. Therefore, a rhombus cannot be uniquely constructed if only the lengths of the four sides are given.

(C) Rectangle: A rectangle has opposite sides equal, say length $l$ and breadth $b$. Knowing the four side lengths {l, b, l, b} tells you the length and breadth. While a rectangle has four $90^\circ$ angles, this angle information is a property of the shape, not explicitly given as one of the "lengths of the four sides". If you were to construct a figure using only side lengths {l, b, l, b} without using the $90^\circ$ property, you could construct various parallelograms with sides $l$ and $b$ and different angles. Thus, a rectangle isn't uniquely determined by *only* the side lengths as numeric values without using its angle property.

Since providing only the four side lengths is not sufficient to uniquely determine the shape (angles and diagonals) for any of these specific quadrilaterals without invoking their inherent angle properties (which are not given as length measurements), none of them can be uniquely constructed based *only* on this information.

Therefore, for unique construction, you need at least one more measurement in addition to the four side lengths (e.g., one diagonal or one angle).

The correct option is (A) Both A and R are true, and R is the correct explanation of A.

Let's evaluate the Assertion (A) and the Reason (R).

Assertion (A): It is possible to construct a unique quadrilateral ABCD with sides $AB = 5 \text{ cm}$, $BC = 6 \text{ cm}$, $CD = 7 \text{ cm}$, $DA = 8 \text{ cm}$, and diagonal $AC = 10 \text{ cm}$.

A quadrilateral with four sides and one diagonal can be constructed uniquely if the given side lengths satisfy the triangle inequality theorem for the two triangles formed by the diagonal.

Consider $\triangle ABC$ with sides $AB = 5$, $BC = 6$, and $AC = 10$.

$AB + BC > AC \implies 5 + 6 > 10 \implies 11 > 10$ (True)

$AB + AC > BC \implies 5 + 10 > 6 \implies 15 > 6$ (True)

$BC + AC > AB \implies 6 + 10 > 5 \implies 16 > 5$ (True)

Since the triangle inequality holds, $\triangle ABC$ can be uniquely constructed.

Consider $\triangle ADC$ with sides $AD = 8$, $CD = 7$, and $AC = 10$.

$AD + CD > AC \implies 8 + 7 > 10 \implies 15 > 10$ (True)

$AD + AC > CD \implies 8 + 10 > 7 \implies 18 > 7$ (True)

$CD + AC > AD \implies 7 + 10 > 8 \implies 17 > 8$ (True)

Since the triangle inequality holds, $\triangle ADC$ can be uniquely constructed.

By constructing $\triangle ABC$ and $\triangle ADC$ on opposite sides of the common diagonal AC, a unique quadrilateral ABCD is formed.

Therefore, Assertion (A) is True.

Reason (R): Four sides and one diagonal are sufficient measurements to construct a unique quadrilateral, provided the triangle inequality holds for the triangles formed by the diagonal.

This statement accurately describes one of the standard and sufficient conditions for the unique construction of a general quadrilateral. The condition about the triangle inequality holding is essential because without it, the two triangles sharing the diagonal as a side cannot be formed, making the quadrilateral construction impossible.

Therefore, Reason (R) is True.

Relationship between A and R:

Assertion (A) provides a specific example of a quadrilateral construction where four sides and one diagonal are given. Reason (R) provides the general geometrical principle that validates this type of construction for a unique quadrilateral, including the necessary condition of the triangle inequality. The specific instance in A is a direct application of the principle stated in R.

Hence, Reason (R) is the correct explanation for Assertion (A).

The correct option is (B) Draw the diagonal of length $10 \text{ cm}$ and use compass arcs of $6 \text{ cm}$ from its endpoints.

A rhombus is a quadrilateral with all four sides equal. We are given the side length ($s = 6 \text{ cm}$) and one diagonal length ($d = 10 \text{ cm}$).

A diagonal of a rhombus divides it into two congruent triangles. For example, if the rhombus is ABCD and the diagonal is AC, then $\triangle ABC$ and $\triangle ADC$ are formed. The sides of $\triangle ABC$ are AB, BC, and AC. Since it's a rhombus, AB = BC = $s = 6 \text{ cm}$, and AC = $d = 10 \text{ cm}$. Similarly, the sides of $\triangle ADC$ are AD, CD, and AC, where AD = CD = $s = 6 \text{ cm}$, and AC = $d = 10 \text{ cm}$.

Thus, we can construct the rhombus by constructing these two triangles using the SSS (Side-Side-Side) criterion.

The most efficient way to start constructing these two triangles sharing the diagonal is to first draw the common side, which is the diagonal.

Step 1: Draw a line segment representing the given diagonal, say AC, of length $10 \text{ cm}$.

Step 2: To locate the vertex B, which is part of $\triangle ABC$, use a compass to draw an arc from A with radius equal to the side length ($6 \text{ cm}$) and another arc from C with radius equal to the side length ($6 \text{ cm}$). The intersection of these two arcs gives the vertex B.

Step 3: To locate the vertex D, which is part of $\triangle ADC$, draw arcs on the opposite side of AC from A and C, both with radius $6 \text{ cm}$. The intersection gives the vertex D.

Step 4: Connect the vertices A, B, C, and D to form the rhombus.

Therefore, the first step is to draw the diagonal and then use compass arcs with the side length from its endpoints to locate the other vertices.

Let's consider why other options are not the best starting point:

(A) Drawing a side of $6 \text{ cm}$ first would then require constructing an angle. Without knowing the angle, this doesn't directly use the diagonal information.

(C) Drawing the diagonal and bisecting it perpendicularly is a step used to find the position of the *other* diagonal, which is useful if you're given the lengths of both diagonals, but less direct when given a side and a diagonal.

(D) Drawing a circle with radius $6 \text{ cm}$ could be a preliminary step to help visualise the side length, but it doesn't directly incorporate the diagonal length to define the vertices of the rhombus.

The correct option is (B) Diagonals are equal and bisect each other.

A rectangle is a quadrilateral with four right angles. Its diagonals have specific properties that are crucial for construction based on their lengths:

1. The diagonals are equal in length.

2. The diagonals bisect each other.

3. The diagonals intersect at the center of the circumscribing circle.

4. The length of each diagonal is equal to the diameter of the circumscribing circle.

If we are given the lengths of the two diagonals ($d_1$ and $d_2$), since it's a rectangle, we know $d_1 = d_2 = d$. So we are effectively given one length, $d$.

The construction process using the diagonal length relies on the property that the diagonals are equal and bisect each other. This allows us to draw the circumscribing circle.

Step 1: Draw a line segment representing one diagonal, say AC, of length $d$.

Step 2: Find the midpoint O of AC. This point O is the center of the circumscribing circle because the diagonals bisect each other.

Step 3: Since the diagonals are equal and are diameters of the circumscribing circle, draw a circle with center O and radius $d/2$. All four vertices of the rectangle lie on this circle.

Step 4: The diagonal AC is a diameter. To form a rectangle, the other diagonal BD must also be a diameter of this circle, passing through O. Any diameter other than AC will define the other two vertices B and D on the circle.

The fact that the vertices lie on a circle with the diagonal as a diameter ensures that the angles at the vertices A, B, C, D are $90^\circ$ (angle in a semicircle).

Let's look at the options again:

(A) Diagonals are equal and bisect each other at right angles: This property is true for a square (a special type of rectangle), but not for all rectangles. If the diagonals intersect at right angles, the rectangle is a square.

(B) Diagonals are equal and bisect each other: This is the correct property of a rectangle's diagonals. This property is essential for using the diagonal length to determine the circumcenter and circumcircle, which in turn guarantees the $90^\circ$ angles of the rectangle.

(C) All angles are $90^\circ$: This is a defining property of a rectangle, but it describes the angles, not the diagonals themselves or how their lengths are used in construction.

(D) Opposite sides are equal and parallel: This is a property of a parallelogram, and also true for a rectangle, but it doesn't directly explain how diagonal lengths are used in construction.

Therefore, the property that the Diagonals are equal and bisect each other is the most essential property that allows us to construct a rectangle using the lengths of its diagonals by relating them to the circumcircle and the $90^\circ$ vertex angles.

The correct option is (B) Square.

We are given the following properties for the quadrilateral ABCD:

1. All four sides are equal in length: $AB=BC=CD=DA=5 \text{ cm}$.

2. One angle is a right angle: $\angle ABC = 90^\circ$.

Let's analyze the properties based on the definitions of the given quadrilaterals:

• A Rhombus is a quadrilateral with all four sides equal. The given figure satisfies this property ($AB=BC=CD=DA=5 \text{ cm}$), so it is a rhombus.
• A Rectangle is a quadrilateral with all four angles equal to $90^\circ$. If one angle of a parallelogram (and a rhombus is a parallelogram) is $90^\circ$, then all its angles are $90^\circ$. Since the figure is a rhombus with one angle $90^\circ$, all its angles must be $90^\circ$. Thus, the given figure is also a rectangle.
• A Parallelogram is a quadrilateral with opposite sides parallel. A rhombus is a special type of parallelogram (with all sides equal). So the given figure is also a parallelogram.
• A Square is a quadrilateral with all four sides equal AND all four angles equal to $90^\circ$. The given figure has all sides equal ($5 \text{ cm}$) and, because one angle is $90^\circ$ in a rhombus, all its angles are $90^\circ$. Thus, the given figure perfectly fits the definition of a square.

The question asks for "what figure are you constructing?". While the figure possesses the properties of a rhombus, a rectangle, and a parallelogram, the most specific and accurate description based on having all sides equal and all angles equal to $90^\circ$ is a Square.

The correct option is (C) Included between.

To uniquely construct a parallelogram using information about its sides and angles, a standard method involves using the lengths of two adjacent sides and the angle formed by these two sides at their common vertex. This angle is referred to as the included angle.

Let the two adjacent sides have lengths $a$ and $b$, and let the angle between them be $\theta$.

Step 1: Draw a line segment of length $a$.

Step 2: At one endpoint of this segment, construct an angle equal to $\theta$.

Step 3: Along the ray forming angle $\theta$, draw a line segment of length $b$.

Step 4: The fourth vertex can be located by drawing lines parallel to the first two sides from their respective endpoints. This relies on the property that opposite sides of a parallelogram are parallel and equal.

The phrase "included between" precisely describes the position of the angle relative to the two adjacent sides that form it.

Options (A), (B), and (D) do not accurately describe the necessary angle for this standard construction method.

Therefore, the correct completion of the sentence is "Included between".

The correct options are (A) Two adjacent sides and one diagonal and (B) Two diagonals and the angle between them.

To construct a unique parallelogram, we need enough information to fix its shape and size. Let's examine each option:

(A) Two adjacent sides and one diagonal: Let the adjacent sides be of lengths $a$ and $b$, and the diagonal be of length $d$. A diagonal of a parallelogram divides it into two congruent triangles. If we consider the triangle formed by the two adjacent sides and the diagonal, its side lengths are $a, b,$ and $d$. By the SSS congruence criterion, a triangle with given side lengths is unique (provided the triangle inequality holds). Since the parallelogram is formed by two such congruent triangles joined along the diagonal, and their relative positions are fixed, the parallelogram is uniquely determined.

This information is sufficient.

(B) Two diagonals and the angle between them: Let the lengths of the two diagonals be $d_1$ and $d_2$, and let the angle between them be $\theta$. The diagonals of a parallelogram bisect each other. Let the intersection point be O. The diagonals divide the parallelogram into four triangles. Consider the triangle formed by half of each diagonal and the angle between them, say with sides $d_1/2$ and $d_2/2$ and included angle $\theta$. By the SAS congruence criterion, this triangle is uniquely determined. The other triangles around the center O are also determined. Since the vertices of the parallelogram are the endpoints of the diagonals, and the diagonals are fixed in position and angle by this information, the parallelogram is uniquely determined.

This information is sufficient.

(C) Two adjacent sides and the altitude to one of these sides: Let the adjacent sides be $a$ and $b$, and the altitude to side $a$ be $h_a$. The area of the parallelogram is $A = a \times h_a$. The area is also given by $A = ab \sin(\theta)$, where $\theta$ is the angle included between sides $a$ and $b$. So, $a h_a = ab \sin(\theta)$, which implies $\sin(\theta) = h_a / b$. If $h_a < b$, there are generally two possible values for the angle $\theta$ between $0^\circ$ and $180^\circ$ (one acute and one obtuse), which would result in two different parallelograms with the same side lengths and altitude. For example, a parallelogram with sides 5 and 10, and altitude 4 to the side of length 10 ($\sin(\theta) = 4/5 = 0.8$) can have an acute angle $\theta \approx 53.13^\circ$ or an obtuse angle $\theta \approx 126.87^\circ$.

This information is generally not sufficient for unique construction.

(D) The lengths of all four sides: For a parallelogram, opposite sides are equal, so knowing all four side lengths essentially means knowing the lengths of two adjacent sides ($a$ and $b$). As discussed in previous questions, knowing only the lengths of the adjacent sides is not sufficient to uniquely determine the parallelogram, as the angles can vary (e.g., different rhombuses can have the same side length). This information defines a family of parallelograms, not a unique one.

This information is not sufficient.

Based on this analysis, the combinations sufficient for constructing a unique parallelogram are (A) and (B).

The correct option is (B) Three sides and two included angles.

Let's identify the given measurements from the case study:

• Side $PQ = 4 \text{ cm}$ (length)
• Side $QR = 6 \text{ cm}$ (length)
• Side $RS = 5 \text{ cm}$ (length)
• Angle $\angle PQR = 70^\circ$ (angle)
• Angle $\angle QRS = 120^\circ$ (angle)

We have a total of $3$ side lengths and $2$ angle measures, making a total of $5$ measurements.

Now, let's check if the angles are included with respect to the given sides:

• The angle $\angle PQR$ is formed by the sides PQ and QR. Both PQ and QR are among the given sides. Thus, $\angle PQR$ is the angle included between sides PQ and QR.
• The angle $\angle QRS$ is formed by the sides QR and RS. Both QR and RS are among the given sides. Thus, $\angle QRS$ is the angle included between sides QR and RS.

Therefore, the information provided consists of Three sides ($PQ, QR, RS$) and two included angles ($\angle PQR, \angle QRS$).

Let's compare this with the given options:

(A) Four sides and one angle: Incorrect, only three sides are given.

(B) Three sides and two included angles: Correct, as identified above.

(C) Two adjacent sides and three angles: Incorrect count of sides and angles.

(D) Three sides and two non-included angles: Incorrect, the angles are included between pairs of the given sides.

The correct option is (A) Yes.

The given measurements in the Case Study (Question 23) are:

$PQ = 4 \text{ cm}$

$QR = 6 \text{ cm}$

$RS = 5 \text{ cm}$

$\angle PQR = 70^\circ$

$\angle QRS = 120^\circ$

This set of measurements consists of the lengths of three sides ($PQ, QR, RS$) and the measures of the two angles included between these sides ($\angle PQR$ is between PQ and QR, $\angle QRS$ is between QR and RS).

This combination of measurements (3 sides and 2 included angles) is one of the standard sets of five independent measurements that is sufficient to construct a unique quadrilateral.

The construction steps would be:

1. Draw the line segment QR of length $6 \text{ cm}$.

2. At point Q, construct an angle of $70^\circ$. Draw a ray along this angle.

3. On this ray, mark point P such that $QP = 4 \text{ cm}$.

4. At point R, construct an angle of $120^\circ$. Draw a ray along this angle.

5. On this ray, mark point S such that $RS = 5 \text{ cm}$.

6. Join points P and S.

Since the positions of points Q, R, P, and S are uniquely determined by these steps, the quadrilateral PQRS is uniquely constructed.

Therefore, a unique quadrilateral can be constructed with the given measurements.

The correct option is (A) $13 \text{ cm}$.

A key property of a rhombus is that its diagonals bisect each other at right angles.

Let the lengths of the two diagonals be $d_1$ and $d_2$. When the diagonals intersect, they form four congruent right-angled triangles.

The lengths of the legs of each of these right-angled triangles are half the lengths of the diagonals, i.e., $\frac{d_1}{2}$ and $\frac{d_2}{2}$. The hypotenuse of each of these right-angled triangles is a side of the rhombus (let the side length be $s$).

We are given $d_1 = 10 \text{ cm}$ and $d_2 = 24 \text{ cm}$.

Half of the diagonals are:

$\frac{d_1}{2} = \frac{10}{2} \text{ cm} = 5 \text{ cm}$

$\frac{d_2}{2} = \frac{24}{2} \text{ cm} = 12 \text{ cm}$

Using the Pythagorean theorem in one of the right-angled triangles, we have:

$(side)^2 = (\frac{d_1}{2})^2 + (\frac{d_2}{2})^2$

$s^2 = (5)^2 + (12)^2$

$s^2 = 25 + 144$

$s^2 = 169$

To find the side length $s$, take the square root of 169:

$s = \sqrt{169}$

$s = 13$

So, the side length of the rhombus is $13 \text{ cm}$.

The correct option is (B) All angles are $90^\circ$.

We are given one side length (say, length $l = 5 \text{ cm}$) and one diagonal length (say, $d = 13 \text{ cm}$) of a rectangle.

Consider a rectangle ABCD. Let AB be the side with length $5 \text{ cm}$ and AC be the diagonal with length $13 \text{ cm}$.

In a rectangle, all interior angles are $90^\circ$. This means that the triangle formed by a side, an adjacent side, and the diagonal connecting their endpoints is a right-angled triangle.

For example, $\triangle ABC$ is a right-angled triangle with $\angle B = 90^\circ$. The sides of this triangle are AB, BC, and the diagonal AC (which is the hypotenuse).

By the Pythagorean theorem, in $\triangle ABC$:

$AB^2 + BC^2 = AC^2$

We have $AB = 5 \text{ cm}$ and $AC = 13 \text{ cm}$. We can find the length of the adjacent side BC (the breadth, $b$):

$5^2 + BC^2 = 13^2$

$25 + BC^2 = 169$

$BC^2 = 169 - 25 = 144$

$BC = \sqrt{144} = 12 \text{ cm}$

The construction of the rectangle proceeds by using the property that the angles are $90^\circ$:

1. Draw the side AB of length $5 \text{ cm}$.

2. At point B, construct a perpendicular line (using the $90^\circ$ property).

3. From point A, draw an arc with radius $13 \text{ cm}$ (the diagonal length).

4. The intersection of the perpendicular line from B and the arc from A locates the vertex C.

5. Complete the rectangle using the fact that opposite sides are equal and angles are $90^\circ$.

The property that all angles are $90^\circ$ is essential because it guarantees that the triangle formed by the side and diagonal is a right-angled triangle, allowing us to determine the other side length or to use perpendicular lines in the construction process based on the given diagonal length. While other properties (A, C, D) are true for a rectangle, the $90^\circ$ angle property is the most directly applied when using a side and a diagonal to locate the vertices and determine the other dimension.

The correct option is (A) Two congruent triangles.

A parallelogram is a quadrilateral where opposite sides are parallel and equal in length.

Any diagonal of a parallelogram divides it into two triangles.

Let the parallelogram be ABCD and consider the diagonal AC.

This diagonal splits the parallelogram into $\triangle ABC$ and $\triangle ADC$.

In a parallelogram ABCD:

By the SSS (Side-Side-Side) congruence criterion, $\triangle ABC \cong \triangle ADC$.

When you are given the lengths of two adjacent sides (say AB and BC) and one diagonal (say AC), you have the three side lengths of $\triangle ABC$. You can uniquely construct this triangle using the SSS method.

Once $\triangle ABC$ is constructed, you effectively have $\triangle ADC$ determined as well because you know its sides (AD=BC, CD=AB, AC=AC). Since $\triangle ADC$ is congruent to $\triangle ABC$, constructing $\triangle ABC$ and then positioning $\triangle ADC$ correctly on the other side of the diagonal AC essentially constructs the parallelogram.

Therefore, the fundamental geometric shapes being constructed when using this information are two congruent triangles.

The correct option is (A) $d = s\sqrt{2}$.

Consider a square with side length $s$. Let the diagonal be $d$. A diagonal of a square divides it into two congruent right-angled triangles.

In any right-angled triangle formed by two adjacent sides and a diagonal of the square, the two sides of the square form the legs of the right triangle, and the diagonal is the hypotenuse.

Let the side lengths be $s$ and the diagonal be $d$. By the Pythagorean theorem, in a right-angled triangle with legs of length $s$ and hypotenuse of length $d$, we have:

$s^2 + s^2 = d^2$

$2s^2 = d^2$

To find the relationship between $d$ and $s$, take the square root of both sides:

$\sqrt{2s^2} = \sqrt{d^2}$

$s\sqrt{2} = d$

Thus, the relationship between the diagonal length $d$ and the side length $s$ of a square is $d = s\sqrt{2}$.

The correct option is (D) There is no single minimum set of 5 measurements for all trapeziums.

A general quadrilateral requires a minimum of five independent measurements for unique construction (e.g., four sides and one diagonal). A trapezium has the property of having at least one pair of parallel sides, which adds a constraint compared to a general quadrilateral.

Let's examine the options and common construction requirements for a unique trapezium:

(A) Four sides: Knowing the lengths of the four sides of a trapezium (even if you know which pair is parallel) is not sufficient for unique construction. Different trapeziums with different angles can be formed with the same four side lengths.

(B) Two parallel sides and the height: This information (3 measurements) is sufficient to determine the area, but not the unique shape of the trapezium. The non-parallel sides can have varying lengths and angles relative to the bases while maintaining the given parallel sides and height.

(C) Three sides and two included angles: This set consists of 5 measurements. A set of 5 measurements, such as three sides and two included angles, is generally sufficient to construct a unique quadrilateral. If the quadrilateral constructed with these measurements happens to satisfy the condition of having a pair of parallel sides, then it is a unique trapezium. This is a valid set of sufficient measurements, but it contains 5 measurements.

However, there are configurations where a unique trapezium can be constructed with fewer than 5 measurements.

Consider a trapezium defined by the lengths of the two parallel sides and the two diagonals. This is a set of four measurements. As shown mathematically (by determining the coordinates of vertices based on these lengths), this set of four measurements is sufficient to construct a unique trapezium (provided the parallel sides have different lengths). This indicates that 5 is not always the minimum number of measurements required.

Given that a unique trapezium can be constructed with 4 measurements in certain cases (like parallel sides and diagonals), the statement that 5 is the minimum number required for *all* trapeziums is false. Also, there isn't one single configuration of measurements (like "Side-Angle-Side" for triangles or "adjacent sides and included angle" for parallelograms) that is universally referred to as the "minimum set of 5 measurements" for all trapeziums. The specific type of 5 measurements needed often varies depending on which sides are parallel and which angles are known.

Therefore, option (D) is the most accurate statement among the choices provided, as 5 is not the universal minimum number of measurements for all trapeziums (some cases require only 4), and there isn't one single canonical type of 5-measurement set for trapezium construction.

The correct option is (D) Constructing a parallelogram using two adjacent sides and included angle.

A rhombus is a special type of parallelogram where all four sides are equal in length.

When you are given the lengths of two sides of a rhombus, since all sides are equal, you are essentially given the side length of the rhombus, let's call it $s$. The fact that you are given "two sides" and the "angle between them" implies these are adjacent sides.

So, the given information is:

• Length of one side = $s$
• Length of an adjacent side = $s$ (since all sides are equal)
• The angle included between these two adjacent sides = $\theta$

This set of information (lengths of two adjacent sides and the included angle) is precisely the standard information needed and used to construct a parallelogram.

Since a rhombus is a type of parallelogram, the construction method for a parallelogram using two adjacent sides and the included angle applies directly to a rhombus when those adjacent sides are of equal length.

Let's look at the options:

(A) SSS construction of a quadrilateral: This refers to using side lengths. While the rhombus has four sides, SSS is a triangle congruence criterion, not a standard method for unique quadrilateral construction from just side lengths.

(B) SAS construction of a quadrilateral: SAS is also a triangle congruence criterion. While the principle is used, the phrasing "SAS construction of a quadrilateral" isn't a standard term for constructing the entire quadrilateral from this specific information.

(C) Constructing two congruent triangles using SAS: A rhombus is indeed made of two congruent triangles, and those triangles can be constructed using SAS if you use a diagonal and the two sides meeting at one end of the diagonal. However, the given information (adjacent sides and *their* included angle) is more directly related to the parallelogram construction method starting from one vertex.

(D) Constructing a parallelogram using two adjacent sides and included angle: This exactly matches the given information for a rhombus (where the adjacent sides are equal) and describes a standard construction method for parallelograms.

Thus, constructing a rhombus with given adjacent sides and included angle is equivalent to constructing a parallelogram with those specific adjacent sides and included angle.

The correct option is (A) Both A and R are true, and R is the correct explanation of A.

Let's evaluate the Assertion (A) and the Reason (R).

Assertion (A): A rectangle with side lengths $4 \text{ cm}$ and $5 \text{ cm}$ is congruent to a parallelogram with adjacent sides $4 \text{ cm}$ and $5 \text{ cm}$ and one angle $90^\circ$.

A rectangle with side lengths $4 \text{ cm}$ and $5 \text{ cm}$ is a unique figure with all four angles being $90^\circ$.

Consider a parallelogram with adjacent sides $4 \text{ cm}$ and $5 \text{ cm}$ and one angle $90^\circ$. Let the adjacent sides be $a = 4 \text{ cm}$ and $b = 5 \text{ cm}$, and the included angle be $90^\circ$. In a parallelogram, adjacent angles are supplementary. So, if one angle is $90^\circ$, the angle adjacent to it is $180^\circ - 90^\circ = 90^\circ$. Opposite angles in a parallelogram are equal. Thus, all four angles of this parallelogram must be $90^\circ$. A parallelogram with all angles equal to $90^\circ$ is a rectangle.

So, a parallelogram with adjacent sides $4 \text{ cm}$ and $5 \text{ cm}$ and one angle $90^\circ$ is precisely a rectangle with side lengths $4 \text{ cm}$ and $5 \text{ cm}$.

Two geometric figures are congruent if they have the same shape and size. Since both descriptions refer to the same unique rectangle, they are congruent.

Therefore, Assertion (A) is True.

Reason (R): A rectangle is a parallelogram with one angle equal to $90^\circ$.

This is a fundamental definition of a rectangle. If a parallelogram has one angle equal to $90^\circ$, because adjacent angles are supplementary and opposite angles are equal, all its angles must be $90^\circ$. Hence, it is a rectangle.

Therefore, Reason (R) is True.

Relationship between A and R:

Reason (R) provides the necessary and sufficient condition for a parallelogram to be a rectangle. This property explains why the parallelogram described in Assertion (A) (with adjacent sides 4 and 5 and one $90^\circ$ angle) is in fact a rectangle with sides 4 and 5, and thus congruent to the rectangle mentioned in Assertion (A). Reason (R) directly justifies Assertion (A).

Hence, Reason (R) is the correct explanation for Assertion (A).

The correct option is (D) The two angles adjacent to the given sides and the third angle anywhere else.

To construct a unique quadrilateral, we typically need a minimum of five independent measurements. One standard case for construction involves using the lengths of two adjacent sides and the measures of three angles.

Let the two adjacent sides be AB and BC. We are given their lengths. The four interior angles of the quadrilateral are $\angle A, \angle B, \angle C, \angle D$. The angle included between the given sides AB and BC is $\angle B$. The angles adjacent to the given sides at their other endpoints are $\angle A$ (adjacent to AB at A) and $\angle C$ (adjacent to BC at C).

A common and direct method for constructing a quadrilateral with two adjacent sides (AB, BC) and three angles involves knowing the angles at the three vertices A, B, and C. That is, we need the measures of $\angle A$, $\angle B$, and $\angle C$.

• $\angle B$ is the angle included between the given sides AB and BC.
• $\angle A$ is adjacent to the given side AB (at the endpoint A).
• $\angle C$ is adjacent to the given side BC (at the endpoint C).

Knowing these three angles and the lengths of AB and BC is sufficient because once vertices A, B, and C are fixed, the directions of the rays from A and C that form the sides AD and CD are determined by $\angle A$ and $\angle C$. The intersection of these rays gives the vertex D, uniquely determining the quadrilateral.

Let's analyze option (D): "The two angles adjacent to the given sides and the third angle anywhere else."

• "The two angles adjacent to the given sides": This can be interpreted as the angles at the 'outer' endpoints of the given sides, i.e., $\angle A$ (adjacent to AB at A) and $\angle C$ (adjacent to BC at C).
• "the third angle anywhere else": This third angle could be $\angle B$ or $\angle D$.

If the third angle is $\angle B$, then the set of angles is $\{\angle A, \angle C, \angle B\}$. This is the set of angles at the vertices A, B, and C, which is sufficient information along with sides AB and BC for unique construction.

Other options:

(A) "The two angles included between the sides and one other angle." There is only one angle included between two adjacent sides. This phrasing is incorrect.

(B) "The two angles adjacent to the given sides ($\angle A, \angle C$) and the angle opposite to the junction of the sides ($\angle D$)." This means knowing $\angle A, \angle C, \angle D$. Given these three angles, the fourth angle $\angle B$ can be calculated ($\angle B = 360^\circ - (\angle A + \angle C + \angle D)$). Thus, this set is equivalent to knowing $\angle A, \angle C, \angle B$. While sufficient, option (D) describes this set more directly based on standard construction steps.

(C) "The angle included between the sides ($\angle B$) and the two angles adjacent to the other two sides (likely $\angle C, \angle D$)." This means knowing $\angle B, \angle C, \angle D$. Similar to (B), this allows calculating $\angle A$. While sufficient, option (D) provides a description that aligns better with starting the construction from the given adjacent sides.

The phrasing in option (D), while slightly informal ("anywhere else"), most accurately describes the set of angles {$\angle A, \angle B, \angle C$} when given adjacent sides AB and BC, which is a standard sufficient set for unique quadrilateral construction.

The correct option is (A) $26 \text{ cm}$.

In a rhombus, the diagonals bisect each other at right angles.

Let the lengths of the diagonals be $d_1 = 48 \text{ cm}$ and $d_2 = 20 \text{ cm}$. When the diagonals intersect at the center of the rhombus, they form four congruent right-angled triangles.

The legs of each right-angled triangle are half the lengths of the diagonals.

Length of half of the first diagonal = $\frac{d_1}{2} = \frac{48}{2} \text{ cm} = 24 \text{ cm}$.

Length of half of the second diagonal = $\frac{d_2}{2} = \frac{20}{2} \text{ cm} = 10 \text{ cm}$.

The hypotenuse of each of these right-angled triangles is the side length of the rhombus, let's call it $s$.

Using the Pythagorean theorem in one of these right-angled triangles:

$(side)^2 = (\text{half diagonal}_1)^2 + (\text{half diagonal}_2)^2$

$s^2 = (24 \text{ cm})^2 + (10 \text{ cm})^2$

$s^2 = 576 \text{ cm}^2 + 100 \text{ cm}^2$

$s^2 = 676 \text{ cm}^2$

To find the side length $s$, take the square root of 676:

$s = \sqrt{676 \text{ cm}^2}$

$s = 26 \text{ cm}$

So, the length of each wooden piece (the side length of the rhombus) is $26 \text{ cm}$.

The correct option is (C) Diagonals are perpendicular.

In the construction method for a rhombus using its diagonals, the diagonals are drawn so that they bisect each other and are perpendicular.

Step 1: Draw the two diagonals intersecting at their midpoints.

Step 2: Ensure that the angle between the two diagonals is $90^\circ$.

Step 3: Connect the endpoints of the diagonals.

The right angle property of the intersecting diagonals creates four right-angled triangles. The sides of the rhombus are the hypotenuses of these right triangles. The vertices of the rhombus are the endpoints of the diagonals.

The fact that the diagonals are perpendicular is precisely what ensures that the angles at the intersection point are $90^\circ$. When the endpoints of the diagonals are connected, the lines forming the sides of the rhombus are formed. The relationship between the sides and the half-diagonals in the resulting right triangles is defined by this perpendicularity.

Let's look at the options:

(A) Diagonals bisect each other: This property is true for all parallelograms (including rhombuses), but it doesn't guarantee the angles between the diagonals are $90^\circ$.

(B) All sides are equal: This is a defining property of a rhombus, but it is a consequence of the diagonals bisecting each other perpendicularly and connecting the endpoints, rather than a property directly used to set the angle between the diagonals during construction from diagonal lengths.

(C) Diagonals are perpendicular: This property specifically states that the angle between the diagonals is $90^\circ$, which is essential for forming the right-angled triangles whose hypotenuses are the sides of the rhombus and whose legs are the half-diagonals. This directly ensures the geometry needed to calculate the side length using Pythagoras and to construct the rhombus correctly from its diagonals.

(D) Opposite angles are equal: This is a property of a parallelogram (including a rhombus), but it doesn't directly relate to the angles formed by the intersecting diagonals.

Therefore, the property that the Diagonals are perpendicular is what ensures the $90^\circ$ angles at the intersection relative to the diagonals, which is key to this construction method.

The correct option is (C) Square.

To construct a unique quadrilateral, we generally need a certain number of independent measurements. The number of measurements required varies depending on the specific type of quadrilateral, as its properties provide inherent constraints.

Let's consider the minimum measurements required for each type:

• Rectangle: Requires at least two independent measurements, such as length and breadth, or one side and a diagonal. (e.g., $l, b$)
• Rhombus: Requires at least two independent measurements, such as side length and one angle, or the lengths of the two diagonals. (e.g., $s, \theta$ or $d_1, d_2$)
• Parallelogram: Requires at least three independent measurements, such as two adjacent sides and the included angle, or two adjacent sides and one diagonal. (e.g., $a, b, \theta$ or $a, b, d$)
• Square: A square is a highly symmetric figure. All its sides are equal, and all its angles are $90^\circ$. Knowing just one measure related to its size is sufficient to define a unique square. For example:
  • Knowing the side length ($s$) allows unique construction.
  • Knowing the diagonal length ($d$) allows unique construction (since $s = d/\sqrt{2}$).
  • Knowing the perimeter ($P$) allows unique construction (since $s = P/4$).
  • Knowing the area ($A$) allows unique construction (since $s = \sqrt{A}$).

Among the given options, only a Square can be uniquely constructed with just one measurement (such as side length, diagonal length, perimeter, or area).

The correct option is (A) Yes.

A square has specific properties related to its diagonals:

1. The diagonals are equal in length.

2. The diagonals bisect each other.

3. The diagonals are perpendicular to each other.

If you are given the lengths of the two diagonals of a square, you are essentially given one length, say $d$, because in a square, the two diagonals must be equal ($d_1 = d_2 = d$). So, knowing "the lengths of the two diagonals" means you know their common length $d$.

As discussed in Question 11, knowing the diagonal length of a square is sufficient for unique construction.

Construction steps using diagonal length $d$:

1. Draw a line segment of length $d$ (representing one diagonal).

2. Find the midpoint of this segment.

3. Draw the perpendicular bisector of the segment.

4. Along the perpendicular bisector, mark points on either side of the midpoint at a distance of $d/2$. These points define the endpoints of the second diagonal.

5. Connect the endpoints of the two diagonals to form the square.

This construction works because we leverage the properties that the diagonals are equal, they bisect each other, and they are perpendicular. Giving the lengths of the two diagonals implicitly provides their common length, and the inherent properties of a square's diagonals (bisecting and perpendicularity) allow for unique construction.

Therefore, knowing the lengths of the two diagonals of a square is sufficient to construct it uniquely.

Option (D) is incorrect because the property that diagonals are perpendicular is inherent to a square; it doesn't need to be given explicitly as separate information if you know it's a square and have its diagonal lengths.

Given:

The question asks for the minimum number of measurements required to construct a unique quadrilateral.

To Find:

The minimum number of measurements needed for a unique quadrilateral construction.

Solution:

To construct a unique geometric figure, we generally need a sufficient number of independent measurements to fix all its degrees of freedom.

A quadrilateral has 4 vertices and 4 sides and 4 angles. However, the sides and angles are related. A quadrilateral can be defined by its 4 side lengths and 2 diagonal lengths, or 4 side lengths and angles, etc.

Consider a triangle. A unique triangle can be constructed with 3 measurements (e.g., SSS, SAS, ASA, AAS, RHS). A quadrilateral is a more complex figure.

For a quadrilateral, simply knowing the lengths of the four sides is not enough, as it can result in multiple quadrilaterals (e.g., a rhombus can have varying angles while keeping side lengths constant).

We typically need measurements that constrain the shape and size definitively.

The minimum number of independent measurements generally required to construct a unique quadrilateral is 5.

These 5 measurements can be in various combinations, such as:

1. Four sides and one diagonal.

2. Three sides and two included angles.

3. Two adjacent sides and three angles.

4. Four sides and one angle.

5. Two diagonals and three angles.

Each of these combinations provides enough information to fix the vertices of the quadrilateral uniquely in a plane (assuming certain conditions like triangle inequality are met for side lengths).

Final Answer:

The minimum number of measurements generally required to construct a unique quadrilateral is 5.

Given:

The question asks for the five common cases of measurements that allow for the construction of a unique quadrilateral.

To Find:

The five common cases for constructing a unique quadrilateral.

Solution:

To construct a unique quadrilateral, we generally need five independent measurements. The five common cases for which we can construct a unique quadrilateral are:

1. When four sides and one diagonal are given: If the lengths of all four sides and one of the diagonals are known, the quadrilateral can be uniquely constructed.

2. When three sides and two included angles are given: If the lengths of three consecutive sides and the measures of the two angles included between them are known, the quadrilateral can be uniquely constructed.

3. When two adjacent sides and three angles are given: If the lengths of two adjacent sides and the measures of three angles (one of which must be the angle between the two given sides, or the two angles adjacent to the given sides) are known, the quadrilateral can be uniquely constructed.

4. When four sides and one angle are given: If the lengths of all four sides and the measure of one angle are known, the quadrilateral can be uniquely constructed (this case is not always straightforward and sometimes falls under case 1 or requires specific angle placement, but is often listed as a standard case, particularly when the angle's position is specified, e.g., an included angle). However, it is more precise to say that four sides and one *diagonal* define a unique quadrilateral (Case 1). The cases involving angles are usually preferred for clarity.

5. When two diagonals and three sides are given: If the lengths of the two diagonals and the lengths of three sides are known, the quadrilateral can be uniquely constructed.

Note: Case 4 (four sides and one angle) can sometimes lead to ambiguity depending on the position of the angle relative to the sides. Cases involving diagonals or included angles/sides tend to be more universally unique.

Final Answer:

The five common cases for constructing a unique quadrilateral are:

1. Four sides and one diagonal.

2. Three sides and two included angles.

3. Two adjacent sides and three angles.

4. Four sides and one angle (with careful consideration of the angle's position).

5. Two diagonals and three sides.

Given:

The question asks to identify the essential tools from a geometry box required for constructing quadrilaterals.

To Find:

The essential geometry tools for constructing quadrilaterals.

Solution:

Constructing a quadrilateral involves drawing line segments of specific lengths and angles. The essential tools from a geometry box that are typically required for these tasks are:

1. Ruler: A ruler is used to draw straight line segments (which represent the sides and diagonals of the quadrilateral) and to measure their lengths accurately.

2. Pencil: A sharp pencil is needed to draw clear and precise lines and points on paper.

3. Compass: A compass is used to draw arcs and circles of specific radii. This is crucial for locating vertices when side lengths or diagonal lengths are given. For example, to find the intersection point of two sides or a side and a diagonal when their lengths from known points are given.

4. Protractor: A protractor is used to measure existing angles and to draw angles of specific degrees. This is necessary when the angles of the quadrilateral are given.

5. Eraser: An eraser is essential for correcting any errors made during the construction process.

While other tools like set squares might be present in a geometry box, the ruler, pencil, compass, and protractor are the fundamental instruments for constructing a unique quadrilateral based on given measurements.

Final Answer:

The essential tools from a geometry box for constructing quadrilaterals are the Ruler, Pencil, Compass, and Protractor.

Given:

The question asks for the name of the construction case where the lengths of all four sides and one diagonal of a quadrilateral are given, and the initial steps for this construction.

To Find:

The name of the construction case and the first few steps involved.

Solution:

The construction case where the lengths of all four sides and one diagonal of a quadrilateral are given is typically referred to as: Construction of a quadrilateral when Four Sides and a Diagonal are given.

Brief description of the first few steps:

The key idea is that the diagonal divides the quadrilateral into two triangles. If we have the lengths of the diagonal and the two sides connected to each endpoint of that diagonal, we can construct the two triangles.

Let the quadrilateral be ABCD, and let the given diagonal be AC. We are given the lengths of AB, BC, CD, DA, and AC.

Here are the initial steps:

1. Draw the given diagonal first. For example, draw a line segment AC of the given length.

2. Consider the triangle formed by this diagonal, say $\triangle$ABC. We know the lengths of AB, BC, and AC. With A and C as centers, and radii equal to the lengths of AB and BC respectively, draw arcs on one side of the diagonal AC. The intersection point of these two arcs will be vertex B. Join AB and BC.

3. Now consider the other triangle formed by the diagonal, say $\triangle$ADC. We know the lengths of AD, CD, and AC. With A and C as centers, and radii equal to the lengths of AD and CD respectively, draw arcs on the other side of the diagonal AC. The intersection point of these two arcs will be vertex D. Join AD and CD.

At this point, you have constructed the four sides and the diagonal, thus forming the quadrilateral ABCD.

Final Answer:

The construction case is Four Sides and a Diagonal.

The first few steps involve drawing the given diagonal and then using compass arcs from the endpoints of the diagonal with the given side lengths to locate the other two vertices, one on each side of the diagonal, forming two triangles.

Given:

The question asks for the name of the construction case where the lengths of three sides and two diagonals of a quadrilateral are given, and the initial steps for this construction.

To Find:

The name of the construction case and the first few steps involved.

Solution:

The construction case where the lengths of three sides and two diagonals of a quadrilateral are given is typically referred to as: Construction of a quadrilateral when Three Sides and Two Diagonals are given.

Brief description of the initial steps:

In this case, we are given a total of five measurements: three side lengths and two diagonal lengths. The strategy involves constructing a triangle using a diagonal and two sides connected to its endpoints, and then using the other diagonal and the remaining known side to locate the fourth vertex.

Let the quadrilateral be ABCD. Suppose the given lengths are AB, BC, CD (three sides) and AC, BD (two diagonals).

Here are the initial steps:

1. Choose one of the diagonals. Let's choose AC. Draw a line segment AC of the given length.

2. Now, consider the triangle $\triangle$ABC. We know the lengths of AC (the diagonal), AB, and BC (two of the given sides). With A as the center and radius equal to the length of AB, draw an arc. With C as the center and radius equal to the length of BC, draw another arc. The intersection of these two arcs gives the vertex B. Join AB and BC.

At this stage, you have constructed triangle ABC, which includes one diagonal AC and two sides AB and BC.

3. Next, we need to find the position of the fourth vertex, D. We know the lengths of the diagonal BD and the side CD. From point B, with radius equal to the length of the diagonal BD, draw an arc. From point C, with radius equal to the length of the side CD, draw an arc. The intersection of these two arcs will give the vertex D. Join AD and CD to complete the quadrilateral ABCD.

Final Answer:

The construction case is Three Sides and Two Diagonals.

The initial steps involve constructing a triangle using one of the given diagonals and two of the given sides, and then using the other diagonal and the third given side to locate the fourth vertex by the intersection of arcs.

Given:

The question asks whether a unique quadrilateral can be constructed solely from the lengths of its four sides and requires an explanation.

To Determine:

If four side lengths are sufficient to construct a unique quadrilateral, and to explain the reason.

Solution:

No, you generally cannot construct a unique quadrilateral if only the lengths of all four sides are given.

Here's the explanation:

A quadrilateral has 4 sides and 4 angles. While the side lengths are fixed, the angles between the sides can vary. Imagine a framework made of four rods connected by flexible joints at the corners. If you fix the lengths of the rods, you can still push or pull the framework, changing the angles and thus the shape of the quadrilateral without changing the lengths of the sides.

For example, consider a rhombus. A rhombus has all four sides equal in length. However, a rhombus can have various shapes depending on its angles. A square is a special type of rhombus where all angles are $90^\circ$. You can have a rhombus with side length 5 cm and angles $60^\circ$ and $120^\circ$, or a square with side length 5 cm and angles $90^\circ$. Both have four sides of length 5 cm, but they are different quadrilaterals.

To fix the shape of a quadrilateral uniquely, you need more information than just the side lengths. You need something that constrains the angles or fixes the relative positions of the vertices. This is why a fifth measurement, such as the length of a diagonal or the measure of one of the angles, is typically required to construct a unique quadrilateral.

Final Answer:

No, a unique quadrilateral cannot generally be constructed if only the lengths of all four sides are given because the angles can vary, leading to different shapes with the same side lengths.

Given:

The question asks about the property of a parallelogram used during its construction when given the lengths of two adjacent sides and one angle.

To Identify:

The property of a parallelogram utilized in this specific construction case.

Solution:

When constructing a parallelogram given the lengths of two adjacent sides and one angle, the primary property used is that opposite sides of a parallelogram are equal in length.

Let the parallelogram be ABCD, with adjacent sides AB and BC and the angle at B, $\angle$ABC, given.

If we know the length of side AB, we automatically know the length of the opposite side CD, because in a parallelogram, AB = CD.

Similarly, if we know the length of side BC, we automatically know the length of the opposite side AD, because in a parallelogram, BC = AD.

Therefore, having the lengths of two adjacent sides and one angle effectively gives us the lengths of all four sides and one angle (the angle between the two known adjacent sides). This falls under a case where a unique quadrilateral (a parallelogram in this instance) can be constructed using four sides and one angle, or more precisely, two pairs of equal opposite sides and one angle.

The steps would typically involve:

1. Drawing one of the given adjacent sides, say AB.

2. At vertex B, constructing the given angle $\angle$ABC.

3. Marking point C on the ray of the angle such that BC has the given length.

4. Now, using the property that opposite sides are equal, from point A, draw an arc with radius equal to BC (since AD = BC). From point C, draw an arc with radius equal to AB (since CD = AB). The intersection of these two arcs gives the vertex D.

5. Join AD and CD to complete the parallelogram.

This process directly uses the equality of opposite sides.

Final Answer:

The property of a parallelogram used is that opposite sides are equal in length (AB = CD and BC = AD).

Given:

The question asks about the property of a parallelogram used during its construction when given the lengths of its two diagonals.

To Identify:

The property of a parallelogram utilized in this specific construction case.

Solution:

When constructing a parallelogram given the lengths of its two diagonals, the key property used is that the diagonals of a parallelogram bisect each other.

Let the parallelogram be ABCD, and let its diagonals be AC and BD. If the diagonals intersect at point O, then the property of bisection means that AO = OC = $\frac{1}{2}$ AC and BO = OD = $\frac{1}{2}$ BD.

Given the lengths of the two diagonals, say $d_1$ (length of AC) and $d_2$ (length of BD), we know the lengths of the segments formed by their intersection: $AO = OC = d_1/2$ and $BO = OD = d_2/2$.

This property allows us to construct the point of intersection of the diagonals (O) by considering segments of half the lengths of the diagonals. Once the point of intersection is established, and a diagonal is drawn, the other diagonal can be drawn bisecting the first one at O.

However, simply knowing the lengths of the diagonals is not enough to construct a unique parallelogram, as the angle between the diagonals can vary. To construct a unique parallelogram using diagonals, you typically need the lengths of the two diagonals and the angle between them, or the lengths of the two diagonals and one side.

If the context implies constructing a parallelogram given the diagonals *and enough information to fix the angles*, the bisection property is still the fundamental concept used for the diagonals themselves.

Assuming the question refers to the common construction case where the lengths of two diagonals and an angle between them (or one side) are given, the bisection property is essential.

Final Answer:

The property of a parallelogram used is that the diagonals bisect each other.

Given:

The question asks for the minimum information required to construct a unique rectangle and the explanation for its sufficiency.

To Determine:

The minimum necessary measurements and the reason they guarantee uniqueness for a rectangle.

Solution:

A rectangle is a special type of parallelogram with specific properties:

1. Opposite sides are equal and parallel.

2. All four angles are right angles ($90^\circ$).

3. Diagonals are equal in length and bisect each other.

Because of these inherent properties, we don't need 5 independent measurements like a general quadrilateral. The right angles provide significant constraints.

The minimum information needed to construct a unique rectangle is the lengths of two adjacent sides.

Let the lengths of the adjacent sides be $l$ and $w$ (length and width).

Explanation of sufficiency:

If we are given the lengths of two adjacent sides, say AB = $l$ and BC = $w$, we can construct a unique rectangle using the properties of a rectangle:

1. Draw a line segment AB of length $l$.

2. At point B, construct a right angle ($90^\circ$).

3. Along the arm of the right angle, mark point C such that BC has length $w$.

4. Now, we know that in a rectangle, the opposite side AD must be parallel to BC and have the same length $w$, and the opposite side CD must be parallel to AB and have the same length $l$. Also, the angles at A, C, and D must be $90^\circ$.

We can locate point D by drawing a line through A parallel to BC (or constructing a $90^\circ$ angle at A) and marking a point at distance $w$. Alternatively, using the property of opposite sides, from A, draw an arc with radius $w$ (length of AD), and from C, draw an arc with radius $l$ (length of CD). The intersection of these arcs will give point D. Since angles A and C will naturally form $90^\circ$ (as AD is parallel to BC and AB is a transversal, and similarly for CD parallel to AB), this uniquely defines the rectangle.

The information of two adjacent sides, combined with the property that all angles are $90^\circ$, is sufficient to fix the position of all four vertices and thus construct a unique rectangle.

Other minimum information sets also exist, for example, the length of one side and the length of a diagonal, or the lengths of both diagonals and the angle between them (which would have to be $90^\circ$ for a rectangle, but that's a property derived from the definition). However, the lengths of two adjacent sides is the most common and fundamental minimum set.

Final Answer:

The minimum information needed to construct a unique rectangle is the lengths of its two adjacent sides.

This is sufficient because knowing the lengths of two adjacent sides ($l$ and $w$) implies that the opposite sides also have these lengths (using the parallelogram property), and the knowledge that all angles are $90^\circ$ fixes the relative positions of the vertices, allowing the construction of a unique figure.

Given:

The question asks about the property of a rectangle that simplifies its construction when given the lengths of two adjacent sides.

To Identify:

The specific property that facilitates this construction.

Solution:

When constructing a rectangle given the lengths of two adjacent sides, say length $l$ and width $w$, the property of a rectangle that makes the construction easy is that all four interior angles are right angles ($90^\circ$).

This property means that if you draw one side (say of length $l$), the adjacent side must be drawn from an endpoint of the first side at an angle of $90^\circ$. You can then simply use a protractor or a set square to draw perpendicular lines.

For example, if you draw side AB of length $l$, you know that $\angle$ABC must be $90^\circ$ and $\angle$BAD must be $90^\circ$. This immediately fixes the direction of the sides BC and AD. You then only need to measure the length $w$ along these perpendicular lines from B and A respectively to find points C and D. Since CD will automatically be parallel and equal to AB, and AD parallel and equal to BC, the rectangle is uniquely determined and easily drawn using right angles.

While the property that opposite sides are equal is also used to determine the lengths of the remaining sides, it is the property of having right angles that dictates the shape and relative orientation of the sides, making the construction straightforward once the adjacent sides and their included angle ($90^\circ$) are established.

Final Answer:

The property of a rectangle that makes it easy to construct given the lengths of two adjacent sides is that all its interior angles are right angles ($90^\circ$).

Given:

The question asks for the minimum information required to construct a unique rhombus and an explanation for its sufficiency.

To Determine:

The minimum necessary measurements and the reason they guarantee uniqueness for a rhombus.

Solution:

A rhombus is a special type of parallelogram where all four sides are equal in length. Due to its specific properties, such as all sides being equal and opposite angles being equal, we don't need as many independent measurements as a general quadrilateral.

The minimum information needed to construct a unique rhombus is the length of one side and the measure of one angle.

Explanation of sufficiency:

Let the given side length be $s$ and the given angle be $\theta$.

1. Since all sides of a rhombus are equal, knowing the length of one side ($s$) means we know the length of all four sides (AB = BC = CD = DA = $s$).

2. A rhombus is a parallelogram, and opposite angles in a parallelogram are equal. Also, adjacent angles are supplementary (sum up to $180^\circ$).

3. If we are given one angle $\theta$, say $\angle$A = $\theta$, then we immediately know that the opposite angle $\angle$C = $\theta$. The adjacent angles, $\angle$B and $\angle$D, must each be $180^\circ - \theta$.

So, by knowing the length of one side and one angle, we effectively know the lengths of all four sides and the measures of all four angles. With all side lengths and all angles determined, the shape and size of the rhombus are uniquely fixed.

A construction based on this information would involve:

1. Drawing a line segment AB of length $s$.

2. At point A, constructing the given angle $\theta$.

3. Along the arm of the angle, marking point D such that AD has length $s$.

4. Now, from point B, draw an arc with radius $s$ (since BC = $s$). From point D, draw an arc with radius $s$ (since CD = $s$). The intersection of these arcs gives point C.

5. Join BC and CD. The resulting figure ABCD is a unique rhombus with side length $s$ and $\angle$A = $\theta$.

Other valid sets of minimum information (two measurements) include the lengths of the two diagonals.

Final Answer:

The minimum information needed to construct a unique rhombus is the length of one side and the measure of one angle.

This is sufficient because knowing one side length determines all four side lengths, and knowing one angle (along with the properties of a parallelogram) determines all four angles. With all side lengths and all angles known, the rhombus is uniquely defined.

Given:

The question asks about the property of a rhombus's diagonals used during its construction when given the lengths of the diagonals.

To Identify:

The property of a rhombus's diagonals utilized in this specific construction case.

Solution:

When constructing a rhombus given the lengths of its two diagonals, the crucial properties of the diagonals that are used are that they bisect each other at right angles.

Let the rhombus be ABCD, and let the diagonals be AC and BD, intersecting at point O.

The properties used are:

1. The diagonals bisect each other: This means that the intersection point O is the midpoint of both diagonals. So, AO = OC = $\frac{1}{2}$ AC and BO = OD = $\frac{1}{2}$ BD. If we are given the lengths of AC and BD, we immediately know the lengths of the segments AO, OC, BO, and OD.

2. The diagonals are perpendicular to each other: This means that the angle formed at the intersection point O is a right angle ($90^\circ$). So, $\angle$AOB = $\angle$BOC = $\angle$COD = $\angle$DOA = $90^\circ$.

To construct the rhombus using the lengths of the diagonals, we typically:

1. Draw one diagonal, say AC, and find its midpoint O.

2. Draw a line perpendicular to AC passing through O. This line is the axis along which the other diagonal BD lies.

3. On this perpendicular line, mark points B and D on opposite sides of O such that BO = OD = half the length of the second diagonal.

4. Join the points A, B, C, and D to form the rhombus.

Both the bisection and perpendicularity properties are essential for this construction method.

Final Answer:

The property of a rhombus's diagonals used is that they bisect each other at right angles.

Given:

The question asks for the minimum information required to construct a unique square and an explanation for its sufficiency.

To Determine:

The minimum necessary measurements and the reason they guarantee uniqueness for a square.

Solution:

A square is a special type of rectangle and a special type of rhombus. It possesses properties from both:

1. All four sides are equal in length.

2. All four interior angles are right angles ($90^\circ$).

3. Diagonals are equal in length and bisect each other at right angles.

Due to these very strict properties, a square is highly constrained.

The minimum information needed to construct a unique square is the length of one side.

Explanation of sufficiency:

If we are given the length of one side, say $s$, then we know the following from the properties of a square:

1. All four sides (AB, BC, CD, DA) must have the same length $s$.

2. All four interior angles ($\angle$A, $\angle$B, $\angle$C, $\angle$D) must be $90^\circ$.

With the length of all sides and the measure of all angles determined solely by the length of one side, the shape and size of the square are completely fixed. There is only one possible square that can be formed with a given side length.

A construction based on this information would involve:

1. Draw a line segment AB of length $s$.

2. At point A, construct a line segment AD perpendicular to AB (forming a $90^\circ$ angle) with length $s$.

3. At point B, construct a line segment BC perpendicular to AB (forming a $90^\circ$ angle) with length $s$.

4. Join points D and C. Since AD is parallel to BC and AD = BC = $s$, the figure ABCD is a parallelogram with a right angle at A and adjacent sides of length $s$. A parallelogram with adjacent sides equal and one right angle is a square. The length of CD will also automatically be $s$, and angles at C and D will be $90^\circ$.

Another set of minimum information could be the length of a diagonal. Knowing the diagonal length $d$, we can find the side length $s$ using the Pythagorean theorem ($s^2 + s^2 = d^2 \implies 2s^2 = d^2 \implies s = d/\sqrt{2}$) and then proceed as above. So, the length of a diagonal is also sufficient.

However, the length of one side is the simplest minimum measurement.

Final Answer:

The minimum information needed to construct a unique square is the length of one side.

This is sufficient because knowing one side length determines the lengths of all four sides, and the inherent property that all angles are $90^\circ$ fixes the shape, leaving only one possible square for that side length.

Given:

The question asks to explain the difference in the construction method for a square when given either the side length or the diagonal length.

To Explain:

How the construction process differs based on the given information (side length vs. diagonal length).

Solution:

The construction of a unique square requires only one measurement (either side length or diagonal length) due to its inherent properties (all sides equal, all angles $90^\circ$, diagonals equal and bisect at $90^\circ$). The construction process differs significantly depending on which measurement is provided:

Case 1: Given the side length ($s$):

In this case, the construction focuses on utilizing the right angles and equal side lengths.

Initial Steps:

1. Draw one side of the square as a line segment of the given length $s$. Let this segment be AB.

2. At one endpoint of this segment (say A), construct a perpendicular line (using a protractor, set square, or compass for a $90^\circ$ angle). This line will contain the adjacent side AD.

3. On this perpendicular line, mark a point D at a distance $s$ from A. AD is the second side of the square.

4. From points B and D, draw arcs with radius $s$. The intersection of these arcs will be the fourth vertex, C. Join BC and CD.

This method directly uses the side length and the $90^\circ$ angle property.

Case 2: Given the diagonal length ($d$):

In this case, the construction focuses on the properties of the diagonals: they are equal in length, bisect each other, and are perpendicular.

Initial Steps:

1. Draw one of the diagonals as a line segment of the given length $d$. Let this segment be AC.

2. Find the midpoint of the diagonal AC. Let this midpoint be O (using a compass to bisect the segment).

3. Draw a line perpendicular to AC passing through its midpoint O. This line will contain the other diagonal BD.

4. Since the diagonals of a square are equal and bisect each other, the other diagonal BD must also have length $d$ and be bisected at O. Mark points B and D on the perpendicular line such that BO = OD = $\frac{d}{2}$.

5. Join the endpoints of the diagonals (A, B, C, and D) to form the sides of the square.

This method uses the length and bisection/perpendicularity properties of the diagonals to locate the vertices.

Difference:

The fundamental difference lies in the starting point and the properties utilized:

• When given the side length, you start by drawing a side and use the $90^\circ$ angle property to find adjacent vertices.
• When given the diagonal length, you start by drawing a diagonal and use the diagonal bisection and perpendicularity properties to find the other diagonal's position and the remaining vertices.

Final Answer:

The construction differs in the initial step and the properties applied. Given the side length, one starts by drawing a side and constructing $90^\circ$ angles. Given the diagonal length, one starts by drawing a diagonal, finding its midpoint, and drawing a perpendicular bisector to locate the other diagonal's endpoints.

Given:

Measurements for a quadrilateral ABCD:

To Identify:

The construction case that applies based on the given measurements.

Solution:

We are given the lengths of the four sides of the quadrilateral (AB, BC, CD, AD) and the length of one diagonal (AC).

Let's examine the given measurements:

• Four side lengths: $5\text{ cm}$, $6\text{ cm}$, $7\text{ cm}$, $8\text{ cm}$.
• One diagonal length: $9\text{ cm}$.

This set of measurements directly corresponds to one of the standard construction cases for a unique quadrilateral.

The standard construction cases for quadrilaterals are generally based on providing 5 independent measurements. The given measurements are 5 lengths.

The case that involves knowing the lengths of all four sides and the length of one of the diagonals is a fundamental method for constructing a unique quadrilateral.

Specifically, the given information allows us to divide the quadrilateral into two triangles (Triangle ABC and Triangle ADC in this case) that share the common diagonal AC. Since we know the lengths of all three sides of each triangle (AB, BC, AC for $\triangle$ABC and AD, CD, AC for $\triangle$ADC), each triangle can be uniquely constructed using the SSS (Side-Side-Side) criterion. Constructing these two unique triangles adjacent to each other along the shared diagonal AC results in a unique quadrilateral ABCD.

Final Answer:

The construction case that applies is Four Sides and a Diagonal.

Given:

Measurements for a quadrilateral PQRS:

To Identify:

The construction case that applies based on the given measurements.

Solution:

We are given the lengths of three consecutive sides of the quadrilateral (PQ, QR, RS) and the measures of two angles ($\angle$Q and $\angle$R) where the side QR is included between these two angles.

Let's list the given measurements:

• Three side lengths: PQ ($4\text{ cm}$), QR ($5\text{ cm}$), RS ($5.5\text{ cm}$).
• Two angles: $\angle$Q ($70^\circ$), $\angle$R ($80^\circ$).

We have a total of 5 measurements, which is the typical number required to construct a unique quadrilateral.

This set of measurements fits the description of one of the standard construction cases:

Specifically, we have the sequence of elements defining the quadrilateral as follows:

Side PQ (4 cm)

Angle at Q ($70^\circ$)

Side QR (5 cm)

Angle at R ($80^\circ$)

Side RS (5.5 cm)

This sequence is often described as having **three sides and two included angles**. While $\angle$Q is included between PQ and QR, and $\angle$R is related to QR and RS, the common naming for this case covers situations where you have a sequence of three consecutive sides and the angles at the two vertices between them.

A typical construction process would involve:

1. Draw the segment QR of length $5\text{ cm}$.

2. At Q, construct an angle of $70^\circ$. Mark point P on the arm of this angle such that QP = $4\text{ cm}$.

3. At R, construct an angle of $80^\circ$. Mark point S on the arm of this angle such that RS = $5.5\text{ cm}$.

4. Join P to S.

This sequence of steps uses the three side lengths and the two angles. This construction is uniquely determined by the given measurements.

Final Answer:

The construction case that applies is Three Sides and Two Included Angles.

Given:

Adjacent sides of a parallelogram have lengths $6\text{ cm}$ and $4\text{ cm}$.

To Determine:

The number of different parallelograms that can be constructed with these side lengths, and the additional information needed for uniqueness.

Solution:

A parallelogram is defined by the lengths of its adjacent sides and the angle between them. While opposite sides are equal, the angles can vary.

If you are only given the lengths of two adjacent sides (6 cm and 4 cm), you know the lengths of all four sides (two sides of length 6 cm and two sides of length 4 cm). However, you do not know the measure of the angles.

Consider drawing a side of length 6 cm. From one endpoint, draw a segment of length 4 cm. The angle between these two segments can be any value between $0^\circ$ and $180^\circ$ (exclusive). As this angle changes, the shape of the parallelogram changes, even though the side lengths remain the same.

For example, you could have a parallelogram with adjacent sides 6 cm and 4 cm and an angle of $30^\circ$, or $60^\circ$, or $90^\circ$ (a rectangle), or $150^\circ$, etc.

Therefore, if only the lengths of two adjacent sides are given, you can construct an **infinite number** of different parallelograms.

To make the parallelogram unique, you need additional information that fixes the angles or the shape. This additional information could be:

1. The measure of one angle (e.g., the angle between the two given adjacent sides). If you know one angle, say $60^\circ$, then the angles of the parallelogram are fixed ($\angle$A = $60^\circ$, $\angle$B = $120^\circ$, $\angle$C = $60^\circ$, $\angle$D = $120^\circ$). With two adjacent sides and the included angle, a unique parallelogram can be constructed.

2. The length of one diagonal. Knowing the lengths of the two adjacent sides and one diagonal allows you to use the 'Four Sides and a Diagonal' construction case (as the opposite sides are equal) to uniquely determine the parallelogram.

Final Answer:

You can construct an infinite number of different parallelograms with adjacent sides 6 cm and 4 cm.

Additional information that would make it unique includes the measure of one angle or the length of one diagonal.

Given:

The question asks about the purpose of using a compass to draw arcs when constructing the sides of triangles or quadrilaterals.

To Explain:

The purpose or function of compass arcs in geometric construction, specifically related to side lengths.

Solution:

When constructing triangles or quadrilaterals, drawing arcs with a compass serves the purpose of accurately locating points (vertices) that are a specific, known distance from another point. This is because a compass is a tool designed to draw circles or arcs of a precise radius, where the radius represents a fixed distance from the center point.

Specifically, when we draw an arc with a compass with the needle at a certain point (say, point A) and the pencil set to a certain radius (say, $r$), every point on that arc is exactly $r$ units away from point A.

In the context of constructing sides of a polygon:

If we know the length of a side, say AB, originating from point A, we can use the compass to mark all possible locations of point B that are the given length away from A. If we know the lengths of two sides originating from two different known points that meet at a third point (say, sides from A and C meet at B), drawing arcs from A with radius AB and from C with radius CB will cause the arcs to intersect at the location of point B. This intersection point is unique (or potentially two points symmetrical across the line connecting the two centers, but in plane geometry construction, we typically consider one solution or context dictates which one is valid) and satisfies the distance requirements from both A and C.

Therefore, drawing arcs with a compass is a method for:

• Transferring lengths accurately from a ruler or given measurement onto the drawing.
• Locating the position of a vertex when its distances from two (or more) other known points are given, using the intersection of circles/arcs.

This is fundamental to constructions like SSS (Side-Side-Side) triangle construction or the quadrilateral construction case involving four sides and a diagonal.

Final Answer:

The purpose of drawing arcs with a compass when constructing sides of a triangle or quadrilateral is to accurately locate vertices based on known distances (side or diagonal lengths) from existing points. The intersection of arcs drawn from two different points with radii equal to the distances to a third unknown point fixes the position of that third point.

Given:

The question asks if a unique quadrilateral can be constructed when only the measures of its four angles are given and requires an explanation.

To Determine:

If knowing only the four angles is sufficient to construct a unique quadrilateral, and to explain why.

Solution:

No, you generally cannot construct a unique quadrilateral if only the measures of all four angles are given.

Here's the explanation:

While the sum of the interior angles of any quadrilateral is always $360^\circ$, knowing the four angles only determines the *shape* of the quadrilateral in terms of its angular properties, but it does not fix its *size*.

Consider the properties of similar figures. Two figures are similar if they have the same shape but potentially different sizes. For polygons, having corresponding angles equal is a condition for similarity (though not always sufficient for quadrilaterals unless side ratios are also considered, but for this argument, the key is that angles alone don't fix size).

For example:

• A square always has four angles of $90^\circ$. You can draw a square with side length $2\text{ cm}$ and another square with side length $5\text{ cm}$. Both have angles $90^\circ, 90^\circ, 90^\circ, 90^\circ$, but they are clearly not the same unique quadrilateral; one is simply a larger version of the other.
• Consider a parallelogram with angles $60^\circ, 120^\circ, 60^\circ, 120^\circ$. You can draw many different parallelograms with these same angle measures but with varying side lengths.

Since you can scale a quadrilateral up or down (making it larger or smaller) while keeping all its angles the same, knowing only the angles is not enough to fix the dimensions (side lengths or diagonal lengths) and thus construct a unique figure.

To construct a unique quadrilateral, you need to fix its size in addition to its shape, which requires at least one linear measurement (like a side length or a diagonal length) in combination with sufficient angular or other linear information (totaling 5 independent measurements for a general quadrilateral, or fewer for special quadrilaterals like rectangles or squares due to their fixed angle properties).

Final Answer:

No, a unique quadrilateral cannot be constructed if only the measures of all four angles are given because knowing the angles determines the shape but not the size of the quadrilateral. Different quadrilaterals with the same angles but different side lengths can exist.

Given:

The task is to describe the initial step for constructing a square when the length of its diagonal is provided.

To Describe:

The first step in the construction process.

Solution:

When constructing a square given the length of its diagonal ($d$), the construction method leverages the properties of the diagonals of a square: they are equal in length, bisect each other, and are perpendicular.

The most logical initial step is to draw one of these diagonals, as its length is the only given measurement.

Initial Step:

1. Draw a line segment equal to the given diagonal length. Let the given diagonal length be $d$. Using a ruler, draw a line segment, say AC, of length $d$. This segment represents one of the diagonals of the square.

Subsequent steps would involve finding the midpoint of this diagonal, constructing a perpendicular line through the midpoint, and marking off half the diagonal length along the perpendicular line on both sides to find the other two vertices (B and D), and finally joining the vertices to form the square.

Final Answer:

The initial step in constructing a square given the length of its diagonal is to draw a line segment that represents one of the diagonals, with a length equal to the given diagonal length.

Given:

The question asks about the angle constructed between adjacent sides originating from the same vertex in a rectangle or a square.

To Identify:

The specific angle measure used in this construction.

Solution:

Both rectangles and squares are quadrilaterals with specific angular properties. A defining characteristic of both shapes is that all their interior angles are right angles.

A right angle measures $90^\circ$.

When constructing a rectangle or a square by drawing adjacent sides from a vertex, you are essentially constructing the angle at that vertex. Since all vertices in a rectangle and a square form right angles, the angle between the two adjacent sides originating from that vertex must be a right angle.

For example, if you are constructing a rectangle ABCD and you draw the side AB and then the adjacent side BC from vertex B, the angle $\angle$ABC must be $90^\circ$. Similarly, if you draw AD from vertex A, the angle $\angle$BAD must be $90^\circ$.

Final Answer:

When drawing sides of a rectangle or a square from a vertex, the angle constructed is a right angle ($90^\circ$).

Given:

The question asks why it is necessary to bisect the diagonals when constructing a parallelogram using information related to its diagonals.

To Explain:

The reason for bisecting the diagonals during parallelogram construction based on diagonals.

Solution:

The necessity of bisecting the diagonals when constructing a parallelogram using their lengths stems directly from a fundamental property of parallelograms:

The diagonals of a parallelogram bisect each other.

This property means that the two diagonals intersect at a single point, and this point is the midpoint of *both* diagonals. If you are given the lengths of the two diagonals, this bisection property tells you how the diagonals relate to each other in terms of their intersection.

To construct the parallelogram, you need to establish the position of this central intersection point. By bisecting each diagonal, you find its exact midpoint. The property guarantees that these midpoints must coincide to form a parallelogram.

Therefore, when you draw one diagonal and find its midpoint, you know that the other diagonal must pass through this same midpoint and be bisected by it. By marking off half the length of the second diagonal on either side of the midpoint, along a line passing through the midpoint, you correctly position the endpoints of the second diagonal relative to the first.

Without using the bisection property (i.e., finding the midpoint), you wouldn't know where along the first diagonal the second diagonal should cross, and thus you couldn't accurately locate the vertices of the parallelogram based on the diagonal lengths.

Final Answer:

It is necessary to bisect the diagonals because a key property of parallelograms is that their diagonals bisect each other. Finding the midpoint (bisection point) of the diagonals allows you to locate their intersection point, which is crucial for correctly positioning the diagonals relative to each other and thus constructing the parallelogram.

Given:

The task is to construct a rhombus with side length $5\text{ cm}$ and one angle $60^\circ$.

To Identify:

The related general quadrilateral construction case.

Solution:

A rhombus is a special type of parallelogram with all four sides equal.

We are given:

• Side length = $5\text{ cm}$. Since it's a rhombus, all four sides are $5\text{ cm}$.
• One angle = $60^\circ$. Let this be one of the interior angles.

The information effectively provides us with:

• Length of two adjacent sides (both $5\text{ cm}$).
• The angle between these two adjacent sides ($60^\circ$).

Consider the general construction cases for a unique quadrilateral:

1. Four sides and one diagonal.

2. Three sides and two included angles.

3. Two adjacent sides and three angles.

4. Four sides and one angle.

5. Two diagonals and three sides.

The given information directly translates to knowing the lengths of two adjacent sides (both 5 cm) and the angle between them (60°). This fits the description of having two adjacent sides and the included angle.

Furthermore, since it's a rhombus, knowing one side length means knowing all four side lengths (all 5 cm). Knowing one angle means knowing all four angles (opposite angles are equal, adjacent angles are supplementary). So, you effectively know the lengths of all four sides and the measure of all four angles. This would relate to the 'Four sides and one angle' case (specifically, all four sides are equal, and one angle is given).

However, the most direct translation of "adjacent sides 5 cm and 5 cm and one angle $60^\circ$" relates to the construction of a parallelogram or a quadrilateral where you establish two adjacent sides and the angle between them, and then use that information to find the remaining vertices.

Let's analyze the options based on the information we have (two adjacent sides and the included angle):

Case 2 mentions "three sides and two included angles". We have only two adjacent sides given, and one angle. Case 3 mentions "Two adjacent sides and three angles". We only have one angle given, not three.

Case 4 mentions "Four sides and one angle". We know all four sides are 5 cm and one angle is 60°. This fits perfectly.

Another common phrasing for parallelogram/rhombus construction is based on adjacent sides and the included angle. This is a specific instance of the general quadrilateral case of "Four sides and one angle" where the four sides happen to be equal in pairs (or all equal). The angle provided serves as the crucial fifth piece of information (along with the four implied side lengths).

Therefore, this construction scenario can be related to the case where the lengths of all four sides and one angle are given (with the specific property that all four sides are equal).

Final Answer:

This construction can be related to the quadrilateral construction case of Four Sides and one Angle, where the specific property of a rhombus (all four sides equal) is used to deduce the lengths of all sides from the length of one side.

Given:

Measurements for a quadrilateral: three angles and two included sides.

To Determine:

If a unique quadrilateral can be constructed with this information, and to identify the corresponding construction case.

Solution:

Let the quadrilateral be ABCD. "Two included sides" in the context of three angles typically means the two sides that are between the three given angles in sequence.

Suppose the given information is:

• Angle A ($\angle$A)
• Side AB
• Angle B ($\angle$B)
• Side BC
• Angle C ($\angle$C)

We are given 5 measurements in total: 3 angles and 2 sides. This is the typical number of measurements required for a unique quadrilateral.

Yes, you can construct a unique quadrilateral if you are given three angles and the two included sides.

The construction process would typically follow these steps:

1. Draw the first included side, AB, of the given length.

2. At point A, construct the given angle $\angle$A. Draw a ray from A.

3. At point B, construct the given angle $\angle$B. Draw a ray from B. Along this ray, mark point C at the given length of side BC.

4. At point C, construct the given angle $\angle$C. Draw a ray from C.

5. The ray drawn from A (step 2) and the ray drawn from C (step 4) will intersect at a point. This intersection point is the fourth vertex, D.

This construction is uniquely determined because the positions of A, B, and C are fixed by the first three measurements, and the directions of the lines AD and CD are fixed by the angles at A and C. The intersection of these two lines uniquely defines point D.

This set of given information corresponds to the construction case often described as: Two Adjacent Sides and Three Angles (where the two sides are adjacent to the middle of the three angles, and the third angle is adjacent to one of the sides).

Looking back at the common cases:

1. Four sides and one diagonal.

2. Three sides and two included angles.

3. Two adjacent sides and three angles.

The given information (Angle-Side-Angle-Side-Angle) matches the structure of Case 3.

Final Answer:

Yes, you can construct a unique quadrilateral if you are given three angles and two included sides.

The construction case that applies is Two Adjacent Sides and Three Angles.

Given:

A quadrilateral ABCD with the following side and diagonal lengths:

To Construct:

A unique quadrilateral ABCD with the given measurements.

Steps of Construction:

This construction falls under the case where four sides and a diagonal are given. The diagonal divides the quadrilateral into two triangles, which can be constructed using the SSS (Side-Side-Side) criterion.

1. Draw a line segment AC of length $7\text{ cm}$. This is the given diagonal.

2. Now, construct triangle ABC above or below AC. With A as the center, draw an arc with a radius equal to the length of AB, i.e., $4\text{ cm}$.

3. With C as the center, draw another arc with a radius equal to the length of BC, i.e., $5\text{ cm}$.

4. Let the two arcs intersect at point B. Join AB and BC. Triangle ABC is now constructed.

5. Next, construct triangle ADC on the opposite side of AC. With A as the center, draw an arc with a radius equal to the length of AD, i.e., $5.5\text{ cm}$.

6. With C as the center, draw another arc with a radius equal to the length of CD, i.e., $4.5\text{ cm}$.

7. Let these two arcs intersect at point D. Join AD and CD. Triangle ADC is now constructed.

8. The figure ABCD is the required quadrilateral.

Given:

A quadrilateral PQRS with the following side and diagonal lengths:

To Construct:

A unique quadrilateral PQRS with the given measurements.

Steps of Construction:

This construction falls under the case where three sides and two diagonals are given. We can start by constructing a triangle using one diagonal and two sides connected to its endpoints, and then use the other diagonal and the third side to locate the remaining vertex.

1. Draw a line segment PR of length $7\text{ cm}$. This is one of the given diagonals.

2. Now, consider triangle PQR. We know PR = $7\text{ cm}$, PQ = $4\text{ cm}$, and QR = $6\text{ cm}$. With P as the center, draw an arc with a radius equal to the length of PQ, i.e., $4\text{ cm}$.

3. With R as the center, draw another arc with a radius equal to the length of QR, i.e., $6\text{ cm}$.

4. Let the two arcs intersect at point Q. Join PQ and QR. Triangle PQR is now constructed.

5. Next, we need to locate point S. We know the length of side RS = $5\text{ cm}$ and the length of diagonal QS = $8\text{ cm}$. With R as the center, draw an arc with a radius equal to the length of RS, i.e., $5\text{ cm}$.

6. With Q as the center, draw another arc with a radius equal to the length of diagonal QS, i.e., $8\text{ cm}$.

7. Let these two arcs intersect at point S. Join RS and PS. (Joining PS completes the fourth side, even though its length wasn't explicitly given, it is determined by the other measurements).

8. The figure PQRS is the required quadrilateral.

Given:

A quadrilateral DEFG with the following measurements:

To Construct:

A unique quadrilateral DEFG with the given measurements.

Steps of Construction:

We are given two adjacent sides (DE and EF) and three angles ($\angle$E, $\angle$F, and $\angle$G). To proceed with the construction, it is helpful to first find the fourth angle of the quadrilateral, $\angle$D. The sum of the interior angles of a quadrilateral is $360^\circ$.

1. Find the measure of the fourth angle, $\angle$D:

2. Draw a line segment EF of length $6.5\text{ cm}$.

3. At point E, construct an angle equal to $\angle$E = $100^\circ$. Draw a ray EX from E. Point D will lie on this ray.

4. With E as the center and radius equal to the length of DE ($4.5\text{ cm}$), draw an arc intersecting the ray EX at point D.

5. At point F, construct an angle equal to $\angle$F = $85^\circ$. Draw a ray FY from F. Point G will lie on this ray.

6. At point D, construct an angle equal to $\angle$D = $100^\circ$. Draw a ray DZ from D. Ensure the angle is constructed such that the ray DZ is on the same side of DE as F.

7. The intersection point of the ray FY (from step 5) and the ray DZ (from step 6) is the point G.

8. Join DG and FG. The quadrilateral DEFG is now constructed.

Given:

A parallelogram ABCD with the following measurements:

To Construct:

A unique parallelogram ABCD with the given measurements.

Properties of a Parallelogram Used:

1. Opposite sides are equal: AB = CD and BC = AD.

2. Opposite angles are equal: $\angle$A = $\angle$C and $\angle$B = $\angle$D.

3. Adjacent angles are supplementary: $\angle$A + $\angle$B = $180^\circ$, $\angle$B + $\angle$C = $180^\circ$, etc.

Specifically, in this construction, we primarily use the property that opposite sides are equal and the fact that we are given two adjacent sides and the included angle.

Steps of Construction:

1. Draw a line segment AB of length $6\text{ cm}$.

2. At point B, construct an angle of $75^\circ$ using a protractor. Draw a ray BY from B.

3. On the ray BY, mark a point C such that BC has a length of $4\text{ cm}$.

4. Now, we need to find point D. We know that AD is parallel and equal to BC (property 1), so AD = $4\text{ cm}$. We also know that CD is parallel and equal to AB (property 1), so CD = $6\text{ cm}$.

5. With A as the center, draw an arc with a radius of $4\text{ cm}$ (length of AD).

6. With C as the center, draw another arc with a radius of $6\text{ cm}$ (length of CD).

7. Let the two arcs intersect at point D. Join AD and CD.

8. The figure ABCD is the required parallelogram.

Explanation of Properties Used:

We used the property that opposite sides of a parallelogram are equal in length (AB = CD and BC = AD) in steps 5 and 6 to determine the lengths of the arcs used to locate the fourth vertex D. By drawing arcs from A and C with radii equal to the lengths of the opposite sides, we ensure that the resulting figure has opposite sides equal, a key characteristic of a parallelogram. The initial steps (1-3) establish two adjacent sides and the included angle, which, combined with the properties of a parallelogram, uniquely defines the figure.

Given:

A rectangle PQRS with adjacent sides of lengths $5\text{ cm}$ and $7\text{ cm}$.

To Construct:

A unique rectangle PQRS with the given measurements.

Steps of Construction:

Let the lengths of the adjacent sides be PQ = $7\text{ cm}$ and QR = $5\text{ cm}$.

1. Draw a line segment PQ of length $7\text{ cm}$.

2. At point Q, construct a perpendicular line (an angle of $90^\circ$) using a compass or set square. Draw a ray QX.

3. On the ray QX, mark a point R such that QR has a length of $5\text{ cm}$.

4. Now, we need to find point S. We know that PS is parallel and equal to QR, so PS = $5\text{ cm}$. We also know that RS is parallel and equal to PQ, so RS = $7\text{ cm}$. Also, $\angle$P and $\angle$R must be $90^\circ$.

5. From point P, construct a perpendicular line (an angle of $90^\circ$) or draw an arc with radius $5\text{ cm}$.

6. From point R, draw an arc with radius $7\text{ cm}$.

7. Let the arcs from steps 5 and 6 intersect at point S. (Alternatively, the intersection of the perpendicular from P and the line parallel to PQ through R gives S). Join PS and RS.

8. The figure PQRS is the required rectangle.

Explanation for not needing an angle measure:

We didn't need to be given any angle measure because a fundamental property of a rectangle is that all its interior angles are right angles ($90^\circ$).

This property is part of the definition of a rectangle. When we are asked to construct a figure and told it is a rectangle, we automatically know that all four angles must be $90^\circ$. Therefore, the angle information is inherent in the type of quadrilateral we are constructing (a rectangle), and does not need to be provided as a separate measurement.

Knowing the lengths of two adjacent sides and the fact that the included angle is $90^\circ$ (due to the figure being a rectangle) is sufficient to construct a unique rectangle.

Final Answer:

The construction steps are provided above.

We didn't need to be given any angle measure because all interior angles of a rectangle are by definition right angles ($90^\circ$).

Given:

A rhombus LMNP with side length $5.5\text{ cm}$ and one angle measuring $60^\circ$.

To Construct:

A unique rhombus LMNP with the given measurements.

Properties of a Rhombus Used:

1. All four sides are equal: LM = MN = NP = PL.

2. Opposite angles are equal: $\angle$L = $\angle$N and $\angle$M = $\angle$P.

3. Adjacent angles are supplementary: $\angle$L + $\angle$M = $180^\circ$, etc.

In this construction, we primarily use the property that all four sides are equal and the given angle.

Steps of Construction:

Let the given side length be $s = 5.5\text{ cm}$ and the given angle be $\angle$L = $60^\circ$.

1. Draw a line segment LM of length $5.5\text{ cm}$.

2. At point L, construct an angle of $60^\circ$ using a protractor or compass. Draw a ray LX from L.

3. On the ray LX, mark a point P such that LP has a length of $5.5\text{ cm}$ (since all sides of a rhombus are equal).

4. Now, we need to find point N. We know that MN is equal to LP (opposite sides are equal) and NP is equal to LM (opposite sides are equal). Both MN and NP have length $5.5\text{ cm}$ (since all sides are equal).

5. With M as the center, draw an arc with a radius of $5.5\text{ cm}$ (length of MN).

6. With P as the center, draw another arc with a radius of $5.5\text{ cm}$ (length of NP).

7. Let the two arcs intersect at point N. Join MN and NP.

8. The figure LMNP is the required rhombus.

Explanation of Properties Used:

We used the property that all four sides of a rhombus are equal in length (LM = LP = MN = NP = $5.5\text{ cm}$) in steps 3, 5, and 6 to determine the lengths of the segments and arcs used to locate the vertices P and N. The given angle ($\angle$L = $60^\circ$) helps to fix the relative position of LP with respect to LM, establishing the shape. Combining the equal side lengths with one angle is sufficient to define a unique rhombus.

Given:

Task 1: Construct a square with side length $6\text{ cm}$.

Task 2: Explain how the construction changes if given a diagonal length of $8.5\text{ cm}$ instead.

To Construct:

A square with side length $6\text{ cm}$ and describe the construction change for a given diagonal length.

Steps of Construction (Given Side Length = $6\text{ cm}$):

This construction uses the properties that all sides are equal and all angles are $90^\circ$.

1. Draw a line segment AB of length $6\text{ cm}$.

2. At point A, construct a perpendicular line (an angle of $90^\circ$) using a compass or set square. Draw a ray AX from A.

3. On the ray AX, mark a point D such that AD has a length of $6\text{ cm}$ (since all sides of a square are equal).

4. At point B, construct a perpendicular line (an angle of $90^\circ$) using a compass or set square. Draw a ray BY from B.

5. On the ray BY, mark a point C such that BC has a length of $6\text{ cm}$ (since all sides of a square are equal).

6. Join points D and C. The figure ABCD is the required square.

Alternatively, after step 3, you could locate C by drawing an arc from D with radius $6\text{ cm}$ and an arc from B with radius $6\text{ cm}$. Their intersection is C. Then join BC and CD.

Change in Construction (Given Diagonal Length = $8.5\text{ cm}$):

If you are given the length of the diagonal (say, $d = 8.5\text{ cm}$) instead of the side length, the construction method changes significantly. This method utilizes the properties that the diagonals of a square are equal, bisect each other, and intersect at right angles.

1. Draw a line segment AC of length $8.5\text{ cm}$. This represents one of the diagonals.

2. Find the midpoint of the segment AC. You can do this by constructing the perpendicular bisector of AC using a compass. The point where the perpendicular bisector intersects AC is the midpoint, let's call it O.

3. The perpendicular bisector is the line along which the other diagonal (BD) lies. Since the diagonals of a square are equal and bisect each other, the length of the other diagonal BD is also $8.5\text{ cm}$, and it is bisected at O. So, BO = OD = $\frac{8.5}{2} = 4.25\text{ cm}$.

4. On the perpendicular bisector passing through O, mark points B and D on opposite sides of O such that OB = OD = $4.25\text{ cm}$.

5. Join the vertices A, B, C, and D to form the sides of the square ABCD.

Summary of Change:

The construction differs based on the given measurement:

• Given side length: Start by drawing a side, then construct $90^\circ$ angles and use the side length to find adjacent vertices.
• Given diagonal length: Start by drawing a diagonal, find its midpoint, construct a perpendicular bisector, and use half the diagonal length to find the other two vertices along the bisector.

Given:

A rhombus with diagonals measuring $10\text{ cm}$ and $8\text{ cm}$.

To Construct:

A unique rhombus with the given diagonal lengths, and explain the crucial properties used.

Crucial Properties of a Rhombus's Diagonals:

The properties of a rhombus's diagonals that are essential for this construction are:

1. The diagonals bisect each other: They cut each other into two equal halves at their intersection point.

2. The diagonals are perpendicular to each other: They intersect at a right angle ($90^\circ$).

Steps of Construction:

Let the lengths of the diagonals be $d_1 = 10\text{ cm}$ and $d_2 = 8\text{ cm}$. The halves of the diagonals will be $\frac{d_1}{2} = \frac{10}{2} = 5\text{ cm}$ and $\frac{d_2}{2} = \frac{8}{2} = 4\text{ cm}$.

1. Draw a line segment representing one of the diagonals, say AC, of length $10\text{ cm}$.

2. Find the midpoint of the line segment AC by constructing its perpendicular bisector. This is done by setting a compass to more than half the length of AC, drawing arcs from A above and below AC, and repeating from C with the same compass setting to intersect the first arcs. Draw a line through the intersection points of the arcs. This line is the perpendicular bisector, and it intersects AC at its midpoint, let's call it O.

3. The perpendicular bisector is the line along which the other diagonal (BD) lies, and it passes through O. Since the diagonals bisect each other, the second diagonal BD is bisected at O. Also, the diagonals are perpendicular, so BD lies on the perpendicular bisector of AC.

4. On the perpendicular bisector (the line drawn in step 2), mark points B and D on opposite sides of O such that OB = OD = half the length of the second diagonal, i.e., $4\text{ cm}$.

5. Join the endpoints of the diagonals: AB, BC, CD, and DA. The figure ABCD is the required rhombus.

Explanation of Properties Used:

We used the property that the diagonals bisect each other when we found the midpoint O of AC and marked off half the length of the second diagonal from O. We used the property that the diagonals are perpendicular to each other by drawing the second diagonal along the perpendicular bisector of the first diagonal. These two properties together guarantee that the resulting figure is a rhombus with the specified diagonal lengths.

Given:

A parallelogram with adjacent sides $7\text{ cm}$ and $5\text{ cm}$, and one diagonal measuring $9\text{ cm}$.

To Construct:

A unique parallelogram with the given measurements.

Steps of Construction:

Let the parallelogram be ABCD, with adjacent sides AB = $7\text{ cm}$ and BC = $5\text{ cm}$. Let the given diagonal be AC = $9\text{ cm}$.

1. Draw a line segment AC of length $9\text{ cm}$. This is the given diagonal.

2. Now, construct triangle ABC using the given side lengths and the diagonal. With A as the center, draw an arc with a radius equal to the length of AB, i.e., $7\text{ cm}$.

3. With C as the center, draw another arc with a radius equal to the length of BC, i.e., $5\text{ cm}$.

4. Let the two arcs intersect at point B. Join AB and BC. Triangle ABC is now constructed.

5. Now we need to find the fourth vertex D. Since ABCD is a parallelogram, we know that opposite sides are equal. So, AD = BC = $5\text{ cm}$ and CD = AB = $7\text{ cm}$.

6. With A as the center, draw an arc with a radius equal to the length of AD, i.e., $5\text{ cm}$. Ensure this arc is on the opposite side of AC from point B.

7. With C as the center, draw another arc with a radius equal to the length of CD, i.e., $7\text{ cm}$. Ensure this arc is on the opposite side of AC from point B.

8. Let the two arcs from steps 6 and 7 intersect at point D. Join AD and CD.

9. The figure ABCD is the required parallelogram.

Explanation:

This construction uses the property that opposite sides of a parallelogram are equal. By constructing the first triangle using two adjacent sides and the diagonal, we establish the position of three vertices. Then, using the lengths of the opposite sides, we locate the fourth vertex using compass arcs, effectively applying the "Four Sides and a Diagonal" construction case, where two pairs of sides are equal due to the parallelogram property.

Given:

A quadrilateral MNOP with the following measurements:

To Construct:

A unique quadrilateral MNOP with the given measurements.

Steps of Construction:

This construction falls under the case where two adjacent sides (NO and OP) and the angle between them ($\angle$O), plus another adjacent side (MN) and the angle at its endpoint ($\angle$N), are given. It's essentially two sides and two angles, arranged in a way that allows step-by-step construction.

1. Draw a line segment NO of length $6\text{ cm}$.

2. At point N, construct an angle equal to $\angle$N = $105^\circ$ using a protractor. Draw a ray NX from N.

3. On the ray NX, mark a point M such that NM has a length of $5\text{ cm}$.

4. At point O, construct an angle equal to $\angle$O = $60^\circ$ using a protractor. Draw a ray OY from O.

5. On the ray OY, mark a point P such that OP has a length of $7\text{ cm}$.

6. Join points M and P. The figure MNOP is the required quadrilateral.

Explanation:

We start by drawing the side NO which is included between the two given angles. Then, we construct the angles at N and O and mark off the lengths of the adjacent sides MN and OP along the respective rays. Joining the endpoints M and P completes the quadrilateral. This sequence of steps, using the given side lengths and angles, uniquely determines the position of all four vertices.

Given:

A rectangle with one side $4\text{ cm}$ and diagonal $5\text{ cm}$.

To Construct:

A unique rectangle with the given measurements, write down the steps, and identify the implicit congruence criterion.

Steps of Construction:

Let the rectangle be ABCD. Let the given side be AB = $4\text{ cm}$ and the diagonal be AC = $5\text{ cm}$.

1. Draw a line segment AB of length $4\text{ cm}$.

2. At point A, construct a perpendicular line (an angle of $90^\circ$). Draw a ray AX from A.

3. Now, consider the right-angled triangle ABC, where $\angle$B is $90^\circ$ (property of a rectangle). We know AB = $4\text{ cm}$ and the hypotenuse AC = $5\text{ cm}$. Point C lies on the ray AX. To find the exact position of C, we use the diagonal length.

4. With A as the center, draw an arc with a radius equal to the length of the diagonal, i.e., $5\text{ cm}$.

5. Let the arc from step 4 intersect the ray AX (the perpendicular line from A) at point C.

6. Join BC. The length of BC represents the other side of the rectangle. By the Pythagorean theorem in $\triangle$ABC, $AB^2 + BC^2 = AC^2$, so $4^2 + BC^2 = 5^2$, which means $16 + BC^2 = 25$, so $BC^2 = 9$, and $BC = 3\text{ cm}$. The other side length is $3\text{ cm}$.

7. Now that we have two adjacent sides (AB = $4\text{ cm}$, BC = $3\text{ cm}$), we can find the fourth vertex D. From A, draw an arc with radius $3\text{ cm}$ (length of AD, equal to BC). From C, draw an arc with radius $4\text{ cm}$ (length of CD, equal to AB). Let these arcs intersect at D.

8. Join AD and CD. The figure ABCD is the required rectangle.

Implicit Congruence Criterion Used:

The construction of the triangle ABC in steps 1-5 involves forming a right-angled triangle where the lengths of one leg (AB) and the hypotenuse (AC) are known. The construction of a unique right-angled triangle using the length of one leg and the hypotenuse implicitly relies on the **RHS (Right angle - Hypotenuse - Side)** congruence criterion.

RHS congruence states that if in two right-angled triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent. In our construction, by drawing AB, the $90^\circ$ angle at B (or A, depending on how you orient it), and the hypotenuse AC, we are essentially fixing the shape and size of the right-angled triangle formed by the side and the diagonal.

Final Answer:

The steps of construction are provided above.

The congruence criterion used implicitly is the RHS (Right angle - Hypotenuse - Side) congruence criterion, applied to the right-angled triangle formed by the given side and the diagonal.

Given:

The length of a square's diagonal, $d$.

To Construct:

A unique square with diagonal length $d$, and explain the steps using properties of a square's diagonals.

Properties of a Square's Diagonals:

The construction relies on these key properties:

1. Diagonals are equal in length.

2. Diagonals bisect each other.

3. Diagonals are perpendicular to each other.

Combining properties 2 and 3, the diagonals of a square bisect each other at right angles. Property 1 means both diagonals have the same given length $d$.

Steps of Construction and Justification:

1. Draw a line segment AC of length $d$.

Justification: This segment represents one of the diagonals of the square. Since the length of a diagonal is given, we start by drawing it.

2. Construct the perpendicular bisector of AC. Find the midpoint of AC, say O.

Justification: The diagonals of a square bisect each other (Property 2). This means the intersection point O is the midpoint of AC. The diagonals are also perpendicular (Property 3). Thus, the other diagonal BD must lie on the line perpendicular to AC passing through its midpoint O. Constructing the perpendicular bisector finds this exact line and the midpoint O.

3. On the perpendicular bisector, mark points B and D on opposite sides of O such that OB = OD = $\frac{d}{2}$.

Justification: The diagonals are equal in length (Property 1), so the other diagonal BD also has length $d$. The diagonals bisect each other at O (Property 2), so O is the midpoint of BD. Therefore, the length from O to each endpoint B and D is half the length of the diagonal BD, which is $\frac{d}{2}$.

4. Join the points A, B, C, and D.

Justification: These are the four vertices of the square. By connecting them in order, we form the sides of the quadrilateral. Since the diagonals AC and BD are equal, bisect each other at right angles, the resulting figure is a square.

Final Answer:

The steps of construction are provided above, with justifications based on the properties that the diagonals of a square are equal in length and bisect each other at right angles.