Chapter 7 Congruence of Triangles (Additional Questions)

Two plane figures are said to be congruent if they are exact copies of each other. This means that one figure can be placed on top of the other so that they match up perfectly. This can only happen if they have the exact same shape and the exact same size.

Option (A) is incorrect because congruent figures must have the same size.

Option (B) is incorrect because congruent figures must have the same shape.

Option (D) is incorrect because two figures can have the same area but different shapes and perimeters (e.g., a square with side 4 and a rectangle with sides 2 and 8 both have area 16, but are not congruent).

Option (C) is correct because congruent figures have the same shape and size.

Therefore, two plane figures are said to be congruent if they have the same shape and size.

The correct answer is (C) Shape and size.

Two geometric figures are said to be congruent if they have exactly the same shape and size. For line segments, the "shape" is always a straight line. Therefore, the congruence of two line segments depends only on their size, which is measured by their length.

If two line segments have the same length, they can be placed on top of each other to coincide completely, meaning they are congruent.

Let's consider the given options:

(A) Endpoint: Two line segments can have the same endpoint but different lengths (e.g., OA and OB where A, O, B are distinct and collinear, O is the common endpoint). So, having the same endpoint does not guarantee congruence.

(B) Length: If two line segments have the same length, they are exact copies of each other in terms of size. They can be moved (translated or rotated) to coincide perfectly. This is the definition of congruent line segments.

(C) Direction: Line segments have a direction if considered as vectors, but for geometric congruence, direction is not a defining property unless specified in a context like directed line segments. Two segments of the same length but different orientations (directions) in space are still congruent as geometric figures.

(D) Position: Having the same position would mean they occupy the exact same space, effectively being the same line segment. Congruent figures can be in different positions but are still congruent if they have the same shape and size.

Thus, two line segments are congruent if and only if they have the same length.

The correct answer is (B) Length.

Two geometric figures are congruent if they have the same shape and size. For angles, the "shape" is determined by the angle itself (the opening between the two rays). The "size" of an angle is its measure.

If two angles have the same measure (e.g., both are $45^\circ$), they can be superimposed onto each other perfectly, meaning they are congruent.

Let's evaluate the options:

(A) Vertex: Two angles can share the same vertex but have different measures (e.g., adjacent angles $\angle$AOB and $\angle$BOC where O is the common vertex). So, having the same vertex does not imply congruence.

(B) Arms: Two angles sharing one or both arms are not necessarily congruent. For example, two angles can share a common arm but have different measures. If they share both arms and the vertex, they are the same angle.

(C) Measure: If two angles have the same measure, say $m\angle A = m\angle B$, then angle A and angle B are congruent. This is the defining condition for congruence of angles.

(D) Orientation: The orientation of an angle in the plane or space does not affect its congruence. An angle with a certain measure is congruent to another angle with the same measure, regardless of how they are rotated or positioned.

Therefore, two angles are congruent if and only if they have the same measure.

The correct answer is (C) Measure.

Two geometric figures are said to be congruent if they have the same shape and the same size. This means that one figure can be transformed (by translation, rotation, or reflection) into the other.

Let's consider each option:

(A) Two squares: A square is defined by the length of its side. If two squares have equal sides, say side length '$s$', then they both have four sides of length '$s$' and four right angles. Any square with side length '$s$' is an exact replica of any other square with side length '$s$'. Therefore, two squares with equal sides are always congruent.

(B) Two rectangles: A rectangle is defined by its length and width. For two rectangles to be congruent, their corresponding lengths and widths must be equal. If a rectangle has sides $l$ and $w$, another rectangle is congruent to it if it also has sides $l$ and $w$ (or $w$ and $l$). The statement "if their sides are equal" is somewhat ambiguous for rectangles. If it implies that their length and width are respectively equal ($l_1=l_2$ and $w_1=w_2$), then they are congruent. However, if it is interpreted more loosely, for example, having one side of equal length, it is not sufficient for congruence.

Consider a rectangle with sides $3$ and $4$ and another rectangle with sides $3$ and $5$. They both have a side equal to $3$. However, they are not congruent as their dimensions are different ($3 \times 4$ vs $3 \times 5$). Thus, two rectangles with "equal sides" are not always congruent unless specifically stating that corresponding lengths and widths are equal.

(C) Two circles: A circle is defined by its radius. The shape of any circle is the same. The size of a circle is determined solely by its radius. If two circles have equal radii, say radius '$r$', then they both have the same size and shape. Any circle with radius '$r$' is congruent to any other circle with radius '$r$'. Therefore, two circles with equal radii are always congruent.

Based on the analysis, two squares with equal sides are always congruent, and two circles with equal radii are always congruent. Two rectangles are not always congruent if their sides are equal, based on the ambiguity of the phrasing which can allow for non-congruent examples.

Thus, the pairs of figures that are always congruent if their sides/radii are equal are two squares and two circles.

The correct answer is (D) Both (A) and (C).

When two triangles, say $\triangle\text{ABC}$ and $\triangle\text{PQR}$, are stated to be congruent, written as $\triangle\text{ABC} \cong \triangle\text{PQR}$, it means that one triangle can be perfectly superimposed onto the other by a sequence of rigid transformations (translation, rotation, or reflection).

The notation $\triangle\text{ABC} \cong \triangle\text{PQR}$ implies a specific correspondence between the vertices:

Vertex A corresponds to Vertex P

Vertex B corresponds to Vertex Q

Vertex C corresponds to Vertex R

Due to this perfect superposition, all corresponding parts of the two triangles are equal. The corresponding parts are the corresponding angles and the corresponding sides.

The corresponding angles are equal:

$\angle\text{A} = \angle\text{P}$

$\angle\text{B} = \angle\text{Q}$

$\angle\text{C} = \angle\text{R}$

The corresponding sides are equal:

AB = PQ

BC = QR

AC = PR

This concept is often summarised by the acronym CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Now let's consider the given options:

(A) Their areas are equal: If two triangles are congruent, they are identical in shape and size. Therefore, they must occupy the same amount of space, meaning their areas are equal. This is a consequence of congruence.

(B) Their perimeters are equal: The perimeter of a triangle is the sum of the lengths of its sides. Since the corresponding sides of congruent triangles are equal (AB=PQ, BC=QR, AC=PR), the sum of the sides must also be equal. Perimeter of $\triangle\text{ABC}$ = AB + BC + AC, and Perimeter of $\triangle\text{PQR}$ = PQ + QR + PR. Since AB=PQ, BC=QR, and AC=PR, it follows that AB + BC + AC = PQ + QR + PR. Thus, their perimeters are equal. This is also a consequence of congruence.

(C) Their corresponding parts are equal: As explained above, the definition and meaning of congruent triangles include the equality of all corresponding angles and sides. This is the fundamental property that results from congruence.

(D) All of the above: Since options (A), (B), and (C) are all true statements about congruent triangles, this is the most comprehensive correct answer.

The congruence of two triangles means that their corresponding parts are equal, which in turn implies that their areas and perimeters are equal.

The correct answer is (D) All of the above.

When two triangles are congruent, written as $\triangle\text{XYZ} \cong \triangle\text{LMN}$, the order of the vertices in the congruence statement is crucial. It indicates the correspondence between the vertices of the two triangles.

In this case, the congruence $\triangle\text{XYZ} \cong \triangle\text{LMN}$ means the vertices correspond as follows:

X $\leftrightarrow$ L

Y $\leftrightarrow$ M

Z $\leftrightarrow$ N

This correspondence implies that the corresponding sides and corresponding angles are equal (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

The corresponding sides are:

Side XY corresponds to side LM, so $XY = LM$.

Side YZ corresponds to side MN, so $YZ = MN$.

Side XZ corresponds to side LN, so $XZ = LN$.

The corresponding angles are:

Angle $\angle\text{X}$ corresponds to angle $\angle\text{L}$, so $\angle\text{X} = \angle\text{L}$.

Angle $\angle\text{Y}$ corresponds to angle $\angle\text{M}$, so $\angle\text{Y} = \angle\text{M}$.

Angle $\angle\text{Z}$ corresponds to angle $\angle\text{N}$, so $\angle\text{Z} = \angle\text{N}$.

Now let's check the given options based on these corresponding parts:

(A) XY = LM: This is a statement about corresponding sides (XY corresponds to LM). This is necessarily true.

(B) $\angle\text{Y} = \angle\text{M}$: This is a statement about corresponding angles ($\angle\text{Y}$ corresponds to $\angle\text{M}$). This is necessarily true.

(C) YZ = LN: YZ is a side of $\triangle\text{XYZ}$. LN is a side of $\triangle\text{LMN}$. According to the correspondence, YZ corresponds to MN, and LN corresponds to XZ. The statement YZ = LN means that the side YZ is equal in length to the side LN. This is not a statement about corresponding parts (YZ corresponds to MN, not LN). Therefore, this statement is NOT necessarily true based on the given congruence.

(D) $\angle\text{X} = \angle\text{L}$: This is a statement about corresponding angles ($\angle\text{X}$ corresponds to $\angle\text{L}$). This is necessarily true.

The statement that is not necessarily true based on the congruence $\triangle\text{XYZ} \cong \triangle\text{LMN}$ is YZ = LN.

The correct answer is (C) YZ = LN.

Triangle congruence criteria are rules that allow us to determine if two triangles are congruent without having to check if all six corresponding parts (three sides and three angles) are equal. These criteria specify a minimum set of corresponding parts that, if equal, guarantee the congruence of the triangles.

Let's examine the given criteria:

(A) SAS (Side-Angle-Side): This criterion states that if two sides and the included angle (the angle between the two sides) of one triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

(B) ASA (Angle-Side-Angle): This criterion states that if two angles and the included side (the side between the two angles) of one triangle are equal to the corresponding two angles and the included side of another triangle, then the triangles are congruent.

(C) SSS (Side-Side-Side): This criterion states that if the three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.

(D) RHS (Right-angle-Hypotenuse-Side): This criterion is specifically for right-angled triangles. It states that if the hypotenuse and one side of a right-angled triangle are equal to the corresponding hypotenuse and one side of another right-angled triangle, then the triangles are congruent.

The question asks for the criterion where three sides of one triangle are equal to the three corresponding sides of another triangle. This description perfectly matches the definition of the SSS congruence criterion.

Therefore, the congruence criterion that states that if three sides of one triangle are equal to the three corresponding sides of another triangle, the triangles are congruent, is SSS.

The correct answer is (C) SSS.

The question describes a specific condition for proving triangle congruence: the equality of two sides and the angle located between those two sides (the included angle) in two triangles.

Let's recall the common triangle congruence criteria:

(A) ASA (Angle-Side-Angle): Two angles and the included side are equal.

(B) SSS (Side-Side-Side): Three sides are equal.

(C) SAS (Side-Angle-Side): Two sides and the included angle are equal.

(D) RHS (Right-angle-Hypotenuse-Side): Hypotenuse and one side of a right triangle are equal.

The description given in the question, "two sides and the included angle", directly corresponds to the SAS congruence criterion.

Therefore, if two sides and the included angle of one triangle are equal to two corresponding sides and the included angle of another triangle, the triangles are congruent by the SAS criterion.

The correct answer is (C) SAS.

The ASA congruence criterion stands for Angle-Side-Angle. This means that the side being considered must be positioned between the two angles being considered. This specific position between the two angles is referred to as the included side.

The ASA congruence criterion states that if two angles and the side included between them of one triangle are equal to the corresponding two angles and the included side of another triangle, then the triangles are congruent.

Let's illustrate with a triangle $\triangle\text{ABC}$. If we consider angles $\angle\text{A}$ and $\angle\text{B}$, the side included between them is the side AB. The sides opposite to these angles are BC (opposite $\angle\text{A}$) and AC (opposite $\angle\text{B}$). Adjacent sides to $\angle\text{A}$ are AB and AC, and adjacent sides to $\angle\text{B}$ are AB and BC.

For the ASA criterion, we must have two angles and the side connecting the vertices of those two angles being equal in both triangles.

Considering the options:

(A) Opposite: The side opposite to one of the angles is not the side included between the two angles.

(B) Adjacent: While the included side is adjacent to both angles, the term "adjacent" is not specific enough in this context.

(C) Included: This term correctly describes the position of the side that lies between the two specified angles.

(D) Any: The ASA criterion specifically requires the included side; any side will not work.

Therefore, the ASA congruence criterion requires two angles and the included side to be equal.

The correct answer is (C) Included.

The RHS congruence criterion is a specific rule used to prove the congruence of triangles. The acronym RHS stands for:

R - Right angle

H - Hypotenuse

S - Side

The RHS congruence criterion states that if in two right-angled triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one corresponding side of the other triangle, then the two triangles are congruent.

This criterion requires the presence of a right angle and refers to the hypotenuse, which is the side opposite the right angle. Both a right angle and a hypotenuse are features exclusive to right-angled triangles.

Let's look at the options:

(A) Equilateral triangles: These have all three sides equal and all three angles equal to $60^\circ$. They do not have a right angle, so the RHS criterion is not applicable.

(B) Isosceles triangles: These have two sides equal. They may or may not have a right angle. The RHS criterion cannot be applied to all isosceles triangles, only those that are also right-angled.

(C) Right-angled triangles: These triangles have one angle equal to $90^\circ$ and thus have a hypotenuse. The RHS criterion is specifically formulated for this type of triangle.

(D) Scalene triangles: These have all three sides of different lengths. They may or may not have a right angle. The RHS criterion cannot be applied to all scalene triangles, only those that are also right-angled.

Therefore, the RHS congruence criterion is applicable specifically to right-angled triangles.

The correct answer is (C) Right-angled triangles.

The RHS congruence criterion is used to prove the congruence of two right-angled triangles. The letters in RHS stand for specific parts of a right-angled triangle.

R stands for Right angle ($90^\circ$).

H stands for Hypotenuse (the side opposite the right angle, which is the longest side in a right-angled triangle).

S stands for Side. In the context of the RHS criterion, this side must be one of the legs of the right-angled triangle (the sides adjacent to the right angle). It cannot be the hypotenuse because the hypotenuse is already represented by 'H'.

The RHS congruence criterion states that two right-angled triangles are congruent if the hypotenuse and one side (leg) of one triangle are equal to the corresponding hypotenuse and one corresponding side (leg) of the other triangle.

Let's examine the options:

(A) Side (any side): While 'S' stands for side, it cannot be *any* side; specifically, it cannot be the hypotenuse.

(B) Corresponding angle: 'S' stands for a side length, not an angle.

(C) Any one side (other than hypotenuse): This accurately describes the 'S' in RHS. It refers to one of the two legs of the right triangle.

(D) Sum of sides: 'S' represents the length of a single side, not the sum of sides (which is the perimeter).

Therefore, in the RHS congruence criterion, 'S' stands for any one side other than the hypotenuse.

The correct answer is (C) Any one side (other than hypotenuse).

We are given two triangles, $\triangle\text{ABC}$ and $\triangle\text{PQR}$. We are provided with information about the lengths of their sides:

AB = PQ

BC = QR

CA = RP

Let's analyze the given equalities and the congruence criteria:

The equality AB = PQ means that a side of $\triangle\text{ABC}$ is equal to a corresponding side of $\triangle\text{PQR}$.

The equality BC = QR means that another side of $\triangle\text{ABC}$ is equal to a corresponding side of $\triangle\text{PQR}$.

The equality CA = RP means that the third side of $\triangle\text{ABC}$ is equal to a corresponding side of $\triangle\text{PQR}$.

We have shown that all three sides of $\triangle\text{ABC}$ are equal to the three corresponding sides of $\triangle\text{PQR}$.

Recall the congruence criteria:

SAS: Two sides and the included angle.

ASA: Two angles and the included side.

SSS: Three sides.

RHS: Right angle, hypotenuse, and one side (for right triangles).

The condition where all three sides of one triangle are equal to the corresponding three sides of another triangle is the definition of the SSS congruence criterion.

Therefore, based on the given information (AB = PQ, BC = QR, CA = RP), the triangles $\triangle\text{ABC}$ and $\triangle\text{PQR}$ are congruent by the SSS criterion.

Note that the order of vertices in the congruence statement should reflect the side equalities. Since AB=PQ, BC=QR, and CA=RP, the correct congruence statement is $\triangle\text{ABC} \cong \triangle\text{PQR}$.

The correct answer is (C) SSS.

We are given two triangles, $\triangle\text{ABC}$ and $\triangle\text{XYZ}$. We are provided with the following equalities:

AB = XY

$\angle\text{B} = \angle\text{Y}$

BC = YZ

Let's analyze these equalities in relation to the vertices of the triangles:

AB is a side in $\triangle\text{ABC}$. XY is a side in $\triangle\text{XYZ}$.

BC is a side in $\triangle\text{ABC}$. YZ is a side in $\triangle\text{XYZ}$.

$\angle\text{B}$ is an angle in $\triangle\text{ABC}$. $\angle\text{Y}$ is an angle in $\triangle\text{XYZ}$.

Observe the relationship between the equal angle and the equal sides in $\triangle\text{ABC}$. The sides AB and BC meet at vertex B, forming the angle $\angle\text{B}$. Thus, $\angle\text{B}$ is the angle included between sides AB and BC.

Similarly, in $\triangle\text{XYZ}$, the sides XY and YZ meet at vertex Y, forming the angle $\angle\text{Y}$. Thus, $\angle\text{Y}$ is the angle included between sides XY and YZ.

We are given that:

Side AB = Side XY

Included Angle $\angle\text{B}$ = Included Angle $\angle\text{Y}$

Side BC = Side YZ

This set of conditions (Side-Angle-Side, where the angle is included between the two sides) corresponds precisely to the SAS congruence criterion.

The SAS criterion states that if two sides and the included angle of one triangle are equal to the two corresponding sides and the included angle of another triangle, then the triangles are congruent.

Based on the given information, $\triangle\text{ABC}$ and $\triangle\text{XYZ}$ satisfy the conditions of the SAS congruence criterion.

The correct congruence statement, following the order of the equal parts, is $\triangle\text{ABC} \cong \triangle\text{XYZ}$.

The correct answer is (A) SAS.

We are given two triangles, $\triangle\text{PQR}$ and $\triangle\text{STU}$. We are provided with the following equal corresponding parts:

1. $\angle\text{Q} = \angle\text{T}$

2. QR = TU

3. $\angle\text{R} = \angle\text{U}$

Let's examine the relationship between these equal parts in each triangle.

In $\triangle\text{PQR}$, we have two angles, $\angle\text{Q}$ and $\angle\text{R}$, and the side QR. The side QR connects the vertices Q and R, which are the vertices where angles $\angle\text{Q}$ and $\angle\text{R}$ are located. Thus, the side QR is the included side between angles $\angle\text{Q}$ and $\angle\text{R}$.

Similarly, in $\triangle\text{STU}$, we have two angles, $\angle\text{T}$ and $\angle\text{U}$, and the side TU. The side TU connects the vertices T and U, which are the vertices where angles $\angle\text{T}$ and $\angle\text{U}$ are located. Thus, the side TU is the included side between angles $\angle\text{T}$ and $\angle\text{U}$.

We are given that $\angle\text{Q} = \angle\text{T}$, the included side QR = the included side TU, and $\angle\text{R} = \angle\text{U}$.

This pattern of equal corresponding parts is: Angle - Included Side - Angle.

Let's review the congruence criteria:

SAS (Side-Angle-Side): Two sides and the included angle are equal.

ASA (Angle-Side-Angle): Two angles and the included side are equal.

SSS (Side-Side-Side): Three sides are equal.

AAS (Angle-Angle-Side): Two angles and a non-included side are equal.

The given conditions (Angle $\angle\text{Q}$ = Angle $\angle\text{T}$, Included Side QR = Included Side TU, Angle $\angle\text{R}$ = Angle $\angle\text{U}$) perfectly match the requirements of the ASA congruence criterion.

Therefore, based on the given information, the triangles $\triangle\text{PQR}$ and $\triangle\text{STU}$ are congruent by the ASA criterion.

The congruence statement $\triangle\text{PQR} \cong \triangle\text{STU}$ indicates the correct correspondence of vertices: P $\leftrightarrow$ S, Q $\leftrightarrow$ T, R $\leftrightarrow$ U.

The correct answer is (B) ASA.

We are given two triangles, $\triangle\text{ABC}$ and $\triangle\text{DEF}$. We are told that both are right-angled triangles, with the right angle at vertex B in $\triangle\text{ABC}$ and at vertex E in $\triangle\text{DEF}$.

This means $\angle\text{B} = 90^\circ$ and $\angle\text{E} = 90^\circ$. Therefore, $\angle\text{B} = \angle\text{E}$ (both are right angles).

We are also given the following equalities regarding their sides:

AC = DF

BC = EF

In a right-angled triangle $\triangle\text{ABC}$ with the right angle at B, the side opposite the right angle (AC) is the hypotenuse.

In a right-angled triangle $\triangle\text{DEF}$ with the right angle at E, the side opposite the right angle (DF) is the hypotenuse.

We are given that AC = DF, which means the hypotenuses of the two triangles are equal.

We are also given that BC = EF. BC is one of the legs (sides forming the right angle) of $\triangle\text{ABC}$, and EF is one of the legs of $\triangle\text{DEF}$. So, one pair of corresponding sides (legs) are equal.

We have identified the following equal corresponding parts in two right-angled triangles:

1. A right angle ($\angle\text{B} = \angle\text{E} = 90^\circ$).

2. The hypotenuse (AC = DF).

3. One side (leg) (BC = EF).

This set of conditions (Right angle - Hypotenuse - Side) is the definition of the RHS congruence criterion.

The RHS criterion specifically states that if the hypotenuse and one side of a right-angled triangle are equal to the corresponding hypotenuse and side of another right-angled triangle, then the triangles are congruent.

Based on the given information and the fact that both triangles are right-angled, the triangles $\triangle\text{ABC}$ and $\triangle\text{DEF}$ are congruent by the RHS criterion.

The correct congruence statement, following the correspondence (A $\leftrightarrow$ D, B $\leftrightarrow$ E, C $\leftrightarrow$ F), is $\triangle\text{ABC} \cong \triangle\text{DEF}$.

The correct answer is (D) RHS.

We need to identify which of the given options is NOT a recognised criterion for proving the congruence of two triangles.

Let's review the standard congruence criteria that are commonly used:

1. SSS (Side-Side-Side): If three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent.

2. SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, the triangles are congruent.

3. ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to the corresponding two angles and the included side of another triangle, the triangles are congruent.

4. AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to the corresponding two angles and the non-included side of another triangle, the triangles are congruent. (This criterion can be derived from ASA and the angle sum property of triangles).

5. RHS (Right-angle-Hypotenuse-Side): If in two right-angled triangles, the hypotenuse and one side are equal to the corresponding hypotenuse and side, the triangles are congruent.

Now let's compare these with the given options:

(A) ASS (Angle-Side-Side) or SSA (Side-Side-Angle): This criterion states that if two sides and a non-included angle of one triangle are equal to the corresponding two sides and a non-included angle of another triangle, the triangles are congruent. However, this is NOT a valid congruence criterion in general. There can be two different triangles with two equal sides and one equal non-included angle (this is known as the ambiguous case of the Law of Sines).

(B) ASA: This is a valid congruence criterion.

(C) SSS: This is a valid congruence criterion.

(D) SAS: This is a valid congruence criterion.

The combination of equal Angle, Side, and non-included Side (ASS or SSA) does not guarantee congruence, except in the special case of right-angled triangles, which is covered by the RHS criterion.

Therefore, ASS is not a valid congruence criterion for triangles.

The correct answer is (A) ASS.

When two triangles are proven to be congruent, it means that they are exact copies of each other and can be perfectly superimposed. As a result of this congruence, all of their corresponding parts are equal.

The corresponding parts of congruent triangles include their corresponding angles and their corresponding sides.

The principle that states that the corresponding parts of congruent triangles are equal is a fundamental concept in geometry.

Let's look at the options:

(A) SSS rule: This is a criterion used to *prove* that triangles are congruent if their three sides are equal.

(B) CPCTC: This acronym stands for "Corresponding Parts of Congruent Triangles are Congruent" (or Equal). This principle directly states that if two triangles are congruent, then their corresponding angles and sides are equal.

(C) SAS rule: This is a criterion used to *prove* that triangles are congruent if two sides and the included angle are equal.

(D) ASA rule: This is a criterion used to *prove* that triangles are congruent if two angles and the included side are equal.

The statement "If two triangles are congruent, then their corresponding angles are equal" (and their corresponding sides are equal) is a direct consequence of the definition of congruence and is formally stated by the CPCTC principle.

The correct answer is (B) CPCTC.

Assertion (A): Two equilateral triangles with the same side length are always congruent.

An equilateral triangle has three sides of equal length and three angles of equal measure, each being $60^\circ$. If two equilateral triangles have the same side length, say $s$, then all three sides of the first triangle measure $s$, and all three sides of the second triangle measure $s$. Since they have the same side lengths and the same angles ($60^\circ$), they have the same shape and size. Therefore, they are congruent.

Thus, Assertion (A) is True.

Reason (R): If all three sides of one triangle are equal to the corresponding three sides of another triangle, they are congruent by SSS criterion.

This statement is the definition of the SSS (Side-Side-Side) congruence criterion. The SSS criterion is a valid and fundamental rule in geometry for proving the congruence of triangles.

Thus, Reason (R) is True.

Now let's check if Reason (R) is the correct explanation for Assertion (A).

Assertion (A) deals with two equilateral triangles having the same side length. Let this side length be $s$. Triangle 1 has sides of lengths $(s, s, s)$. Triangle 2 has sides of lengths $(s, s, s)$.

The condition in Reason (R) is that the three sides of one triangle are equal to the corresponding three sides of another triangle. In the case of the two equilateral triangles with side length $s$, this condition is met (side $s$ in Triangle 1 equals corresponding side $s$ in Triangle 2, and so on for all three sides).

Reason (R) states that if this condition (equality of three sides) is met, the triangles are congruent by the SSS criterion. This directly explains why the two equilateral triangles described in Assertion (A) are congruent.

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

Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).

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

Assertion (A): Two squares with the same perimeter are congruent.

Let the side length of a square be $s$. The perimeter of a square is given by $P = 4s$.

If two squares have the same perimeter, let's say $P_1 = P_2$. Then $4s_1 = 4s_2$, which implies $s_1 = s_2$.

Two squares with equal side lengths are congruent because they have the same shape (square) and the same size (determined by the side length). All corresponding sides are equal, and all angles are $90^\circ$.

Thus, Assertion (A) is True.

Reason (R): Congruent figures have the same shape and size.

This statement is the definition of congruent figures. Congruence means that two figures are identical in shape and size, allowing one to be placed exactly over the other.

Thus, Reason (R) is True.

Now we examine if Reason (R) is the correct explanation for Assertion (A).

Assertion (A) states that equal perimeter implies congruence for squares. The reason why this is true is that the perimeter of a square uniquely determines its side length ($s = P/4$), and the side length of a square uniquely determines its shape and size.

Reason (R) simply defines what congruence means (having the same shape and size). It does not explain the specific link between the perimeter of squares and their congruence. The explanation for Assertion (A) lies in the relationship $P=4s$ and the SSS congruence criterion for triangles (a square can be divided into two congruent right triangles, and the sides of the square determine the sides of these triangles, and thus their congruence by SSS or RHS).

While both statements are true, Reason (R) is the definition of congruence and does not specifically explain why equal perimeter leads to equal side length, which is the direct reason for congruence in the case of squares.

Therefore, both A and R are true, but R is not the correct explanation of A.

The correct answer is (B) Both A and R are true, but R is not the correct explanation of A.

We are given a figure and the following information about a quadrilateral ABCD:

AD = CD

AB = CB

We need to determine the congruence criterion that can be used to prove $\triangle\text{ABD} \cong \triangle\text{CBD}$.

Let's consider the two triangles, $\triangle\text{ABD}$ and $\triangle\text{CBD}$. We can list the given equal parts and any common parts:

1. AB = CB

This is a side in $\triangle\text{ABD}$ and a corresponding side in $\triangle\text{CBD}$.

2. AD = CD

This is another side in $\triangle\text{ABD}$ and a corresponding side in $\triangle\text{CBD}$.

3. BD is common to both triangles.

This is the third side in both $\triangle\text{ABD}$ and $\triangle\text{CBD}$.

We have found that all three sides of $\triangle\text{ABD}$ are equal to the corresponding three sides of $\triangle\text{CBD}$ (AB=CB, AD=CD, and BD=BD).

Recall the triangle congruence criteria:

SAS: Two sides and the included angle.

ASA: Two angles and the included side.

SSS: Three sides.

RHS: Right angle, hypotenuse, and one side (for right triangles).

The condition where three sides of one triangle are equal to the three corresponding sides of another triangle is the definition of the SSS congruence criterion.

Therefore, we can prove $\triangle\text{ABD} \cong \triangle\text{CBD}$ using the SSS congruence criterion.

The correct answer is (C) SSS.

From the case study in Question 20, we were given that AD = CD and AB = CB in quadrilateral ABCD, and we proved that $\triangle\text{ABD} \cong \triangle\text{CBD}$ using the SSS congruence criterion (AD=CD, AB=CB, BD=BD).

Once two triangles are proven to be congruent, their Corresponding Parts of Congruent Triangles are Congruent (or Equal). This is the principle known as CPCTC.

The congruence statement $\triangle\text{ABD} \cong \triangle\text{CBD}$ establishes the correspondence between the vertices:

A $\leftrightarrow$ C

B $\leftrightarrow$ B

D $\leftrightarrow$ D

Based on this correspondence, the corresponding angles are equal due to CPCTC:

1. The angle at vertex A in $\triangle\text{ABD}$ corresponds to the angle at vertex C in $\triangle\text{CBD}$. So, $\angle\text{BAD} = \angle\text{BCD}$, which is the same as $\angle\text{DAB} = \angle\text{DCB}$.

2. The angle at vertex B in $\triangle\text{ABD}$ corresponds to the angle at vertex B in $\triangle\text{CBD}$. So, $\angle\text{ABD} = \angle\text{CBD}$.

3. The angle at vertex D in $\triangle\text{ABD}$ corresponds to the angle at vertex D in $\triangle\text{CBD}$. So, $\angle\text{ADB} = \angle\text{CDB}$.

Let's check the given options against these equalities:

(A) $\angle\text{DAB} = \angle\text{DCB}$: This equality corresponds to the angles at the first and third vertices in the congruence statement ($\angle\text{BAD}$ corresponds to $\angle\text{BCD}$). This is true by CPCTC.

(B) $\angle\text{ADB} = \angle\text{CDB}$: This equality corresponds to the angles at the third vertex in the congruence statement ($\angle\text{ADB}$ corresponds to $\angle\text{CDB}$). This is true by CPCTC.

(C) $\angle\text{ABD} = \angle\text{CBD}$: This equality corresponds to the angles at the second vertex in the congruence statement ($\angle\text{ABD}$ corresponds to $\angle\text{CBD}$). This is true by CPCTC.

(D) All of the above: Since options (A), (B), and (C) are all true statements derived from the congruence $\triangle\text{ABD} \cong \triangle\text{CBD}$ using CPCTC, this option is correct.

Therefore, all the listed equalities are true because of CPCTC.

The correct answer is (D) All of the above.

The question asks to complete the statement that is commonly abbreviated as CPCTC.

The acronym CPCTC stands for:

Corresponding

Parts

of

Congruent

Triangles

are

Congruent

In the context of lengths of sides and measures of angles, being "congruent" for these parts means having the same measure or size. So, while "congruent" is technically correct for the parts themselves (e.g., segment AB is congruent to segment PQ), when referring to their measures, we say they are "equal" (e.g., length of AB equals length of PQ, measure of $\angle\text{A}$ equals measure of $\angle\text{P}$).

In the options provided, "Equal" is the term that signifies that the measures of corresponding sides and angles are the same.

Let's consider the options:

(A) Different: This is incorrect. Corresponding parts of congruent triangles are the same.

(B) Similar: Similar figures have the same shape but not necessarily the same size. Corresponding angles are equal, but corresponding sides are proportional. Congruent figures are a special case of similar figures where the ratio of corresponding sides is 1:1.

(C) Equal: This is correct. Corresponding sides have equal lengths, and corresponding angles have equal measures.

(D) Proportional: This applies to corresponding sides of similar triangles, not congruent triangles (unless the proportionality constant is 1).

Therefore, Corresponding Parts of Congruent Triangles are Equal.

The correct answer is (C) Equal.

We are given that two triangles, $\triangle\text{PQR}$ and $\triangle\text{RSP}$, are congruent, written as $\triangle\text{PQR} \cong \triangle\text{RSP}$.

The congruence statement establishes the correspondence between the vertices of the two triangles based on their order:

P $\leftrightarrow$ R

Q $\leftrightarrow$ S

R $\leftrightarrow$ P

According to the CPCTC principle (Corresponding Parts of Congruent Triangles are Congruent, or Equal), the corresponding sides and angles of these two triangles must be equal.

Let's determine the corresponding sides based on the vertex correspondence:

1. Side PQ (formed by the 1st and 2nd vertices of $\triangle\text{PQR}$) corresponds to side RS (formed by the 1st and 2nd vertices of $\triangle\text{RSP}$).

2. Side QR (formed by the 2nd and 3rd vertices of $\triangle\text{PQR}$) corresponds to side SP (formed by the 2nd and 3rd vertices of $\triangle\text{RSP}$).

3. Side PR (formed by the 1st and 3rd vertices of $\triangle\text{PQR}$) corresponds to side RP (formed by the 1st and 3rd vertices of $\triangle\text{RSP}$).

Note that PR and RP refer to the same line segment, just read in opposite directions. This side is common to both triangles.

Now let's check the given options:

(A) PQ = RS: This equality matches the first pair of corresponding sides derived from the congruence statement. This is true by CPCTC.

(B) QR = SP: This equality matches the second pair of corresponding sides derived from the congruence statement. This is true by CPCTC.

(C) PR = RP (common side): This equality matches the third pair of corresponding sides derived from the congruence statement. It is true by CPCTC, and the note about it being a common side provides geometric context.

(D) All of the above: Since options (A), (B), and (C) are all verified to be true consequences of the congruence $\triangle\text{PQR} \cong \triangle\text{RSP}$ by CPCTC, this is the correct answer.

Therefore, all the listed equalities are true when $\triangle\text{PQR} \cong \triangle\text{RSP}$.

The correct answer is (D) All of the above.

The question describes the conditions required by a specific triangle congruence criterion: equality of two sides and the angle situated between these two sides in two different triangles. The angle between the two sides is called the included angle.

Let's examine the given congruence criteria:

(A) SSS (Side-Side-Side): Requires the equality of three sides.

(B) ASA (Angle-Side-Angle): Requires the equality of two angles and the included side.

(C) AAS (Angle-Angle-Side): Requires the equality of two angles and a non-included side.

(D) SAS (Side-Angle-Side): Requires the equality of two sides and the included angle (the angle between the two sides).

The condition mentioned in the question, "two sides and the angle between them equal in two triangles," perfectly matches the description of the SAS congruence criterion.

Therefore, the congruence criterion used when you have two sides and the angle between them equal in two triangles is SAS.

The correct answer is (D) SAS.

We are given that the triangle $\triangle\text{LMN}$ is congruent to the triangle $\triangle\text{XYZ}$, written as $\triangle\text{LMN} \cong \triangle\text{XYZ}$.

When two triangles are congruent, the order of the vertices in the congruence statement is very important as it indicates the one-to-one correspondence between the vertices, and consequently, between the corresponding sides and angles.

From the statement $\triangle\text{LMN} \cong \triangle\text{XYZ}$, the correspondence is as follows:

Vertex L corresponds to Vertex X.

Vertex M corresponds to Vertex Y.

Vertex N corresponds to Vertex Z.

According to the principle of CPCTC (Corresponding Parts of Congruent Triangles are Congruent, or Equal), the corresponding angles and corresponding sides are equal.

Based on the vertex correspondence, the corresponding angles are:

Angle at L corresponds to Angle at X: $\angle\text{L} = \angle\text{X}$.

Angle at M corresponds to Angle at Y: $\angle\text{M} = \angle\text{Y}$.

Angle at N corresponds to Angle at Z: $\angle\text{N} = \angle\text{Z}$.

Now let's check the given options:

(A) $\angle\text{L}$ and $\angle\text{Y}$: $\angle\text{L}$ corresponds to $\angle\text{X}$, and $\angle\text{Y}$ corresponds to $\angle\text{M}$. This pair is not a corresponding pair.

(B) $\angle\text{M}$ and $\angle\text{Z}$: $\angle\text{M}$ corresponds to $\angle\text{Y}$, and $\angle\text{Z}$ corresponds to $\angle\text{N}$. This pair is not a corresponding pair.

(C) $\angle\text{N}$ and $\angle\text{X}$: $\angle\text{N}$ corresponds to $\angle\text{Z}$, and $\angle\text{X}$ corresponds to $\angle\text{L}$. This pair is not a corresponding pair.

(D) $\angle\text{M}$ and $\angle\text{Y}$: $\angle\text{M}$ is the angle at the second vertex of $\triangle\text{LMN}$, and $\angle\text{Y}$ is the angle at the second vertex of $\triangle\text{XYZ}$. These are corresponding angles according to the congruence statement $\triangle\text{LMN} \cong \triangle\text{XYZ}$. Therefore, $\angle\text{M}$ must be equal to $\angle\text{Y}$ by CPCTC.

The pair of angles that must be equal due to the congruence $\triangle\text{LMN} \cong \triangle\text{XYZ}$ is $\angle\text{M}$ and $\angle\text{Y}$.

The correct answer is (D) $\angle\text{M}$ and $\angle\text{Y}$.

We need to match each triangle congruence criterion with the required equal parts.

Let's recall what each criterion stands for and requires:

(i) SSS stands for Side-Side-Side. This criterion requires that the three sides of one triangle are equal to the three corresponding sides of another triangle.

Matching (i) with the descriptions: It matches with (c) Three sides.

(ii) SAS stands for Side-Angle-Side. This criterion requires that two sides and the included angle (the angle between the two sides) of one triangle are equal to the corresponding two sides and the included angle of another triangle.

Matching (ii) with the descriptions: It matches with (d) Two sides and included angle.

(iii) ASA stands for Angle-Side-Angle. This criterion requires that two angles and the included side (the side between the two angles) of one triangle are equal to the corresponding two angles and the included side of another triangle.

Matching (iii) with the descriptions: It matches with (a) Two angles and included side.

(iv) RHS stands for Right-angle-Hypotenuse-Side. This criterion is specifically for right triangles and requires that the hypotenuse and one side of a right triangle are equal to the corresponding hypotenuse and side of another right triangle.

Matching (iv) with the descriptions: It matches with (b) Hypotenuse and one side (right triangle).

So, the correct matching is:

(i) SSS $\leftrightarrow$ (c)

(ii) SAS $\leftrightarrow$ (d)

(iii) ASA $\leftrightarrow$ (a)

(iv) RHS $\leftrightarrow$ (b)

Now let's check the options provided:

(A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b) - This matches our derived mapping.

(B) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b) - Incorrect.

(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b) - Incorrect.

(D) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a) - Incorrect.

The correct answer is (A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b).

We are given two triangles, $\triangle\text{ABC}$ and $\triangle\text{DEF}$, with the following information:

1. $\angle\text{B} = \angle\text{E} = 90^\circ$. This tells us that both triangles are right-angled triangles, with the right angle at B in $\triangle\text{ABC}$ and at E in $\triangle\text{DEF}$.

2. AC = DF. In $\triangle\text{ABC}$ with the right angle at B, AC is the side opposite the right angle, so AC is the hypotenuse. In $\triangle\text{DEF}$ with the right angle at E, DF is the side opposite the right angle, so DF is the hypotenuse. Thus, the hypotenuses are equal.

3. AB = DE. AB is one of the sides (legs) forming the right angle in $\triangle\text{ABC}$. DE is one of the sides (legs) forming the right angle in $\triangle\text{DEF}$. Thus, one pair of corresponding sides (legs) are equal.

We have the following equal corresponding parts in two right-angled triangles:

- A right angle ($\angle\text{B} = \angle\text{E}$).

- The hypotenuse (AC = DF).

- One side (leg) (AB = DE).

This set of conditions matches the requirements of the RHS (Right-angle-Hypotenuse-Side) congruence criterion.

The RHS criterion states that if the hypotenuse and one side of a right-angled triangle are equal to the corresponding hypotenuse and side of another right-angled triangle, then the two triangles are congruent.

Based on the given information, $\triangle\text{ABC}$ and $\triangle\text{DEF}$ are congruent by the RHS criterion.

The correct answer is (D) RHS.

Two plane figures are said to be congruent if they have the exact same shape and size. For circles, the shape is always the same (a perfect round form). The size of a circle is determined by a single measurement: its radius or its diameter.

Let's consider the relationship between the different measurements of a circle:

- Radius ($r$): The distance from the center to any point on the circle.

- Diameter ($d$): The distance across the circle passing through the center ($d = 2r$).

- Circumference ($C$): The distance around the circle ($C = 2\pi r = \pi d$).

- Area ($A$): The space enclosed by the circle ($A = \pi r^2 = \pi (d/2)^2 = \pi d^2/4$).

If two circles are congruent, they must have the same size. This means their radii must be equal ($r_1 = r_2$). If their radii are equal, then their diameters will also be equal ($d_1 = 2r_1 = 2r_2 = d_2$), their circumferences will be equal ($C_1 = 2\pi r_1 = 2\pi r_2 = C_2$), and their areas will be equal ($A_1 = \pi r_1^2 = \pi r_2^2 = A_2$).

Conversely, if any one of these measures is equal for two circles, then all the other measures will also be equal, and the circles will be congruent.

- If radii are equal ($r_1 = r_2$), the circles are congruent.

- If diameters are equal ($d_1 = d_2$), since $d=2r$, it implies $2r_1 = 2r_2$, so $r_1 = r_2$. Thus, the circles are congruent.

- If circumferences are equal ($C_1 = C_2$), since $C=2\pi r$, it implies $2\pi r_1 = 2\pi r_2$, so $r_1 = r_2$. Thus, the circles are congruent.

- If areas are equal ($A_1 = A_2$), since $A=\pi r^2$, it implies $\pi r_1^2 = \pi r_2^2$. Assuming radii are positive, this means $r_1 = r_2$. Thus, the circles are congruent.

The question asks what must be equal for two circles to be congruent. While equality of radii, diameters, or circumferences (or areas) all guarantee congruence, the options provided include all three fundamental measures (diameter, radius, circumference).

Let's check the options:

(A) Diameters: If diameters are equal, radii are equal, so circles are congruent. This is true.

(B) Radii: If radii are equal, the circles are congruent. This is the most direct measure determining the size. This is true.

(C) Circumferences: If circumferences are equal, radii are equal, so circles are congruent. This is true.

(D) All of the above: Since options (A), (B), and (C) are all true statements that guarantee the congruence of two circles, this is the most accurate answer.

Therefore, two circles are congruent if their diameters, radii, or circumferences (among other related measures) are equal.

The correct answer is (D) All of the above.

We are given that $\triangle\text{ABC} \cong \triangle\text{PQR}$. This congruence statement establishes the correspondence between the vertices:

A $\leftrightarrow$ P

B $\leftrightarrow$ Q

C $\leftrightarrow$ R

We are given information about $\triangle\text{ABC}$:

AB = 5 cm

$\angle\text{B} = 60^\circ$

$\angle\text{C} = 70^\circ$

Since the triangles are congruent, their corresponding parts are equal (CPCTC).

Using the vertex correspondence:

1. Side AB in $\triangle\text{ABC}$ corresponds to side PQ in $\triangle\text{PQR}$ (connecting the 1st and 2nd corresponding vertices). So, AB = PQ.

Since AB = 5 cm, we must have PQ = 5 cm.

2. Angle $\angle\text{B}$ in $\triangle\text{ABC}$ corresponds to angle $\angle\text{Q}$ in $\triangle\text{PQR}$ (angle at the 2nd corresponding vertex). So, $\angle\text{B} = \angle\text{Q}$.

Since $\angle\text{B} = 60^\circ$, we must have $\angle\text{Q} = 60^\circ$.

3. Angle $\angle\text{C}$ in $\triangle\text{ABC}$ corresponds to angle $\angle\text{R}$ in $\triangle\text{PQR}$ (angle at the 3rd corresponding vertex). So, $\angle\text{C} = \angle\text{R}$.

Since $\angle\text{C} = 70^\circ$, we must have $\angle\text{R} = 70^\circ$.

Therefore, in $\triangle\text{PQR}$, we must have PQ = 5 cm, $\angle\text{Q} = 60^\circ$, and $\angle\text{R} = 70^\circ$.

Now let's check the options:

(A) PQ = 5 cm, $\angle\text{Q} = 60^\circ$, $\angle\text{R} = 70^\circ$: This matches our findings exactly.

(B) PQ = 5 cm, $\angle\text{Q} = 60^\circ$, $\angle\text{P} = 70^\circ$: $\angle\text{P}$ corresponds to $\angle\text{A}$, not $\angle\text{C}$.

(C) QR = 5 cm, $\angle\text{R} = 60^\circ$, $\angle\text{Q} = 70^\circ$: QR corresponds to BC, not AB. $\angle\text{R}$ corresponds to $\angle\text{C}$, and $\angle\text{Q}$ corresponds to $\angle\text{B}$.

(D) PR = 5 cm, $\angle\text{P} = 60^\circ$, $\angle\text{R} = 70^\circ$: PR corresponds to AC, not AB. $\angle\text{P}$ corresponds to $\angle\text{A}$, and $\angle\text{R}$ corresponds to $\angle\text{C}$.

The only option that correctly lists the corresponding parts based on the congruence statement is (A).

The correct answer is (A) PQ = 5 cm, $\angle\text{Q} = 60^\circ$, $\angle\text{R} = 70^\circ$.

The question asks whether having two sides and one angle (which is not the included angle) of one triangle equal to the corresponding two sides and one angle of another triangle is sufficient to prove congruence. This condition is often referred to as ASS or SSA (Angle-Side-Side or Side-Side-Angle).

Let's consider this condition. Suppose we have two triangles, $\triangle\text{ABC}$ and $\triangle\text{DEF}$, and we know that two sides are equal (e.g., AC = DF and BC = EF) and one non-included angle is equal (e.g., $\angle\text{A} = \angle\text{D}$).

In geometry, it is a known fact that the ASS or SSA condition does not always guarantee that two triangles are congruent. This is because there can be situations where, with the given two side lengths and the non-included angle, two different non-congruent triangles can be formed. This is often called the ambiguous case of the Law of Sines in trigonometry.

For example, consider sides of length 6 and 8, and a non-included angle of $30^\circ$ opposite the side of length 6. It is possible to construct two distinct triangles with these measurements.

However, there is a special case where the ASS/SSA condition *does* imply congruence. This occurs when the non-included angle is a right angle ($90^\circ$). In a right-angled triangle, if we have the hypotenuse and one leg equal to the corresponding hypotenuse and leg of another right-angled triangle, the triangles are congruent. This is the RHS (Right-angle-Hypotenuse-Side) congruence criterion. The RHS criterion is essentially a special case of SSA where the angle is $90^\circ$, which removes the ambiguity.

Therefore, while the ASS or SSA condition is not a valid congruence criterion for all triangles, it can prove congruence in the specific case of right-angled triangles (RHS).

Looking at the options:

(A) Yes, always: This is incorrect, as ASS/SSA does not always prove congruence.

(B) No, never: This is incorrect, as it works in the case of right triangles.

(C) Sometimes, but not always (e.g., ambiguous case): This accurately reflects that the condition does not always lead to congruence due to the ambiguous case.

(D) Only if it's a right-angled triangle (RHS): This highlights the specific case where it does work and identifies it by the RHS criterion.

Option (C) captures the general principle that ASS/SSA is not a universal congruence criterion because of the possibility of forming two triangles (the ambiguous case). Option (D) states the specific exception where it *does* lead to congruence for right triangles under the RHS rule. The question asks if you *can prove* congruence using this condition. Since it doesn't *always* work, you cannot rely on it as a general proof. The existence of the ambiguous case (as mentioned in C) shows why it is not a reliable criterion in general.

The most appropriate answer reflecting the limitation of the ASS/SSA condition for proving congruence generally is that it works only in specific cases, not always, and can lead to ambiguity. Option (D) specifies the primary case where it works, linking it to RHS.

Let's refine the interpretation of the question and options. The question asks "Can you prove that two triangles are congruent if... (i.e., ASS or SSA)". This means, is ASS/SSA itself a *sufficient* criterion for congruence in general? The answer is no, because of the ambiguous case. The only time the specific sides/angle configurations given in ASS/SSA *do* guarantee congruence is under specific constraints (like the angle being 90 degrees, leading to RHS, or the side opposite the angle being greater than or equal to the other given side).

Option (C) explains *why* it doesn't always work ("ambiguous case"). Option (D) states *when* it does work universally for a class of triangles. The question is whether the condition *itself* is a valid general criterion. Since it fails in the ambiguous case, it is not a valid general criterion.

Therefore, the statement "Can you prove that two triangles are congruent if two sides and one angle (not included between the sides) of one triangle are equal to the corresponding two sides and one angle of the other triangle?" is answered by understanding that this condition is not always sufficient.

The ambiguity means you cannot *always* prove congruence using just ASS/SSA. Thus, the answer isn't "Yes, always". It also isn't "No, never", because it *does* work sometimes (like in the RHS case). So it must be "Sometimes, but not always" or related to the right triangle case.

Option (C) directly addresses the general failure and the reason (ambiguous case). Option (D) gives the successful special case. Given the phrasing, "Can you prove that two triangles are congruent if...", the answer that highlights the *limitation* and the reason for it seems more fitting than just stating the special case where it works.

The core issue with ASS/SSA as a general congruence criterion is the existence of the ambiguous case, meaning it is only reliable sometimes, not always.

The correct answer is (C) Sometimes, but not always (e.g., ambiguous case).

The question states that $\triangle\text{PQR}$ and $\triangle\text{LMN}$ are congruent by the SAS (Side-Angle-Side) criterion.

The SAS congruence criterion requires that two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.

The included angle is the angle formed by the two sides mentioned in the criterion.

We are given the following information about the corresponding sides:

1. PQ corresponds to LM.

2. PR corresponds to LN.

In $\triangle\text{PQR}$, the two sides mentioned are PQ and PR. The angle that is included between sides PQ and PR is the angle at their common vertex, which is vertex P. So, the included angle in $\triangle\text{PQR}$ is $\angle\text{P}$.

In $\triangle\text{LMN}$, the sides corresponding to PQ and PR are LM and LN, respectively. The angle that is included between sides LM and LN is the angle at their common vertex, which is vertex L. So, the included angle in $\triangle\text{LMN}$ is $\angle\text{L}$.

For the congruence to be established by the SAS criterion using sides PQ and PR (and their corresponding sides LM and LN), the angles included between these pairs of sides must be equal.

Therefore, the included angles that must be equal are $\angle\text{P}$ and $\angle\text{L}$.

Alternatively, from the congruence statement $\triangle\text{PQR} \cong \triangle\text{LMN}$, the corresponding vertices are P$\leftrightarrow$L, Q$\leftrightarrow$M, and R$\leftrightarrow$N. By CPCTC, the corresponding angles are equal:

$\angle\text{P} = \angle\text{L}$

$\angle\text{Q} = \angle\text{M}$

$\angle\text{R} = \angle\text{N}$

The given corresponding sides are PQ and PR in $\triangle\text{PQR}$, and LM and LN in $\triangle\text{LMN}$. The included angle for sides PQ and PR is $\angle\text{P}$. Its corresponding angle in $\triangle\text{LMN}$ is $\angle\text{L}$. The included angle for sides LM and LN is $\angle\text{L}$. Its corresponding angle in $\triangle\text{PQR}$ is $\angle\text{P}$. For SAS congruence using these sides, $\angle\text{P}$ must equal $\angle\text{L}$.

Let's check the options:

(A) $\angle\text{P}$ and $\angle\text{L}$: This matches our finding that the included angles between the given pairs of sides must be equal.

(B) $\angle\text{Q}$ and $\angle\text{M}$: These are corresponding angles, and they are equal by CPCTC, but they are not the angles included between the given sides PQ, PR and LM, LN.

(C) $\angle\text{R}$ and $\angle\text{N}$: These are corresponding angles, and they are equal by CPCTC, but they are not the angles included between the given sides.

(D) $\angle\text{P}$ and $\angle\text{M}$: These are not corresponding angles based on the congruence statement.

Therefore, if the congruence $\triangle\text{PQR} \cong \triangle\text{LMN}$ is established using the SAS criterion with the given sides PQ, PR and LM, LN, then the included angles $\angle\text{P}$ and $\angle\text{L}$ must be equal.

The correct answer is (A) $\angle\text{P}$ and $\angle\text{L}$.

The question asks which of the given options demonstrates the use of the CPCTC principle.

CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent" (or Equal).

CPCTC is used *after* two triangles have been proven to be congruent using one of the congruence criteria (like SSS, SAS, ASA, AAS, RHS). It allows us to conclude that any pair of corresponding sides or corresponding angles are equal.

Let's analyze each option:

(A) Proving triangles congruent using SSS: The SSS criterion is used to *establish* that triangles are congruent. CPCTC is used *after* congruence is established to deduce properties of the triangles. So, this is not a use of CPCTC; it's a use of a congruence criterion.

(B) Stating that corresponding sides of congruent triangles are equal: If we have already proven that two triangles are congruent, then we can use CPCTC to say that their corresponding sides are equal in length and their corresponding angles are equal in measure. This statement is a direct application of the CPCTC principle.

(C) Identifying the included angle in the SAS criterion: Identifying the included angle is part of the process of *using* the SAS criterion to prove congruence. It's a requirement of the criterion itself, not a consequence of congruence proven by CPCTC.

(D) Checking if the hypotenuse is equal in RHS criterion: Checking for equal hypotenuses is a requirement when *using* the RHS criterion to prove congruence in right triangles. It is part of the input conditions for applying the criterion, not a deduction made after congruence is proven by CPCTC.

Therefore, the statement that demonstrates the use of CPCTC is the conclusion that corresponding sides (or angles) of congruent triangles are equal.

The correct answer is (B) Stating that corresponding sides of congruent triangles are equal.

We are given an isosceles triangle $\triangle\text{ABC}$ where AB = AC. AD is the median to BC, which means that D is the midpoint of BC, and therefore, BD = CD.

We need to prove that $\triangle\text{ABD} \cong \triangle\text{ACD}$ and identify the congruence criterion used.

Let's consider the two triangles, $\triangle\text{ABD}$ and $\triangle\text{ACD}$, and list the equal corresponding parts:

1. AB = AC

This is a side in $\triangle\text{ABD}$ and a corresponding side in $\triangle\text{ACD}$.

2. BD = CD

This is another side in $\triangle\text{ABD}$ and a corresponding side in $\triangle\text{ACD}$.

3. AD is common to both triangles.

This is the third side in both $\triangle\text{ABD}$ and $\triangle\text{ACD}$.

We have found that all three sides of $\triangle\text{ABD}$ are equal to the three corresponding sides of $\triangle\text{ACD}$ (AB=AC, BD=CD, and AD=AD).

Recall the triangle congruence criteria:

SAS: Two sides and the included angle.

ASA: Two angles and the included side.

SSS: Three sides.

RHS: Right angle, hypotenuse, and one side (for right triangles).

The condition where three sides of one triangle are equal to the three corresponding sides of another triangle is the definition of the SSS congruence criterion.

Therefore, $\triangle\text{ABD} \cong \triangle\text{ACD}$ by the SSS criterion.

Note about option (D): While AD in an isosceles triangle to the base is also the altitude (making $\angle\text{ADB} = \angle\text{ADC} = 90^\circ$), and we have AB=AC (hypotenuses) and BD=CD (sides), we could use RHS. However, the most direct proof using only the given information (AB=AC, AD is median implies BD=CD, and AD is common) utilizes the SSS criterion. Without explicitly stating that it's a right triangle or that the median is also the altitude, SSS is the immediate conclusion from the side equalities.

The correct answer is (C) SSS.

We are given two triangles, $\triangle\text{ABC}$ and $\triangle\text{DEF}$, with the following equal corresponding parts:

1. $\angle\text{A} = \angle\text{D}$

2. $\angle\text{B} = \angle\text{E}$

3. BC = EF

Let's analyze the relationship between these parts within each triangle and relate them to the congruence criteria.

We have two angles and one side given as equal to their corresponding parts in the other triangle. The side BC in $\triangle\text{ABC}$ is opposite to angle $\angle\text{A}$ and adjacent to angle $\angle\text{B}$ and $\angle\text{C}$. The side EF in $\triangle\text{DEF}$ is opposite to angle $\angle\text{D}$ and adjacent to angle $\angle\text{E}$ and $\angle\text{F}$.

The side BC is not included between angles $\angle\text{A}$ and $\angle\text{B}$ in $\triangle\text{ABC}$. The side included between $\angle\text{A}$ and $\angle\text{B}$ is AB. The side BC is included between $\angle\text{B}$ and $\angle\text{C}$.

The given information consists of two angles and a non-included side. This matches the description of the AAS (Angle-Angle-Side) congruence criterion.

The AAS congruence criterion states that if two angles and a non-included side of one triangle are equal to the corresponding two angles and the non-included side of another triangle, then the triangles are congruent.

Let's see if the AAS criterion is always valid.

If we have $\angle\text{A} = \angle\text{D}$ and $\angle\text{B} = \angle\text{E}$, then by the angle sum property of a triangle ($\angle\text{A} + \angle\text{B} + \angle\text{C} = 180^\circ$ and $\angle\text{D} + \angle\text{E} + \angle\text{F} = 180^\circ$), the third angles must also be equal:

$\angle\text{C} = 180^\circ - (\angle\text{A} + \angle\text{B})$

$\angle\text{F} = 180^\circ - (\angle\text{D} + \angle\text{E})$

Since $\angle\text{A} = \angle\text{D}$ and $\angle\text{B} = \angle\text{E}$, we have $\angle\text{A} + \angle\text{B} = \angle\text{D} + \angle\text{E}$, which implies $\angle\text{C} = \angle\text{F}$.

Now we have $\angle\text{B} = \angle\text{E}$, the included side BC = EF (which is the given side), and $\angle\text{C} = \angle\text{F}$ (which we just deduced). This is the condition for ASA congruence (Angle-Included Side-Angle).

Thus, if we have two angles and a non-included side equal, we can deduce that the third angles are equal, which then gives us two angles and the included side being equal, allowing us to use the ASA criterion.

So, the AAS criterion is a valid congruence criterion, derivable from ASA and the angle sum property.

The given information ($\angle\text{A} = \angle\text{D}$, $\angle\text{B} = \angle\text{E}$, BC = EF) fits the AAS criterion, where the side BC is opposite to $\angle\text{A}$ and EF is opposite to $\angle\text{D}$, and $\angle\text{B}$ and $\angle\text{E}$ are another pair of corresponding angles.

Let's examine the options:

(A) Yes, by ASA: While ASA is related (as shown by deriving the third angle), the given parts are not directly in the ASA configuration (Angle-Included Side-Angle), as BC is not included between $\angle\text{A}$ and $\angle\text{B}$.

(B) Yes, by AAS: The given parts directly fit the AAS criterion (Angle $\angle\text{A}$, Angle $\angle\text{B}$, Side BC not included between them).

(C) Yes, by SAS: SAS requires two sides and the included angle. We have two angles and one side.

(D) No, not necessarily congruent: Since AAS is a valid criterion, the triangles are necessarily congruent.

Therefore, the triangles are congruent by the AAS criterion.

The correct answer is (B) Yes, by AAS.

The question asks to identify the triangle congruence criterion that requires a right angle, the hypotenuse, and one other side to be equal in two triangles.

Let's look at the common congruence criteria:

(A) SSS (Side-Side-Side): Requires equality of three sides.

(B) ASA (Angle-Side-Angle): Requires equality of two angles and the included side.

(C) SAS (Side-Angle-Side): Requires equality of two sides and the included angle.

(D) RHS (Right-angle-Hypotenuse-Side): This criterion is specifically for right-angled triangles. It requires that the right angle is present in both triangles, the hypotenuses are equal, and one pair of corresponding sides (legs) are equal.

The description "requires a right angle, hypotenuse, and one side" precisely matches the definition of the RHS congruence criterion.

The letters R, H, and S in RHS stand for Right angle, Hypotenuse, and Side (specifically, a leg of the right triangle).

Therefore, the criterion that requires a right angle, hypotenuse, and one side is RHS.

The correct answer is (D) RHS.

Two figures are defined as congruent if they have the exact same shape and the exact same size. This means that one figure can be moved (translated, rotated, or reflected) and placed exactly on top of the other figure so that they coincide perfectly.

Let's consider the given options:

(A) They have the same area: Since congruent figures occupy the exact same amount of space (because they are identical in shape and size), their areas must be equal. This is a consequence of congruence.

(B) They have the same perimeter: The perimeter is the total length of the boundary of a figure. If two figures are congruent, their boundaries have the same shape and total length. Therefore, their perimeters must be equal. This is also a consequence of congruence.

(C) They can be made to coincide by superposition: This is essentially the definition of congruence. If two figures are congruent, there exists a rigid motion (translation, rotation, reflection) that maps one figure onto the other, making them coincide perfectly when superimposed.

(D) All of the above: Since options (A), (B), and (C) are all true statements about congruent figures, this is the comprehensive correct answer.

Therefore, if two figures are congruent, they must have the same area, the same perimeter, and they can be made to coincide by superposition.

The correct answer is (D) All of the above.

We are given that $\triangle\text{PQR} \cong \triangle\text{STU}$. This congruence statement implies a specific correspondence between the vertices of the two triangles:

P $\leftrightarrow$ S

Q $\leftrightarrow$ T

R $\leftrightarrow$ U

According to the CPCTC principle (Corresponding Parts of Congruent Triangles are Equal), the corresponding angles are equal.

$\angle\text{P} = \angle\text{S}$

$\angle\text{Q} = \angle\text{T}$

$\angle\text{R} = \angle\text{U}$

We are given the measures of two angles in $\triangle\text{PQR}$:

$\angle\text{P} = 50^\circ$

$\angle\text{Q} = 60^\circ$

We need to find the measure of $\angle\text{U}$ in $\triangle\text{STU}$. From the correspondence, $\angle\text{U}$ corresponds to $\angle\text{R}$. So, if we find the measure of $\angle\text{R}$, we will know the measure of $\angle\text{U}$.

The sum of angles in any triangle is $180^\circ$. In $\triangle\text{PQR}$:

Substitute the given values:

Since $\triangle\text{PQR} \cong \triangle\text{STU}$, the corresponding angles are equal. We have $\angle\text{R}$ corresponding to $\angle\text{U}$.

Therefore, the measure of $\angle\text{U}$ is $70^\circ$.

The correct answer is (C) $70^\circ$.

We are given two triangles, $\triangle\text{ABC}$ and $\triangle\text{PQR}$, with the following information:

AB = PQ

BC = QR

$\angle\text{B} \neq \angle\text{Q}$

We are comparing two sides (AB and BC) and the angle included between them ($\angle\text{B}$) in $\triangle\text{ABC}$ with two sides (PQ and QR) and the angle included between them ($\angle\text{Q}$) in $\triangle\text{PQR}$.

The SAS (Side-Angle-Side) congruence criterion states that if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

In this case, we have AB = PQ and BC = QR, but the included angles are not equal ($\angle\text{B} \neq \angle\text{Q}$). This violates the condition of the SAS criterion.

If two triangles have two pairs of equal sides, but the included angles are not equal, the triangles cannot be congruent. This is because the third side (AC in $\triangle\text{ABC}$ and PR in $\triangle\text{PQR}$) would have different lengths if the included angles are different, given that the two adjacent sides are the same length.

Consider the Law of Cosines: For $\triangle\text{ABC}$, $AC^2 = AB^2 + BC^2 - 2(AB)(BC)\cos(\angle\text{B})$. For $\triangle\text{PQR}$, $PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)\cos(\angle\text{Q})$.

Given AB=PQ and BC=QR, we have $AC^2 = AB^2 + BC^2 - 2(AB)(BC)\cos(\angle\text{B})$ and $PR^2 = AB^2 + BC^2 - 2(AB)(BC)\cos(\angle\text{Q})$.

If $\angle\text{B} \neq \angle\text{Q}$, then $\cos(\angle\text{B}) \neq \cos(\angle\text{Q})$ (for angles in the range $0^\circ$ to $180^\circ$). Therefore, $AC^2 \neq PR^2$, which means $AC \neq PR$.

If the third pair of sides are not equal (AC $\neq$ PR), the triangles cannot be congruent by the SSS criterion. Also, if the triangles are not congruent, their corresponding angles are not all equal, so ASA and AAS cannot apply. RHS applies only to right triangles, and even then, if two sides are equal but the included right angle is not equal (which is impossible for right angles, but conceptually), they wouldn't be congruent.

Therefore, if two triangles have two corresponding sides equal, but the included angles are not equal, the triangles cannot be congruent.

Let's look at the options:

(A) Yes, by SSS if AC=PR: If AC=PR, and we are given AB=PQ and BC=QR, then the triangles are congruent by SSS. However, the question states $\angle\text{B} \neq \angle\text{Q}$. As shown above, if $\angle\text{B} \neq \angle\text{Q}$ (with AB=PQ and BC=QR), then AC $\neq$ PR. So, the condition "AC=PR" contradicts the given information in the context of the question.

(B) Yes, by ASA if two angles are equal: The ASA criterion involves two angles and the included side. While congruent triangles have equal angles, the given condition $\angle\text{B} \neq \angle\text{Q}$ implies they cannot be congruent by ASA (as corresponding angles would be unequal).

(C) No, they cannot be congruent: Based on the analysis using the SAS criterion and the Law of Cosines, if two sides and the included angle of two triangles are not all correspondingly equal, the triangles cannot be congruent.

(D) Only if they are right-angled triangles: If they were right-angled at B and Q, then $\angle\text{B} = \angle\text{Q} = 90^\circ$. But the question states $\angle\text{B} \neq \angle\text{Q}$, so they cannot both be right-angled at those vertices.

The fact that AB = PQ, BC = QR, and $\angle\text{B} \neq \angle\text{Q}$ is a sufficient condition to conclude that the triangles are not congruent.

The correct answer is (C) No, they cannot be congruent.

Consider the rectangle ABCD with diagonal AC, which divides it into $\triangle\text{ABC}$ and $\triangle\text{ADC}$.

In $\triangle\text{ABC}$ and $\triangle\text{ADC}$:

AB = DC

BC = AD

AC = AC

Since all three sides of $\triangle\text{ABC}$ are equal to the corresponding three sides of $\triangle\text{ADC}$, the triangles are congruent by the SSS congruence criterion.

The criterion used is SSS.

The correct answer is (B) SSS.

Let's analyze each statement regarding congruent figures and similar figures.

Congruent figures have the same shape and the same size. One figure can be transformed into the other by a rigid motion (translation, rotation, reflection).

Similar figures have the same shape but not necessarily the same size. Corresponding angles are equal, and corresponding sides are proportional.

(A) Congruent figures are similar: If two figures are congruent, they have the same shape and the same size. Figures with the same shape are similar. Since they also have the same size, the ratio of corresponding sides is 1:1, which is a case of proportionality. Therefore, congruent figures are a special case of similar figures where the scale factor is 1. This statement is True.

(B) Similar figures are always congruent: Similar figures only require the same shape, not the same size. For example, a small square and a large square are similar (same shape), but they are not congruent (different sizes). Thus, this statement is False.

(C) Congruent figures have equal area: If two figures are congruent, they are identical and occupy the exact same amount of space. Therefore, their areas are equal. This statement is True.

(D) Congruent figures have equal perimeter: If two figures are congruent, their boundaries are identical in shape and length. Therefore, their perimeters are equal. This statement is True.

The statement that is FALSE is that similar figures are always congruent.

The correct answer is (B) Similar figures are always congruent.

Two geometric figures are said to be congruent if they have exactly the same shape and the exact same size. This means that one figure can be perfectly superimposed on the other figure by a rigid motion, which includes translation (sliding), rotation (turning), or reflection (flipping).

Essentially, congruent figures are identical copies of each other, just possibly located in different positions or orientations in space.

When two figures are congruent, all their corresponding parts (such as sides, angles, vertices) are equal in measure or length.

Real-life example of two congruent objects:

Consider two identical standard chairs produced from the same manufacturing line. If they are truly identical models and haven't been modified or damaged, they are congruent. You could pick up one chair and place it exactly on top of the other, and all their parts would line up perfectly.

Other examples include:

- Two sheets of standard A4 paper from the same pack.

- Two identical wheels of a bicycle.

- Two identical coins of the same denomination and year.

- The left and right shoes of a pair of shoes (although they are reflections, not direct superpositions without flipping).

Two line segments are said to be congruent if they have the same size. For line segments, the "size" is determined by their length. The "shape" of a line segment is always a straight line, so the shape aspect is constant.

Therefore, two line segments are congruent if and only if they have the same length.

If line segment AB is congruent to line segment CD, written as $\text{AB} \cong \text{CD}$, it means that they are identical in size. The size of a line segment is its length.

Thus, if $\text{AB} \cong \text{CD}$, we can say that their lengths are equal.

This is often simply written as AB = CD, where AB and CD denote the lengths of the segments.

Two angles are said to be congruent if they have the same size. For angles, the "size" is determined by their measure. The "shape" of an angle is the opening between its two rays from a common vertex, and this shape is consistent for a given measure.

Therefore, two angles are congruent if and only if they have the same measure.

If angle PQR is congruent to angle XYZ, written as $\angle\text{PQR} \cong \angle\text{XYZ}$, it means that they are identical in size. The size of an angle is its measure, typically expressed in degrees or radians.

Thus, if $\angle\text{PQR} \cong \angle\text{XYZ}$, we can say that their measures are equal.

This is often written using the equality sign for the angle notation itself, like $\angle\text{PQR} = \angle\text{XYZ}$, when equality of measures is implied.

Two geometric figures are congruent if they have the same shape and size. For circles, the shape is always the same (a set of points equidistant from a central point). Therefore, the congruence of circles depends solely on their size.

The size of a circle is determined by its radius or its diameter. If two circles have the same radius, they have the same size, and thus they are congruent. Similarly, if they have the same diameter (which is twice the radius), they are also congruent.

Two circles are said to be congruent if they have the same radius (or the same diameter).

What determines the congruence of circles is a single linear dimension: the radius (or the diameter). If the radii of two circles are equal, they are congruent. If the radii are different, they are not congruent (they are similar, but not congruent).

Two geometric figures are congruent if they have the same shape and size. For circles, the shape is always the same (a set of points equidistant from a central point). Therefore, the congruence of circles depends solely on their size.

The size of a circle is determined by its radius or its diameter. If two circles have the same radius, they have the same size, and thus they are congruent. Similarly, if they have the same diameter (which is twice the radius), they are also congruent.

Two circles are said to be congruent if they have the same radius (or the same diameter).

What determines the congruence of circles is a single linear dimension: the radius (or the diameter). If the radii of two circles are equal, they are congruent. If the radii are different, they are not congruent (they are similar, but not congruent).

The symbol used to denote congruence between two geometric figures is the equality sign with a tilde above it.

The symbol is: $\cong$

For example:

- To state that line segment AB is congruent to line segment CD, we write $\text{AB} \cong \text{CD}$.

- To state that angle PQR is congruent to angle XYZ, we write $\angle\text{PQR} \cong \angle\text{XYZ}$.

- To state that triangle ABC is congruent to triangle PQR, we write $\triangle\text{ABC} \cong \triangle\text{PQR}$.

The symbol $\cong$ signifies that the figures have the same shape and the same size.

Definition of Congruent Triangles:

Two triangles are said to be congruent if they have exactly the same shape and the exact same size. This means that one triangle can be perfectly superimposed on the other triangle by a sequence of rigid transformations (translation, rotation, or reflection). If two triangles are congruent, all their corresponding sides are equal in length, and all their corresponding angles are equal in measure.

Condition for two triangles to be congruent:

For two triangles to be congruent, there must be a correspondence between their vertices such that all three pairs of corresponding sides are equal in length and all three pairs of corresponding angles are equal in measure.

If $\triangle\text{ABC}$ is congruent to $\triangle\text{PQR}$ (written as $\triangle\text{ABC} \cong \triangle\text{PQR}$), it means that with the correspondence A $\leftrightarrow$ P, B $\leftrightarrow$ Q, and C $\leftrightarrow$ R, the following conditions hold true:

Corresponding Sides are equal:

AB = PQ

BC = QR

AC = PR

Corresponding Angles are equal:

$\angle\text{A} = \angle\text{P}$

$\angle\text{B} = \angle\text{Q}$

$\angle\text{C} = \angle\text{R}$

In practice, we use congruence criteria (like SSS, SAS, ASA, AAS, RHS) which require fewer conditions to be checked, as these sets of conditions are sufficient to guarantee that all corresponding parts are equal and thus the triangles are congruent.

The SSS congruence criterion is one of the fundamental rules used to determine if two triangles are congruent.

SSS Congruence Criterion:

If the three sides of one triangle are equal in length to the three corresponding sides of another triangle, then the two triangles are congruent.

In other words, if in $\triangle\text{ABC}$ and $\triangle\text{PQR}$, we have AB = PQ, BC = QR, and AC = PR, then $\triangle\text{ABC} \cong \triangle\text{PQR}$. The congruence statement must reflect the correct correspondence of vertices based on the equal sides (e.g., since AB corresponds to PQ, A corresponds to P and B corresponds to Q, etc.).

The SAS (Side-Angle-Side) Congruence Criterion for triangles states the following:

If two triangles have two sides and the included angle of one triangle equal to the corresponding two sides and the included angle of the other triangle, then the two triangles are congruent.

More formally, consider two triangles, say $\triangle ABC$ and $\triangle PQR$.

If:

Then, by the SAS congruence criterion,

The term "included angle" is crucial; it refers to the angle formed by the two sides that are given to be equal.

The ASA (Angle-Side-Angle) Congruence Criterion for triangles states the following:

If two triangles have two angles and the included side of one triangle equal to the corresponding two angles and the included side of the other triangle, then the two triangles are congruent.

More formally, consider two triangles, say $\triangle ABC$ and $\triangle PQR$.

If:

Then, by the ASA congruence criterion,

The term "included side" is crucial; it refers to the side that lies between the two angles that are given to be equal.

The RHS (Right Angle-Hypotenuse-Side) Congruence Criterion for right-angled triangles states the following:

If in two right-angled triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle, then the two triangles are congruent.

More formally, consider two right-angled triangles, say $\triangle ABC$ and $\triangle PQR$, where $\angle B = 90^\circ$ and $\angle Q = 90^\circ$.

If:

Then, by the RHS congruence criterion,

This criterion is applicable only to right-angled triangles.

Given the sides of $\triangle ABC$ and $\triangle PQR$ are related as follows:

Since all three sides of $\triangle ABC$ are equal to the corresponding three sides of $\triangle PQR$, the triangles are congruent by the SSS (Side-Side-Side) Congruence Criterion.

To write the congruence statement, we need to match the corresponding vertices.

AB corresponds to PQ.

BC corresponds to QR.

CA corresponds to RP.

Matching the vertices based on the equal sides: A corresponds to P, B corresponds to Q, and C corresponds to R.

Therefore, the congruence statement is:

Given:

In $\triangle XYZ$ and $\triangle LMN$:

We are given that two sides (XY and YZ) and the included angle ($\angle Y$) of $\triangle XYZ$ are equal to the corresponding two sides (LM and MN) and the included angle ($\angle M$) of $\triangle LMN$.

This set of conditions matches the requirements for the SAS (Side-Angle-Side) Congruence Criterion.

Therefore, by the SAS congruence criterion, $\triangle XYZ$ is congruent to $\triangle LMN$.

The congruence statement, matching the corresponding vertices based on the equal sides and included angle, is:

Given:

In $\triangle DEF$ and $\triangle STU$:

We are given that two angles ($\angle D$ and $\angle E$) and the included side (DE) of $\triangle DEF$ are equal to the corresponding two angles ($\angle S$ and $\angle T$) and the included side (ST) of $\triangle STU$.

This set of conditions matches the requirements for the ASA (Angle-Side-Angle) Congruence Criterion.

Therefore, by the ASA congruence criterion, $\triangle DEF$ is congruent to $\triangle STU$.

To write the congruence statement, we match the corresponding vertices based on the equal parts:

$\angle D = \angle S$, so D corresponds to S.

$\angle E = \angle T$, so E corresponds to T.

The side DE corresponds to ST.

The remaining vertex F must correspond to U.

The congruence statement is:

Given:

In right-angled $\triangle ABC$ (with $\angle B = 90^\circ$) and right-angled $\triangle PQR$ (with $\angle Q = 90^\circ$):

We are given that in two right-angled triangles, the hypotenuse and one side of $\triangle ABC$ are equal to the hypotenuse and the corresponding side of $\triangle PQR$.

This set of conditions matches the requirements for the RHS (Right Angle-Hypotenuse-Side) Congruence Criterion.

Therefore, by the RHS congruence criterion, $\triangle ABC$ is congruent to $\triangle PQR$.

To write the congruence statement, we match the corresponding vertices based on the equal parts:

The right angles are at B and Q, so B corresponds to Q.

The hypotenuse AC corresponds to PR, so the vertices opposite the right angles (A and P, C and R) must correspond such that the hypotenuses match. Since AB = PQ, vertex A is connected to B by side AB, and P is connected to Q by side PQ. Given AB = PQ, A must correspond to P.

The remaining vertex C must correspond to R.

The congruence statement is:

Given:

When two triangles are congruent, their corresponding parts (sides and angles) are equal. The congruence statement $\triangle ABC \cong \triangle XYZ$ indicates the correspondence of the vertices.

Vertex A corresponds to Vertex X.

Vertex B corresponds to Vertex Y.

Vertex C corresponds to Vertex Z.

Pairs of Corresponding Sides:

Pairs of Corresponding Angles:

Yes, two squares with the same perimeter are congruent.

Justification:

Let the two squares be $S_1$ and $S_2$. Let the side length of square $S_1$ be $s_1$ and the side length of square $S_2$ be $s_2$.

The perimeter of a square with side length $s$ is given by the formula $P = 4s$.

The perimeter of square $S_1$ is $P_1 = 4s_1$.

The perimeter of square $S_2$ is $P_2 = 4s_2$.

We are given that the two squares have the same perimeter.

Substituting the formulas for the perimeters:

Dividing both sides of the equation by 4:

This shows that the side lengths of the two squares are equal.

Since all sides of a square are equal and all angles are $90^\circ$, having equal side lengths means that all corresponding sides and all corresponding angles of the two squares are equal.

Therefore, by the properties of congruence, the two squares are congruent as they are identical in both shape and size.

No, two rectangles with the same area are not always congruent.

Justification:

For two figures to be congruent, they must have the same shape and the same size. Rectangles with the same area have the same size (in terms of the space they cover), but they can have different dimensions (length and width), which means they can have different shapes.

Example:

Consider two rectangles, Rectangle A and Rectangle B.

Let Rectangle A have length $l_A = 4$ units and width $w_A = 3$ units.

The area of Rectangle A is $A_A = l_A \times w_A = 4 \times 3 = 12$ square units.

Let Rectangle B have length $l_B = 6$ units and width $w_B = 2$ units.

The area of Rectangle B is $A_B = l_B \times w_B = 6 \times 2 = 12$ square units.

In this example, Rectangle A and Rectangle B have the same area ($12$ square units).

However, their dimensions are different ($4 \times 3$ compared to $6 \times 2$).

A $4 \times 3$ rectangle has a different shape than a $6 \times 2$ rectangle.

Therefore, even though they have the same area, Rectangle A and Rectangle B are not congruent.

No, two triangles are not necessarily congruent if only two pairs of corresponding angles are equal.

Justification:

If two pairs of corresponding angles of two triangles are equal, say $\angle A = \angle X$ and $\angle B = \angle Y$ in $\triangle ABC$ and $\triangle XYZ$, then by the Angle Sum Property of a triangle ($180^\circ$), the third pair of angles must also be equal:

Since $\angle A = \angle X$ and $\angle B = \angle Y$, it follows that $\angle C = \angle Z$.

This means that if two pairs of corresponding angles are equal, then all three pairs of corresponding angles are equal. Triangles with all three pairs of corresponding angles equal are called similar triangles. Similar triangles have the same shape but not necessarily the same size.

For two triangles to be congruent, they must have the same shape and the same size. Having equal corresponding angles only guarantees the same shape (similarity), not necessarily the same size (congruence).

Example:

Consider an equilateral triangle with side length $2$ units and another equilateral triangle with side length $4$ units.

In both triangles, all angles are $60^\circ$. So, all pairs of corresponding angles are equal.

However, the side lengths are different ($2 \ne 4$). Therefore, the triangles are similar but not congruent.

To prove congruence using angles, at least one pair of corresponding sides must also be equal (as in the ASA or AAS congruence criteria).

No, two triangles are not necessarily congruent if only two pairs of corresponding sides are equal.

Justification:

While having two pairs of corresponding sides equal is a necessary condition for congruence, it is not sufficient on its own. For two triangles to be congruent based on side lengths, all three pairs of corresponding sides must be equal (SSS criterion), or the equal sides must form an equal included angle (SAS criterion).

If only two pairs of corresponding sides are equal, the angle between these sides can vary, which will result in a different third side and a different shape for the triangle. Different shapes and/or sizes mean the triangles are not congruent.

Example:

Consider two triangles, $\triangle ABC$ and $\triangle PQR$.

Let AB = 5 cm and BC = 7 cm.

Let PQ = 5 cm and QR = 7 cm.

So, we have two pairs of corresponding sides equal: AB = PQ and BC = QR.

However, the angle between these sides can be different.

Case 1: Let $\angle B = 30^\circ$. This defines a unique triangle $\triangle ABC$.

Case 2: Let $\angle Q = 90^\circ$. This defines a unique triangle $\triangle PQR$.

In this example, AB = PQ and BC = QR, but $\angle B \ne \angle Q$. The third side AC and PR will be different lengths (by the Law of Cosines, or simply by observation). Thus, $\triangle ABC$ and $\triangle PQR$ are not congruent.

Therefore, having only two pairs of corresponding sides equal is not enough to guarantee congruence.

Yes, the triangles $\triangle ABC$ and $\triangle PQR$ are congruent.

Given:

In $\triangle ABC$:

In $\triangle PQR$:

Comparing the corresponding parts:

We observe that two sides (AB and BC) and the included angle ($\angle B$) of $\triangle ABC$ are equal to the corresponding two sides (PQ and QR) and the included angle ($\angle Q$) of $\triangle PQR$.

This matches the conditions of the SAS (Side-Angle-Side) Congruence Criterion.

Therefore, by the SAS congruence criterion, $\triangle ABC$ is congruent to $\triangle PQR$.

The congruence statement is:

Yes, the triangles $\triangle LMN$ and $\triangle XYZ$ are congruent.

Given:

In $\triangle LMN$:

In $\triangle XYZ$:

Comparing the side lengths of the two triangles, we have:

Since all three sides of $\triangle LMN$ are equal to the corresponding three sides of $\triangle XYZ$, the triangles are congruent by the SSS (Side-Side-Side) Congruence Criterion.

To write the correct congruence statement, we need to match the vertices based on the equal sides:

LM (7 cm) corresponds to XY (7 cm).

MN (8 cm) corresponds to ZX (8 cm).

NL (9 cm) corresponds to YZ (9 cm).

Vertex L is formed by sides LM (7) and NL (9). The corresponding vertex in $\triangle XYZ$ is formed by sides XY (7) and YZ (9), which is Y. So, L corresponds to Y.

Vertex M is formed by sides LM (7) and MN (8). The corresponding vertex in $\triangle XYZ$ is formed by sides XY (7) and ZX (8), which is X. So, M corresponds to X.

Vertex N is formed by sides MN (8) and NL (9). The corresponding vertex in $\triangle XYZ$ is formed by sides ZX (8) and YZ (9), which is Z. So, N corresponds to Z.

Therefore, the congruence statement is:

The ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and AAA (Angle-Angle-Angle) describe conditions involving angles and sides of triangles. The key difference lies in which of these conditions guarantee congruence (identical triangles in shape and size).

1. ASA (Angle-Side-Angle) Criterion:

This criterion states that if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of another triangle, then the triangles are congruent.

2. AAS (Angle-Angle-Side) Criterion:

This criterion states that if two angles and a non-included side of one triangle are equal to the corresponding two angles and the corresponding non-included side of another triangle, then the triangles are congruent. The AAS criterion is a valid congruence criterion.

Difference between ASA and AAS:

The difference between ASA and AAS is the position of the side relative to the two equal angles. In ASA, the equal side is located between the two equal angles (the included side). In AAS, the equal side is located next to one of the equal angles and opposite the other equal angle (a non-included side).

However, if two angles of a triangle are equal to two angles of another triangle, the third angles must also be equal (due to the Angle Sum Property of triangles, where the sum of angles is $180^\circ$). Therefore, if AAS holds, we actually have all three corresponding angles equal and one side equal, which effectively implies congruence.

3. AAA (Angle-Angle-Angle) Condition:

This condition means that all three corresponding angles of two triangles are equal.

Difference from ASA/AAS:

The crucial difference is that the AAA condition does not guarantee congruence. Triangles with all three corresponding angles equal are called similar triangles. Similar triangles have the same shape but can be of different sizes.

For example, an equilateral triangle with side length 1 cm has angles $60^\circ, 60^\circ, 60^\circ$. An equilateral triangle with side length 10 cm also has angles $60^\circ, 60^\circ, 60^\circ$. They satisfy the AAA condition, but they are clearly not congruent because their side lengths are different.

In summary, ASA and AAS are specific criteria that, along with angle equalities, require a corresponding side equality in a particular position to ensure that the triangles are not only the same shape but also the same size (congruent). AAA only guarantees the same shape (similarity), not necessarily the same size, and thus is not a congruence criterion.

Given:

The congruence is by the SAS (Side-Angle-Side) criterion.

The SAS congruence criterion states that two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.

In this case, the given equal angles are $\angle Q$ in $\triangle PQR$ and $\angle M$ in $\triangle LMN$. For the SAS criterion to apply with these angles being the "A" (Angle) part, they must be the included angles between the two pairs of equal sides.

In $\triangle PQR$, the angle $\angle Q$ is included between sides PQ and QR.

In $\triangle LMN$, the angle $\angle M$ is included between sides LM and MN.

For the congruence $\triangle PQR \cong \triangle LMN$ by SAS with $\angle Q = \angle M$, the sides including these angles must be equal to their corresponding sides.

Based on the congruence statement $\triangle PQR \cong \triangle LMN$, the correspondence of vertices is P $\leftrightarrow$ L, Q $\leftrightarrow$ M, R $\leftrightarrow$ N.

Therefore, the side PQ must be equal to the corresponding side LM, and the side QR must be equal to the corresponding side MN.

The sides that must be equal are:

The concept of congruence in geometry refers to the property of two geometric figures being exactly the same in both shape and size. If two figures are congruent, it means that one can be placed directly on top of the other, by using rigid transformations (like translation, rotation, or reflection) such that they perfectly coincide. Essentially, congruent figures are identical copies of each other.

Examples of Congruent Geometric Figures:

1. Congruent Line Segments:

Two line segments are congruent if and only if they have the same length.

Example: If line segment AB has a length of 5 cm and line segment CD has a length of 5 cm, then AB is congruent to CD. We denote this as AB $\cong$ CD.

So, AB $\cong$ CD.

2. Congruent Angles:

Two angles are congruent if and only if they have the same measure.

Example: If $\angle$PQR has a measure of 45$^\circ$ and $\angle$XYZ has a measure of 45$^\circ$, then $\angle$PQR is congruent to $\angle$XYZ. We denote this as $\angle$PQR $\cong$ $\angle$XYZ.

So, $\angle$PQR $\cong$ $\angle$XYZ.

3. Congruent Circles:

Two circles are congruent if and only if they have the same radius (or diameter).

Example: If Circle 1 has a radius of 3 cm and Circle 2 has a radius of 3 cm, then Circle 1 is congruent to Circle 2.

So, Circle 1 is congruent to Circle 2.

Why is Congruence a Key Concept in Geometry?

Congruence is a fundamental concept in geometry for several reasons:

1. Foundation for Proofs: Many geometric theorems and proofs rely on establishing the congruence of triangles or other figures. The congruence criteria (like SSS, SAS, ASA, AAS, RHS) provide specific conditions under which we can rigorously prove that two figures are identical.

2. Solving Geometric Problems: Congruence allows us to deduce unknown lengths, angles, or other properties of a figure by relating it to a congruent figure whose properties are known. If $\triangle ABC \cong \triangle XYZ$, then we know that corresponding sides and angles are equal (e.g., AB = XY, $\angle$A = $\angle$X) without needing to measure them directly if one triangle's measurements are known.

3. Understanding Transformations: Congruence is directly related to rigid transformations (translation, rotation, reflection). Congruent figures are precisely those that can be mapped onto each other using one or more of these transformations. This links the study of shape and size to the study of movement in geometry.

4. Building Block for Similarity: The concept of congruence ($same shape$ and $same size$) provides a basis for understanding similarity ($same shape$ but possibly different size$), where corresponding angles are equal, and corresponding sides are proportional. Congruence is a special case of similarity where the ratio of corresponding sides is 1.

5. Practical Applications: Congruence is essential in fields like architecture, engineering, manufacturing, and design, where identical parts or structures are required.

The SSS (Side-Side-Side) Congruence Criterion for triangles states the following:

If the three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent.

Illustration with Triangles:

Imagine two triangles, $\triangle ABC$ and $\triangle PQR$.

(Since I cannot draw, visualize $\triangle ABC$ with vertices A, B, C and sides AB, BC, CA, and $\triangle PQR$ with vertices P, Q, R and sides PQ, QR, RP).

Given:

In $\triangle ABC$ and $\triangle PQR$, we assume their corresponding sides are equal:

Conclusion:

Based on the SSS congruence criterion, since the three sides of $\triangle ABC$ are equal to the corresponding three sides of $\triangle PQR$, the two triangles are congruent.

Meaning in terms of Constructing a Unique Triangle:

The SSS criterion means that if you are given three specific lengths for the sides of a triangle, there is only one possible triangle that can be constructed with those side lengths. Any other triangle constructed with the exact same three side lengths will be an identical copy (congruent) of the first one. The side lengths uniquely determine the shape and size of the triangle.

For example, if you are asked to construct a triangle with side lengths 3 cm, 4 cm, and 5 cm, you will always get a right-angled triangle, and any two such triangles constructed will be congruent.

The SAS (Side-Angle-Side) Congruence Criterion for triangles states the following:

If two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, then the two triangles are congruent.

Illustration with Triangles:

Imagine two triangles, $\triangle ABC$ and $\triangle PQR$.

(Since I cannot draw, visualize $\triangle ABC$ with vertices A, B, C and sides AB, BC, CA, and $\triangle PQR$ with vertices P, Q, R and sides PQ, QR, RP).

Given:

In $\triangle ABC$ and $\triangle PQR$, let's assume the following conditions for SAS congruence:

Note that the angle $\angle$ABC is included between sides AB and BC. Similarly, the angle $\angle$PQR is included between sides PQ and QR.

Conclusion:

Based on the SAS congruence criterion, since two sides and the included angle of $\triangle ABC$ are equal to the corresponding two sides and the included angle of $\triangle PQR$, the two triangles are congruent.

Importance of the angle being 'included':

It is crucial that the angle is the 'included' angle because if the angle is not between the two sides, the two sides and a non-included angle do not uniquely determine a triangle (unless it is a right-angled triangle, leading to the RHS criterion, which is a special case). Different triangles can be formed with the same two side lengths and the same non-included angle.

For example, consider two triangles with sides of length 5 cm and 7 cm, and an angle of $30^\circ$.

If the $30^\circ$ angle is included between the 5 cm and 7 cm sides, there is only one possible triangle (SAS criterion applies).

However, if the $30^\circ$ angle is not included (e.g., opposite the 5 cm side or opposite the 7 cm side), it is possible to construct two different, non-congruent triangles with these dimensions. This is related to the ambiguous case of the Law of Sines in trigonometry.

Thus, the position of the equal angle relative to the two equal sides is essential for guaranteeing congruence under the SAS criterion.

The ASA (Angle-Side-Angle) Congruence Criterion for triangles states the following:

If two angles and the included side of one triangle are equal to the corresponding two angles and the included side of another triangle, then the two triangles are congruent.

Illustration with Triangles:

Imagine two triangles, $\triangle ABC$ and $\triangle PQR$.

(Since I cannot draw, visualize $\triangle ABC$ with vertices A, B, C and angles $\angle A, \angle B, \angle C$ and side BC, and $\triangle PQR$ with vertices P, Q, R and angles $\angle P, \angle Q, \angle R$ and side QR).

Given:

In $\triangle ABC$ and $\triangle PQR$, let's assume the following conditions for ASA congruence:

Note that the side BC is included between angles $\angle$ABC and $\angle$BCA. Similarly, the side QR is included between angles $\angle$PQR and $\angle$QRP.

Conclusion:

Based on the ASA congruence criterion, since two angles and the included side of $\triangle ABC$ are equal to the corresponding two angles and the included side of $\triangle PQR$, the two triangles are congruent.

Importance of the side being 'included':

It is important that the side is the 'included' side because fixing two angles and the length of the segment between their vertices uniquely determines the triangle. The included side dictates the scale of the triangle formed by the two angles.

If the side were not included, say we had two equal angles and a side opposite one of them (AAS case), this also guarantees congruence. However, the ASA criterion specifies the side *between* the two angles. This particular arrangement of information (two angles and the side joining their vertices) provides exactly enough information to fix the size and shape of the triangle uniquely.

For example, if you have angles of $30^\circ$ and $70^\circ$ and an included side of length 10 cm, there is only one way to construct the triangle. The $10$ cm side forms the base, and the other two sides are determined by the angles at its endpoints. If the 10 cm side was opposite the $30^\circ$ angle (AAS case), that would be a different set of conditions, though it also leads to a unique (congruent) triangle.

The RHS (Right Angle-Hypotenuse-Side) Congruence Criterion is a special congruence criterion applicable only to right-angled triangles. It states the following:

If in two right-angled triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle, then the two triangles are congruent.

The "R" stands for the Right Angle ($90^\circ$), the "H" stands for the Hypotenuse (the side opposite the right angle), and the "S" stands for one of the other two sides (a leg).

Why this criterion is specifically for right-angled triangles:

This criterion relies on the unique properties of right-angled triangles, specifically the relationship between their sides as described by the Pythagorean theorem. If we know the lengths of the hypotenuse and one leg in a right-angled triangle, the length of the other leg is uniquely determined by the Pythagorean theorem ($a^2 + b^2 = c^2$, where $c$ is the hypotenuse). If two right triangles have equal hypotenuses and one pair of corresponding legs equal, then their other pair of legs must also be equal in length.

Let the hypotenuse be $c$ and the given leg be $a$. The other leg $b$ can be found using $b = \sqrt{c^2 - a^2}$. Since $c$ and $a$ are the same for both triangles, $b$ will also be the same. This means all three sides are equal, satisfying the SSS congruence criterion. Thus, the RHS criterion is essentially a consequence of the Pythagorean theorem combined with the SSS criterion.

For non-right-angled triangles, knowing two sides and a non-included angle (which RHS is conceptually similar to, as the right angle is not included between the hypotenuse and a leg) does not guarantee congruence (this is the ambiguous case of the Law of Sines). The presence of the fixed $90^\circ$ angle is crucial for this criterion to work.

Illustration with Triangles:

Imagine two right-angled triangles, $\triangle ABC$ and $\triangle PQR$, where $\angle B$ and $\angle Q$ are the right angles.

(Since I cannot draw, visualize $\triangle ABC$ right-angled at B, and $\triangle PQR$ right-angled at Q).

Given:

In $\triangle ABC$ and $\triangle PQR$:

Conclusion:

Based on the RHS congruence criterion, since both triangles are right-angled, their hypotenuses are equal, and one pair of corresponding sides is equal, the two triangles are congruent.

Given:

1. AD and BC are perpendiculars to line segment AB. This means $\angle$DAO and $\angle$CBO are right angles.

2. AD = BC (The perpendiculars are equal in length).

3. CD intersects AB at point O.

To Prove:

CD bisects AB, i.e., AO = OB.

Proof:

Consider triangles $\triangle AOD$ and $\triangle BOC$.

We have two pairs of equal angles and one pair of equal corresponding sides which is not included between the angles (the side AD is opposite to $\angle$AOD and the side BC is opposite to $\angle$BOC).

Therefore, by the AAS (Angle-Angle-Side) Congruence Criterion,

Since the triangles are congruent, their corresponding parts are equal (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

The corresponding side to AO in $\triangle AOD$ is BO in $\triangle BOC$.

Since AO = BO, this means that point O is the midpoint of the line segment AB. The line segment CD passes through O, thus dividing AB into two equal parts. Therefore, CD bisects AB.

Given:

In $\triangle ABC$,

D is a point on BC such that AD $\perp$ BC.

To Prove:

1. $\triangle ABD \cong \triangle ACD$

2. BD = CD

3. $\angle$B = $\angle$C

Proof:

Consider the triangles $\triangle ABD$ and $\triangle ACD$.

In right-angled triangles $\triangle ABD$ and $\triangle ACD$, AB and AC are the hypotenuses (opposite the right angles $\angle$ADB and $\angle$ADC respectively).

AD is a side common to both triangles.

Therefore, by the RHS (Right Angle-Hypotenuse-Side) Congruence Criterion, we have:

Since the triangles $\triangle ABD$ and $\triangle ACD$ are congruent, their corresponding parts are equal (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

The side BD in $\triangle ABD$ corresponds to the side CD in $\triangle ACD$.

This shows that D is the midpoint of the side BC.

The angle $\angle$B in $\triangle ABD$ corresponds to the angle $\angle$C in $\triangle ACD$.

Hence, we have shown that $\triangle ABD \cong \triangle ACD$, BD = CD, and $\angle$B = $\angle$C.

Given:

1. AB is a line segment and P is its midpoint.

2. D and E are points on the same side of AB.

3. $\angle$BAD = $\angle$ABE

4. $\angle$EPA = $\angle$DPB

To Prove:

$\triangle DAP \cong \triangle EBP$

Proof:

Consider the triangles $\triangle DAP$ and $\triangle EBP$.

We are given that $\angle$EPA = $\angle$DPB.

Add $\angle$EPD to both sides of this equation:

From the figure, $\angle$EPA + $\angle$EPD forms $\angle$APD, and $\angle$DPB + $\angle$EPD forms $\angle$BPE.

Now, consider $\triangle DAP$ and $\triangle EBP$.

We have two pairs of equal angles and the included side between them equal in both triangles.

Therefore, by the ASA (Angle-Side-Angle) Congruence Criterion,

Hence proved.

CPCTC is an acronym that stands for Corresponding Parts of Congruent Triangles are Congruent.

It is a theorem or a principle used in geometry proofs. Once it has been proven that two triangles are congruent using one of the congruence criteria (SSS, SAS, ASA, AAS, RHS), CPCTC allows us to conclude that all pairs of corresponding sides and all pairs of corresponding angles are also equal in measure.

Importance of CPCTC:

CPCTC is a very important tool in geometric proofs because it allows us to deduce equalities of parts of triangles after establishing the congruence of the whole triangles. Often, the goal of a proof is to show that specific sides or angles are equal. By proving that the triangles containing these parts are congruent, we can then use CPCTC to immediately state the required equality without further steps.

Specific Equalities from $\triangle LMN \cong \triangle PQR$ using CPCTC:

Given that $\triangle LMN \cong \triangle PQR$, the congruence statement tells us the correspondence between the vertices:

L corresponds to P

M corresponds to Q

N corresponds to R

Using CPCTC, we can state that the corresponding sides are equal and the corresponding angles are equal.

Corresponding Sides:

Corresponding Angles:

These six equalities (three sides and three angles) can be stated as a direct consequence of the congruence $\triangle LMN \cong \triangle PQR$ by using CPCTC.

Yes, the triangles Rohan and Priya drew are congruent.

Given:

In $\triangle ABC$:

In $\triangle PQR$:

Justification using Congruence Criterion:

Consider $\triangle ABC$ and $\triangle PQR$.

We have the following pairs of corresponding parts that are equal:

In $\triangle ABC$, the side AB is the side included between angles $\angle$A and $\angle$B.

In $\triangle PQR$, the side PQ is the side included between angles $\angle$P and $\angle$Q.

Since two angles and the included side of $\triangle ABC$ are equal to the corresponding two angles and the included side of $\triangle PQR$, the triangles are congruent by the ASA (Angle-Side-Angle) Congruence Criterion.

Measure of the third angle:

Using the Angle Sum Property of a triangle, the sum of the angles in any triangle is $180^\circ$.

In $\triangle ABC$:

In $\triangle PQR$:

The measure of the third angle in both triangles is $80^\circ$.

The SSS (Side-Side-Side) and ASA (Angle-Side-Angle) congruence criteria are two of the fundamental ways to prove that two triangles are congruent. They specify different sets of corresponding equal parts required for congruence.

SSS (Side-Side-Side) Congruence Criterion:

This criterion states that if the three sides of one triangle are equal in length to the corresponding three sides of another triangle, then the triangles are congruent.

ASA (Angle-Side-Angle) Congruence Criterion:

This criterion states that if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of another triangle, then the triangles are congruent.

Comparison (Similarities):

Contrast (Differences):

When to Use Each Criterion (with Examples):

Using SSS:

You would use the SSS criterion when you know (or can prove) that all three sides of one triangle are equal to the corresponding three sides of another triangle. This is common when dealing with properties of geometric shapes where side lengths are key.

Example: Proving that the diagonal of a parallelogram divides it into two congruent triangles. Consider parallelogram ABCD with diagonal AC. In $\triangle ABC$ and $\triangle CDA$:

Therefore, by SSS congruence, $\triangle ABC \cong \triangle CDA$.

Using ASA:

You would use the ASA criterion when you know (or can prove) that two angles and the side between them in one triangle are equal to the corresponding parts in another triangle. This is useful when dealing with angles formed by intersecting lines, parallel lines, or angle bisectors.

Example: In the problem of AD and BC being equal perpendiculars to AB, and CD intersecting AB at O (Question 6 Redo from a previous request). We aimed to show $\triangle AOD \cong \triangle BOC$.

Given $\angle$DAO = $\angle$CBO = $90^\circ$, $\angle$AOD = $\angle$BOC (vertically opposite angles), and AD = BC. While the original problem used AAS, if we had a situation where the *included* side was known, we'd use ASA.

Consider proving $\triangle ABC \cong \triangle PQR$ if $\angle A = \angle P$, AB = PQ, and $\angle B = \angle Q$. Here, the known side (AB) is included between the two known angles ($\angle A$ and $\angle B$).

Therefore, by ASA congruence, $\triangle ABC \cong \triangle PQR$.

Choosing between SSS and ASA (or other criteria) depends directly on which corresponding parts of the triangles you are given or can most easily prove to be equal.

Yes, the two triangular flower beds will be congruent.

Given:

Sides of Flower Bed 1: 7 m, 8 m, 10 m.

Sides of Flower Bed 2: 8 m, 7 m, 10 m.

We can consider the sides of Flower Bed 1 as $s_1, s_2, s_3$ and the sides of Flower Bed 2 as $t_1, t_2, t_3$.

For Flower Bed 1:

For Flower Bed 2:

We can see that the set of side lengths for Flower Bed 1 $\{7m, 8m, 10m\}$ is the same as the set of side lengths for Flower Bed 2 $\{8m, 7m, 10m\}$. Although the order is different in the listing, the actual lengths of the sides are the same for both triangles.

Comparing the corresponding side lengths, we can match them up:

7 m = 7 m

8 m = 8 m

10 m = 10 m

Since all three sides of the first triangle are equal to the corresponding three sides of the second triangle, the triangles are congruent by the SSS (Side-Side-Side) Congruence Criterion.

The SSS criterion guarantees that if the side lengths of two triangles are equal, their angles must also be equal. This means the triangles have the exact same shape and size, making them identical copies of each other. Therefore, the two flower beds will be congruent.