Additional Questions for each Class with Solution
6th	7th	8th	9th	10th	11th	12th

Class 6th Chapters
1. Knowing Our Numbers	2. Whole Numbers	3. Playing With Numbers
4. Basic Geometrical Ideas	5. Understanding Elementary Shapes	6. Integers
7. Fractions	8. Decimals	9. Data Handling
10. Mensuration	11. Algebra	12. Ratio And Proportion
13. Symmetry	14. Practical Geometry

Content On This Page
Objective Type Questions	Short Answer Type Questions	Long Answer Type Questions

Chapter 5 Understanding Elementary Shapes (Additional Questions)

This supplementary section provides a valuable opportunity to further explore and solidify your understanding of the concepts presented in the Class 6th chapter, Understanding Elementary Shapes. Moving beyond the most basic geometric ideas, that chapter introduced crucial methods for comparing and classifying fundamental shapes in both two and three dimensions. These extra practice questions are designed to reinforce those classifications, enhance your measurement skills, and build a stronger visual and analytical foundation for future geometric studies. Engaging with these problems will help ensure you can confidently identify, describe, and differentiate various geometric figures based on their defining characteristics.

One key area reinforced here is the measurement and comparison of line segments and, more extensively, angles. You will find ample opportunities to practice measuring angles accurately using a protractor and classifying them based on their magnitude. Remember the categories:

Acute Angle: Measure is greater than $0^\circ$ and less than $90^\circ$.
Right Angle: Measure is exactly $90^\circ$.
Obtuse Angle: Measure is greater than $90^\circ$ but less than $180^\circ$.
Straight Angle: Measure is exactly $180^\circ$.
Reflex Angle: Measure is greater than $180^\circ$ but less than $360^\circ$.

Practice questions will involve not just static angles but also dynamic scenarios, like determining the type of angle formed by the hands of a clock at various times. The concepts of perpendicular lines (lines intersecting at a perfect $90^\circ$ angle) and perpendicular bisectors (lines that cut a segment in half at a $90^\circ$ angle) will also be revisited through identification and application exercises.

The classification of triangles is another major focus. You'll practice identifying triangles based on their side lengths (Scalene - all sides different, Isosceles - two sides equal, Equilateral - all sides equal) and based on their angle measures (Acute-angled - all angles acute, Obtuse-angled - one angle obtuse, Right-angled - one angle is $90^\circ$). Questions will require you to classify triangles from diagrams or given properties. Similarly, the diverse world of quadrilaterals (four-sided polygons) is explored further. Expect exercises asking you to differentiate between and identify types like the Trapezium, Parallelogram, Rectangle, Square, and Rhombus based on descriptions of their sides, angles, and diagonal properties (e.g., identifying "a parallelogram with four right angles" as a rectangle). Identifying other named polygons based on their number of sides will also be practiced.

Venturing into three dimensions, this practice section reinforces your familiarity with basic 3D shapes. You will encounter exercises focused on identifying common solids like the Cube, Cuboid, Cylinder, Cone, Sphere, and Pyramid from diagrams or descriptions. A crucial skill practiced here is counting the number of Faces (flat or curved surfaces), Edges (where faces meet), and Vertices (corners) for various polyhedrons (solids with flat faces). Matching shapes to their names or key properties will further solidify your understanding. By working through these varied supplementary exercises, you enhance your spatial reasoning, hone your classification skills, improve tool proficiency (like using a protractor), and build the robust conceptual groundwork necessary for tackling more advanced geometry and mensuration topics in subsequent classes.

Objective Type Questions

Question 1. Comparing line segments by mere observation can be inaccurate because:

(A) We cannot see the endpoints clearly.

(B) Our eyes can be deceived by the lengths.

(D) We need a ruler to compare accurately.

Answer:

Comparing line segments by mere observation means using our eyes to judge their lengths. While this might work for segments with significantly different lengths, it becomes unreliable when the lengths are close or when the segments are presented in certain visual contexts.

Human visual perception is subject to various optical illusions, where the surrounding elements or the orientation of objects can make lengths appear different from their actual measurements. A classic example is the Müller-Lyer illusion, where two lines of the same length appear different due to the arrowheads at their ends.

Therefore, relying solely on observation can lead to incorrect conclusions about the relative lengths of line segments because our eyes and brains can misinterpret the visual information.

Let's analyze the given options:

(A) "We cannot see the endpoints clearly." This might happen in poorly printed diagrams but is not the fundamental reason why observation is generally inaccurate for comparison.

(B) "Our eyes can be deceived by the lengths." This directly addresses the issue of visual perception and optical illusions, which make visual comparison unreliable.

(C) "The line segments might not be straight." The question refers to "line segments," which are defined as straight paths between two points. So, this option contradicts the definition of the objects being compared.

(D) "We need a ruler to compare accurately." This is true, a ruler provides an objective measure, but it states a necessity for accuracy, not the reason why observation is inaccurate. The reason observation is inaccurate is *why* a ruler is needed for accuracy.

Based on the analysis, the most accurate reason for the inaccuracy of comparing line segments by mere observation is that our visual perception can be easily misled or deceived.

The correct option is (B) Our eyes can be deceived by the lengths.

Question 2. Which is the most accurate method for comparing two line segments?

(A) Observation

(B) Tracing

(D) Using a divider

Answer:

Solution:

Let's evaluate the given methods for comparing two line segments:

(A) Observation: Comparing by just looking is the least accurate method as it is based on visual estimation and prone to errors.

(B) Tracing: Tracing one line segment and placing the tracing on the other segment for comparison is better than observation, but the thickness of the trace line and slight inaccuracies during tracing can still lead to errors.

(C) Using a ruler: Measuring both line segments with a ruler and then comparing the measurements is a common method. However, potential errors can occur while aligning the zero mark, reading the scale, or due to the thickness of the scale markings.

(D) Using a divider: A divider allows direct comparison of the lengths without reading a scale. You set the points of the divider to match the length of one segment and then move the divider to check if it matches the length of the other segment. This method minimizes errors related to reading scales and alignment, making it the most accurate among the given options for direct comparison.

Comparing two line segments using a divider allows for the most accurate determination of whether one segment is longer than, shorter than, or equal to the other.

The most accurate method for comparing two line segments is using a divider.

The correct option is (D) Using a divider.

Question 3. A line segment of length 5 cm is drawn. If another line segment is twice as long, its length is:

(A) 2.5 cm

(B) 5 cm

(D) 15 cm

Answer:

Solution:

Given:

Length of the first line segment = 5 cm.

To Find:

Length of the second line segment.

Solution:

The second line segment is twice as long as the first line segment.

Length of the second line segment = 2 $\times$ Length of the first line segment

Length of the second line segment = 2 $\times$ 5 cm

Length of the second line segment = 10 cm

The length of the second line segment is 10 cm.

The correct option is (C) 10 cm.

Question 4. What is the measure of a right angle?

(A) $45^\circ$

(B) $90^\circ$

(D) $360^\circ$

Answer:

Solution:

An angle is formed by two rays sharing a common endpoint. Angles are measured in degrees ($^\circ$).

There are different types of angles based on their measure:

- An acute angle measures between $0^\circ$ and $90^\circ$.

- A right angle measures exactly $90^\circ$.

- An obtuse angle measures between $90^\circ$ and $180^\circ$.

- A straight angle measures exactly $180^\circ$.

- A reflex angle measures between $180^\circ$ and $360^\circ$.

- A complete angle measures exactly $360^\circ$.

Based on the definition, the measure of a right angle is $90^\circ$.

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

Question 5. An angle measuring $135^\circ$ is a/an:

(A) Acute angle

(B) Right angle

(D) Straight angle

Answer:

Solution:

Angles are classified based on their measure. The different types of angles are:

Acute angle: An angle measuring between $0^\circ$ and $90^\circ$ ($0^\circ < \theta < 90^\circ$).
Right angle: An angle measuring exactly $90^\circ$ ($\theta = 90^\circ$).
Obtuse angle: An angle measuring between $90^\circ$ and $180^\circ$ ($90^\circ < \theta < 180^\circ$).
Straight angle: An angle measuring exactly $180^\circ$ ($\theta = 180^\circ$).
Reflex angle: An angle measuring between $180^\circ$ and $360^\circ$ ($180^\circ < \theta < 360^\circ$).
Complete angle: An angle measuring exactly $360^\circ$ ($\theta = 360^\circ$).

The given angle measures $135^\circ$.

Comparing this measure with the definitions:

$90^\circ < 135^\circ < 180^\circ$

This fits the definition of an obtuse angle.

Therefore, an angle measuring $135^\circ$ is an obtuse angle.

The correct option is (C) Obtuse angle.

Question 6. How many right angles make a straight angle?

(A) 1

(B) 2

(D) 4

Answer:

Solution:

A right angle is defined as an angle that measures exactly $90^\circ$.

A straight angle is defined as an angle that measures exactly $180^\circ$.

To find out how many right angles make a straight angle, we can divide the measure of a straight angle by the measure of a right angle.

Number of right angles = $\frac{\text{Measure of a straight angle}}{\text{Measure of a right angle}}$

Number of right angles = $\frac{180^\circ}{90^\circ}$

Number of right angles = 2

Therefore, two right angles make a straight angle ($90^\circ + 90^\circ = 180^\circ$).

The correct option is (B) 2.

Question 7. A complete angle measures:

(A) $90^\circ$

(B) $180^\circ$

(D) $360^\circ$

Answer:

Solution:

Angles are measured in degrees ($^\circ$). Different types of angles are defined by their measures:

Acute angle: An angle measuring between $0^\circ$ and $90^\circ$.
Right angle: An angle measuring exactly $90^\circ$.
Obtuse angle: An angle measuring between $90^\circ$ and $180^\circ$.
Straight angle: An angle measuring exactly $180^\circ$.
Reflex angle: An angle measuring between $180^\circ$ and $360^\circ$.
Complete angle: An angle measuring exactly $360^\circ$.

The question asks for the measure of a complete angle.

By definition, a complete angle represents a full rotation, and its measure is $360^\circ$.

Therefore, a complete angle measures $360^\circ$.

The correct option is (D) $360^\circ$.

Question 8. Two lines are perpendicular if they intersect at an angle of:

(A) $45^\circ$

(B) $60^\circ$

(D) $180^\circ$

Answer:

Solution:

Two lines are considered perpendicular if they intersect each other at a specific angle.

The angle of intersection for perpendicular lines is a right angle.

A right angle is defined as an angle that measures exactly $90^\circ$.

Therefore, two lines are perpendicular if they intersect at an angle of $90^\circ$.

The correct option is (C) $90^\circ$.

Question 9. In the English alphabet, which letter pair represents perpendicular lines?

(A) V

(B) L

(D) Z

Answer:

Solution:

Perpendicular lines are lines that intersect each other at a right angle ($90^\circ$).

Let's examine the shapes of the given letters in the English alphabet:

(A) V: The two strokes of the letter 'V' meet at an angle less than $90^\circ$ (an acute angle). They are not perpendicular.

(B) L: The two strokes of the letter 'L' meet at an angle of exactly $90^\circ$ (a right angle). This represents perpendicular lines.

(C) N: The strokes of the letter 'N' form acute and obtuse angles where they meet or cross. They do not intersect at a $90^\circ$ angle.

(D) Z: The strokes of the letter 'Z' form acute and obtuse angles where they meet or cross. They do not intersect at a $90^\circ$ angle.

Among the given options, the letter 'L' clearly shows two line segments that are perpendicular to each other.

The correct option is (B) L.

Question 10. A triangle with all three sides of different lengths is called:

(A) Isosceles triangle

(B) Equilateral triangle

(D) Right-angled triangle

Answer:

Solution:

Triangles can be classified based on the lengths of their sides:

Equilateral Triangle: A triangle in which all three sides are of equal length. In an equilateral triangle, all three angles are also equal, each measuring $60^\circ$.
Isosceles Triangle: A triangle in which at least two sides are of equal length. In an isosceles triangle, the angles opposite the two equal sides are also equal.
Scalene Triangle: A triangle in which all three sides are of different lengths. In a scalene triangle, all three angles are also of different measures.

Triangles can also be classified based on their angles, such as acute-angled, right-angled, and obtuse-angled triangles. However, the question specifically asks about the classification based on side lengths.

The question describes a triangle with all three sides of different lengths.

According to the definitions, a triangle with all three sides of different lengths is called a scalene triangle.

Therefore, a triangle with all three sides of different lengths is called a scalene triangle.

The correct option is (C) Scalene triangle.

Question 11. A triangle with one angle measuring $90^\circ$ is called a/an:

(A) Acute-angled triangle

(B) Obtuse-angled triangle

(D) Equilateral triangle

Answer:

Solution:

Triangles can be classified based on the measure of their angles:

Acute-angled triangle: A triangle in which all three interior angles are acute (measure less than $90^\circ$).
Right-angled triangle: A triangle in which one of the interior angles is a right angle (measures exactly $90^\circ$).
Obtuse-angled triangle: A triangle in which one of the interior angles is an obtuse angle (measures greater than $90^\circ$).

The question describes a triangle with one angle measuring $90^\circ$.

According to the definitions, a triangle with one angle measuring $90^\circ$ is called a right-angled triangle.

Therefore, a triangle with one angle measuring $90^\circ$ is called a right-angled triangle.

The correct option is (C) Right-angled triangle.

Question 12. Which of the following statements is true for an equilateral triangle?

(A) All sides are equal, and all angles are $60^\circ$.

(B) Two sides are equal, and two angles are equal.

(D) One angle is greater than $90^\circ$.

Answer:

Solution:

An equilateral triangle is a triangle that has specific properties regarding its sides and angles.

By definition, an equilateral triangle is a triangle in which all three sides are of equal length.

A fundamental property of triangles is that if two sides are equal, the angles opposite to those sides are also equal. In an equilateral triangle, since all three sides are equal, the angles opposite to these sides are also equal.

Let the measure of each angle be $x$. The sum of the interior angles in any triangle is always $180^\circ$.

So, for an equilateral triangle:

$x + x + x = 180^\circ$

$3x = 180^\circ$

$x = \frac{180^\circ}{3}$

$x = 60^\circ$

Thus, in an equilateral triangle, all three angles measure $60^\circ$.

Let's examine the given options:

(A) All sides are equal, and all angles are $60^\circ$. This statement accurately describes the properties of an equilateral triangle.

(B) Two sides are equal, and two angles are equal. This statement describes an isosceles triangle. While an equilateral triangle is a type of isosceles triangle, this statement is not exclusive to equilateral triangles.

(C) All angles are less than $90^\circ$. This statement describes an acute-angled triangle. An equilateral triangle is always acute-angled ($60^\circ < 90^\circ$), but this property is not unique to equilateral triangles.

(D) One angle is greater than $90^\circ$. This statement describes an obtuse-angled triangle. An equilateral triangle cannot be obtuse-angled as all its angles are $60^\circ$.

The statement that is true specifically and completely for an equilateral triangle among the given options is that all sides are equal, and all angles are $60^\circ$.

The correct option is (A) All sides are equal, and all angles are $60^\circ$.

Question 13. A quadrilateral is a polygon with:

(A) 3 sides

(B) 4 sides

(D) 6 sides

Answer:

Solution:

A polygon is a closed two-dimensional shape made up of straight line segments.

Polygons are classified based on the number of sides they have:

A polygon with 3 sides is called a triangle.
A polygon with 4 sides is called a quadrilateral.
A polygon with 5 sides is called a pentagon.
A polygon with 6 sides is called a hexagon.
And so on.

The question asks for the number of sides a quadrilateral has.

By definition, a quadrilateral is a polygon with 4 sides.

Therefore, a quadrilateral is a polygon with 4 sides.

The correct option is (B) 4 sides.

Question 14. Which of the following quadrilaterals has exactly one pair of parallel sides?

(A) Parallelogram

(B) Rhombus

(D) Rectangle

Answer:

Solution:

Let's examine the properties of the given quadrilaterals regarding parallel sides:

Parallelogram: A quadrilateral with two pairs of parallel sides. Opposite sides are parallel.
Rhombus: A special type of parallelogram where all four sides are equal. Like a parallelogram, it has two pairs of parallel sides.
Trapezium (or Trapezoid): A quadrilateral with at least one pair of parallel sides. In many contexts, especially in elementary geometry, it is defined as a quadrilateral with exactly one pair of parallel sides.
Rectangle: A special type of parallelogram where all four angles are right angles. Like a parallelogram, it has two pairs of parallel sides.

The question asks for the quadrilateral that has exactly one pair of parallel sides.

Based on the definitions, the trapezium (when defined as having exactly one pair of parallel sides) fits this description.

Therefore, the quadrilateral that has exactly one pair of parallel sides is a trapezium.

The correct option is (C) Trapezium.

Question 15. A parallelogram with all four sides equal is called a:

(A) Rectangle

(B) Square

(D) Rhombus

Answer:

Solution:

A parallelogram is a quadrilateral with opposite sides parallel and equal.

We are given a parallelogram with an additional property: all four sides are equal.

Let's consider the definitions of the given quadrilaterals:

A Rectangle is a parallelogram with all four angles equal to $90^\circ$. It does not necessarily have all four sides equal.
A Square is a parallelogram with all four sides equal and all four angles equal to $90^\circ$. While a square has all four sides equal, it also has the property of having right angles, which is not specified as a requirement in the question, only that it's a parallelogram with four equal sides.
A Trapezium is a quadrilateral with at least one pair of parallel sides. It is not necessarily a parallelogram, and its sides are generally not equal.
A Rhombus is a parallelogram with all four sides equal. This definition precisely matches the description given in the question.

Therefore, a parallelogram with all four sides equal is called a rhombus.

The correct option is (D) Rhombus.

Question 16. Which of the following 3D shapes has 6 rectangular faces?

(A) Cube

(B) Cylinder

(D) Cone

Answer:

Solution:

Let's consider the properties of the given 3D shapes, specifically the number and shape of their faces:

Cube: A cube is a 3D shape with 6 square faces. A square is a specific type of rectangle where all sides are equal. So, a cube does have 6 rectangular faces (specifically square ones).
Cylinder: A cylinder typically has 2 circular bases and one curved surface. It does not have 6 rectangular faces.
Cuboid: A cuboid is a 3D shape with 6 rectangular faces. The opposite faces are congruent rectangles. This is the general definition of a cuboid.
Cone: A cone has one circular base and one curved surface ending in a point (apex). It does not have rectangular faces.

The question asks which shape has 6 rectangular faces. Both a cube and a cuboid have 6 rectangular faces.

However, a cuboid is defined by having 6 rectangular faces, whereas a cube is a special case of a cuboid where all faces are squares.

Among the given options, 'Cuboid' is the general term for a shape with 6 rectangular faces, making it the most direct answer to the question as phrased.

Therefore, a cuboid is a 3D shape with 6 rectangular faces.

The correct option is (C) Cuboid.

Question 17. How many vertices does a cube have?

(A) 6

(B) 8

(D) 12

Answer:

Solution:

A cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In geometry, a vertex (plural: vertices) is a point where two or more edges meet.

Let's count the vertices of a cube. Imagine a standard cube like a die or a box:

- There are 4 vertices on the top face.

- There are 4 vertices on the bottom face.

Each vertex on the top is connected to a corresponding vertex on the bottom by a vertical edge. No two vertices from the top plane share a vertex with each other in the bottom plane, so the total number of vertices is the sum of vertices on the top and bottom faces.

Total number of vertices = Number of vertices on top face + Number of vertices on bottom face

Total number of vertices = 4 + 4

Total number of vertices = 8

Alternatively, using Euler's formula for polyhedra, $V - E + F = 2$, where $V$ is the number of vertices, $E$ is the number of edges, and $F$ is the number of faces. For a cube, $F=6$ and $E=12$.

$V - 12 + 6 = 2$

$V - 6 = 2$

$V = 2 + 6$

$V = 8$

A cube has 8 vertices.

The correct option is (B) 8.

Question 18. A matchbox is an example of which 3D shape?

(A) Cube

(B) Cuboid

(D) Prism

Answer:

Solution:

A matchbox is a common example of a rectangular solid.

Let's look at the characteristics of the given 3D shapes:

A Cube is a 3D shape with 6 square faces. All edges are of equal length.
A Cuboid is a 3D shape with 6 rectangular faces. The opposite faces are congruent rectangles. The edges can be of different lengths (length, width, height).
A Cylinder is a 3D shape with two circular bases and a curved surface connecting them.
A Prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases. A cuboid is a specific type of rectangular prism where the base is a rectangle.

A typical matchbox has the shape of a rectangular box. Its faces are rectangles, and it has a length, width, and height that are usually different. This description perfectly matches the definition of a cuboid.

Therefore, a matchbox is an example of a cuboid.

The correct option is (B) Cuboid.

Question 19. Which of the following shapes has no vertices and no edges?

(A) Cube

(B) Cone

(D) Cylinder

Answer:

Solution:

In three-dimensional geometry, a vertex is a point where three or more edges meet, and an edge is a line segment where two faces meet.

Let's examine the properties of the given 3D shapes:

Cube: A cube has 8 vertices and 12 edges.
Cone: A cone has one vertex (the apex) and one circular edge (the boundary of the base).
Sphere: A sphere is a perfectly round three-dimensional object. It has a single continuous curved surface. It does not have any flat faces, straight edges, or vertices.
Cylinder: A cylinder has no vertices, but it has two circular edges (the boundaries of the top and bottom bases).

The question asks for the shape that has no vertices and no edges.

Based on the properties listed above, the sphere is the only shape among the options that has neither vertices nor edges.

Therefore, the sphere is the shape that has no vertices and no edges.

The correct option is (C) Sphere.

Question 20. Match the type of angle with its measure:

(i) Acute Angle

(ii) Obtuse Angle

(iii) Reflex Angle

(iv) Straight Angle

(a) Measures between $90^\circ$ and $180^\circ$

(b) Measures exactly $180^\circ$

(d) Measures between $0^\circ$ and $90^\circ$

(A) (i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)

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

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

Answer:

Solution:

Let's match the types of angles with their corresponding measures based on their definitions:

(i) Acute Angle: An angle that measures between $0^\circ$ and $90^\circ$. This corresponds to option (d).

(ii) Obtuse Angle: An angle that measures between $90^\circ$ and $180^\circ$. This corresponds to option (a).

(iii) Reflex Angle: An angle that measures between $180^\circ$ and $360^\circ$. This corresponds to option (c).

(iv) Straight Angle: An angle that measures exactly $180^\circ$. This corresponds to option (b).

So the correct matches are:

(i) $\rightarrow$ (d)

(ii) $\rightarrow$ (a)

(iii) $\rightarrow$ (c)

(iv) $\rightarrow$ (b)

Comparing these matches with the given options:

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

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

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

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

Question 21. Assertion (A): A square is a special type of rectangle.
Reason (R): All rectangles have four right angles and opposite sides equal, and a square fits this description while having all sides equal.

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

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

(D) A is false but R is true.

Answer:

Solution:

Analysis of Assertion (A):

Assertion (A) states: A square is a special type of rectangle.

Let's recall the definitions:

A rectangle is a quadrilateral with four right angles.

A square is a quadrilateral with four equal sides and four right angles.

For a shape to be a rectangle, it must satisfy the property of having four right angles. A square has four right angles. Therefore, a square fits the definition of a rectangle. The term "special type" indicates that a square has an additional property (all sides equal) that not all rectangles have.

Thus, Assertion (A) is True.

Analysis of Reason (R):

Reason (R) states: All rectangles have four right angles and opposite sides equal, and a square fits this description while having all sides equal.

Let's break down the statement:

"All rectangles have four right angles": This is true by the definition of a rectangle.

"and opposite sides equal": This is a property of parallelograms. Since a rectangle has four right angles, its opposite sides are parallel, making it a parallelogram. Thus, the opposite sides of a rectangle are equal.

"and a square fits this description": A square has four right angles and its opposite sides are equal (in fact, all its sides are equal, which implies opposite sides are equal). So, a square does fit the description of a rectangle.

"while having all sides equal": This part highlights the distinguishing feature of a square compared to a general rectangle.

Thus, Reason (R) is a correct statement about the properties of rectangles and squares. Reason (R) is True.

Relationship between A and R:

Assertion (A) claims that a square is a special type of rectangle.

Reason (R) provides the defining properties of a rectangle (four right angles and opposite sides equal) and explains that a square possesses these properties while having the additional property of all sides being equal.

The fact that a square satisfies the properties of a rectangle (as explained in R) is precisely the reason why it is considered a type of rectangle (as stated in A). The additional property mentioned in R (all sides equal) is what makes it a *special* type.

Therefore, Reason (R) correctly explains Assertion (A).

Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

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

Question 22. Which of the following are types of quadrilaterals? (Select all that apply)

(A) Pentagon

(B) Parallelogram

(D) Triangle

Answer:

Solution:

A quadrilateral is a polygon with four sides and four vertices.

Let's examine each of the given options:

(A) Pentagon: A pentagon is a polygon with five sides. Since it has 5 sides, it is not a quadrilateral.

(B) Parallelogram: A parallelogram is a quadrilateral with two pairs of parallel sides. It has 4 sides and 4 vertices, so it is a type of quadrilateral.

(C) Rhombus: A rhombus is a parallelogram with all four sides of equal length. Since a rhombus is a parallelogram, and a parallelogram is a type of quadrilateral, a rhombus is also a type of quadrilateral. It has 4 sides and 4 vertices.

(D) Triangle: A triangle is a polygon with three sides. Since it has 3 sides, it is not a quadrilateral.

Based on the definitions, both a parallelogram and a rhombus are types of quadrilaterals.

The correct options are (B) Parallelogram and (C) Rhombus.

Question 23. A triangle has angles measuring $40^\circ$, $60^\circ$, and $80^\circ$. This is a/an:

(A) Acute-angled triangle

(B) Right-angled triangle

(D) Equilateral triangle

Answer:

Solution:

Given:

The angles of the triangle are $40^\circ$, $60^\circ$, and $80^\circ$.

To Classify:

The type of triangle based on its angles.

Solution:

Triangles are classified based on their angle measures as follows:

An acute-angled triangle has all three interior angles measuring less than $90^\circ$.
A right-angled triangle has exactly one interior angle measuring $90^\circ$.
An obtuse-angled triangle has exactly one interior angle measuring greater than $90^\circ$.

Let's examine the given angles:

First angle = $40^\circ$. Since $40^\circ < 90^\circ$, this is an acute angle.

Second angle = $60^\circ$. Since $60^\circ < 90^\circ$, this is an acute angle.

Third angle = $80^\circ$. Since $80^\circ < 90^\circ$, this is an acute angle.

Since all three angles of the triangle ($40^\circ$, $60^\circ$, and $80^\circ$) are less than $90^\circ$, the triangle is an acute-angled triangle.

The correct option is (A) Acute-angled triangle.

Question 24. Case Study: Rahul is looking at different objects around his house. He observes the following:
1. A brick
2. A cricket ball
3. A traffic cone
4. A cylindrical water bottle

Identify the 3D shapes corresponding to these objects in order:

(A) Cuboid, Sphere, Cone, Cylinder

(B) Cube, Circle, Triangle, Rectangle

(D) Cube, Sphere, Pyramid, Bottle

Answer:

Solution:

We need to identify the common 3D geometric shape that each object represents.

1. A brick is typically rectangular in shape with flat faces at right angles. This shape is known as a cuboid.

2. A cricket ball is a perfectly round solid object. This shape is known as a sphere.

3. A traffic cone has a circular base and tapers to a point at the top. This shape is known as a cone.

4. A cylindrical water bottle has a circular base and a circular top of the same size, connected by a smooth curved side. This shape is known as a cylinder.

The 3D shapes corresponding to the objects in order are Cuboid, Sphere, Cone, and Cylinder.

Let's match this list with the given options:

(A) Cuboid, Sphere, Cone, Cylinder - This matches our findings.

(B) Cube, Circle, Triangle, Rectangle - Incorrect shapes and dimensions.

(D) Cube, Sphere, Pyramid, Bottle - Incorrect for the brick (typically cuboid) and traffic cone (cone, not pyramid).

The correct sequence of 3D shapes is Cuboid, Sphere, Cone, Cylinder.

The correct option is (A) Cuboid, Sphere, Cone, Cylinder.

Question 25. Which type of angle is formed by the hands of a clock at 3:00 p.m.?

(A) Acute angle

(B) Right angle

(D) Straight angle

Answer:

Solution:

At 3:00 p.m. on a clock, the hour hand points exactly at the number 3, and the minute hand points exactly at the number 12.

A clock face is a circle, which represents a complete angle of $360^\circ$. The clock face is divided into 12 hours.

The angle between any two consecutive hour marks on the clock face is $\frac{360^\circ}{12} = 30^\circ$.

At 3:00 p.m., the hour hand is at 3 and the minute hand is at 12. The number of hour intervals between 12 and 3 is 3 (from 12 to 1, 1 to 2, and 2 to 3).

The angle between the hour hand and the minute hand is the number of intervals multiplied by the angle per interval:

Angle = Number of intervals $\times$ Angle per interval

Angle = $3 \times 30^\circ$

Angle = $90^\circ$

An angle that measures exactly $90^\circ$ is called a right angle.

Therefore, the angle formed by the hands of a clock at 3:00 p.m. is a right angle.

The correct option is (B) Right angle.

Question 26. The instrument used to measure angles is called a:

(A) Ruler

(B) Compass

(D) Divider

Answer:

Solution:

Let's consider the primary use of each instrument listed:

A Ruler is used for measuring the length of straight line segments and for drawing straight lines.
A Compass is used for drawing circles and arcs and for transferring or comparing lengths.
A Protractor is a geometric instrument used specifically for measuring angles and for drawing angles of a given measure.
A Divider is used for comparing lengths of line segments or for marking off equal distances.

The question asks for the instrument used to measure angles.

Based on the uses of the instruments, the protractor is the instrument used to measure angles.

Therefore, the instrument used to measure angles is called a protractor.

The correct option is (C) Protractor.

Question 27. Which of the following statements is false?

(A) All squares are rhombuses.

(B) All rhombuses are parallelograms.

(D) All rectangles are parallelograms.

Answer:

Solution:

Let's analyze each statement based on the definitions and properties of quadrilaterals:

Definitions:

Parallelogram: A quadrilateral with opposite sides parallel.
Rectangle: A parallelogram with four right angles.
Rhombus: A parallelogram with all four sides equal in length.
Square: A quadrilateral with four equal sides and four right angles. Equivalently, a square is a rhombus that is also a rectangle.
Trapezium (or Trapezoid): A quadrilateral with at least one pair of parallel sides. (Note: Some definitions require exactly one pair of parallel sides, while others require at least one. We'll consider the "at least one" definition which is more common in broader contexts and typically used when comparing with parallelograms.)

Let's evaluate each statement:

(A) All squares are rhombuses. A rhombus is a parallelogram with all sides equal. A square has all sides equal and is also a parallelogram. Thus, a square fits the definition of a rhombus. This statement is True.

(B) All rhombuses are parallelograms. A rhombus is defined as a parallelogram with all four sides equal. The definition explicitly states it is a parallelogram. This statement is True.

(C) All parallelograms are trapeziums. A parallelogram has two pairs of parallel sides. A trapezium has at least one pair of parallel sides. Since having two pairs of parallel sides ($AB || DC$ and $AD || BC$) implies having at least one pair of parallel sides (e.g., $AB || DC$), a parallelogram satisfies the condition for being a trapezium (under the "at least one pair" definition). This statement is True.

If using the "exactly one pair" definition of trapezium, then this statement would be false, as a parallelogram has two pairs. However, in the context of comparing hierarchies of quadrilaterals, the "at least one" definition is more standard for this type of statement. Let's assume the "at least one" definition for now.

(D) All rectangles are parallelograms. A rectangle is a quadrilateral with four right angles. Having four right angles implies that opposite sides are parallel. For example, if $\angle A = \angle B = \angle C = \angle D = 90^\circ$, then lines AD and BC are both perpendicular to AB, making them parallel. Similarly, AB and DC are both perpendicular to AD, making them parallel. Since a rectangle has opposite sides parallel, it fits the definition of a parallelogram. This statement is True.

Let's re-examine option (C) assuming the "exactly one pair" definition of a trapezium. If a trapezium must have exactly one pair of parallel sides, then a parallelogram, which has two pairs of parallel sides, would NOT be a trapezium. In this case, statement (C) "All parallelograms are trapeziums" would be False.

Given that this is an objective question with a single false statement, and statements (A), (B), and (D) are consistently true under standard geometric definitions, it is highly probable that the definition of trapezium assumed for this question is one with exactly one pair of parallel sides.

Under the definition that a trapezium has exactly one pair of parallel sides:

(C) All parallelograms are trapeziums. A parallelogram has two pairs of parallel sides. Therefore, a parallelogram is not a trapezium under this definition. This statement is False.

Thus, the false statement is (C).

The correct option is (C) All parallelograms are trapeziums.

Question 28. A triangle with angles $30^\circ$, $60^\circ$, $90^\circ$ is a:

(A) Acute-angled scalene triangle

(B) Right-angled isosceles triangle

(D) Obtuse-angled triangle

Answer:

Solution:

Given:

The angles of the triangle are $30^\circ$, $60^\circ$, and $90^\circ$.

To Classify:

The type of triangle based on its angles and sides.

Classification by Angles:

We examine the measure of the angles:

$30^\circ < 90^\circ$ (Acute angle)
$60^\circ < 90^\circ$ (Acute angle)
$90^\circ$ (Right angle)

Since the triangle has one angle that measures exactly $90^\circ$, it is a right-angled triangle.

Classification by Sides:

The relationship between angle measures and side lengths in a triangle is that the side opposite a larger angle is longer than the side opposite a smaller angle. Also, if two angles are equal, the sides opposite them are equal (Isosceles triangle). If all three angles are equal, all three sides are equal (Equilateral triangle).

The given angles are $30^\circ$, $60^\circ$, and $90^\circ$. These three angles are all different.

Since all three angles are different, the sides opposite these angles must also be of different lengths.

A triangle with all three sides of different lengths is called a scalene triangle.

Combined Classification:

Based on the angles, the triangle is right-angled.

Based on the side lengths (inferred from the angles), the triangle is scalene.

Combining these classifications, the triangle is a right-angled scalene triangle.

Let's match this with the given options:

(A) Acute-angled scalene triangle - Incorrect, as it has a $90^\circ$ angle.

(B) Right-angled isosceles triangle - Incorrect, as all angles are different, implying all sides are different.

(D) Obtuse-angled triangle - Incorrect, as it has a $90^\circ$ angle, not an angle greater than $90^\circ$.

The correct option is (C) Right-angled scalene triangle.

Question 29. How many faces does a triangular prism have?

(A) 3

(B) 4

(D) 6

Answer:

Solution:

A prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases.

A triangular prism has a triangle as its base (and top face). The number of sides on the base polygon is $n=3$.

The faces of a triangular prism consist of:

1. Two bases: These are the two identical triangular faces at the top and bottom.

2. Lateral faces: These are the faces connecting the corresponding sides of the two bases. Since the base is a triangle (3 sides), there are 3 lateral faces. In a right prism, these lateral faces are rectangles.

Total number of faces = Number of base faces + Number of lateral faces

Total number of faces = 2 + 3

Total number of faces = 5

A triangular prism has 5 faces: 2 triangular faces and 3 rectangular faces.

The correct option is (C) 5.

Question 30. Complete the statement: A line segment has a definite _____.

(A) Direction

(B) Endpoint

(D) Width

Answer:

Solution:

Let's understand the characteristics of a line segment.

A line segment is a part of a line that consists of two distinct endpoints and all the points on the line between them.

Now let's evaluate the given options in the context of a line segment having a "definite" property:

(A) Direction: A line segment lies along a line, which has direction. However, the segment itself doesn't have a single definite direction in the same way a ray does (starting at one point and extending infinitely in one direction). A segment can be traversed from one endpoint to the other, or vice versa.
(B) Endpoint: A line segment has two definite endpoints, not just one definite endpoint. The statement implies a singular definite property.
(C) Length: A line segment is the part of a line between two fixed points. The distance between these two endpoints is a fixed, measurable value. This value is the definite length of the line segment.
(D) Width: In geometry, a line segment is considered a one-dimensional object. It has length but no width or thickness. Therefore, it does not have a definite width (or any width at all).

Based on the properties, a line segment has a specific, measurable distance between its two endpoints, which is its length. This length is definite.

Therefore, a line segment has a definite length.

The correct option is (C) Length.

Question 31. Two distinct lines can intersect at most at _____ point.

(A) Zero

(B) One

(D) Infinite

Answer:

Solution:

In Euclidean geometry, a line is a straight, one-dimensional figure that has no thickness and extends infinitely in both directions.

We are considering two distinct lines. This means they are not the same line.

There are two possible scenarios for two distinct lines in a plane:

1. The lines are parallel. Parallel lines are lines that are always the same distance apart and never intersect. In this case, they intersect at zero points.

2. The lines are not parallel. If two distinct lines are not parallel, they must intersect. Since they are straight and distinct, they can cross each other at only one point.

The question asks for the maximum number of points at which two distinct lines can intersect.

Comparing the two scenarios, the maximum number of intersection points is one (when they are not parallel).

If two lines were to intersect at two or more points, they would have to curve or coincide, which contradicts the definition of distinct straight lines.

Therefore, two distinct lines can intersect at most at one point.

The correct option is (B) One.

Question 32. Which of the following angles is a reflex angle? (Select all that apply)

(A) $100^\circ$

(B) $200^\circ$

(D) $30^\circ$

Answer:

Solution:

A reflex angle is an angle whose measure is greater than $180^\circ$ but less than $360^\circ$.

In mathematical terms, an angle $\theta$ is a reflex angle if $180^\circ < \theta < 360^\circ$.

Let's examine the measure of each given angle:

(A) $100^\circ$: Is $100^\circ > 180^\circ$? No. This is an obtuse angle, as $90^\circ < 100^\circ < 180^\circ$.

(B) $200^\circ$: Is $180^\circ < 200^\circ < 360^\circ$? Yes. This is a reflex angle.

(D) $30^\circ$: Is $30^\circ > 180^\circ$? No. This is an acute angle, as $0^\circ < 30^\circ < 90^\circ$.

The angles that satisfy the condition for being a reflex angle are $200^\circ$ and $270^\circ$.

The correct options are (B) $200^\circ$ and (C) $270^\circ$.

Question 33. What is the sum of angles in a triangle?

(A) $90^\circ$

(B) $180^\circ$

(D) $360^\circ$

Answer:

Solution:

The sum of the interior angles of any triangle is a fundamental property in Euclidean geometry.

Consider a triangle with interior angles $\angle A$, $\angle B$, and $\angle C$.

The sum of these angles is always constant, regardless of the type or size of the triangle.

Sum of interior angles of a triangle = $\angle A + \angle B + \angle C$

This sum is equal to $180^\circ$.

Therefore, the sum of angles in a triangle is $180^\circ$.

The correct option is (B) $180^\circ$.

Question 34. A pyramid with a square base has how many faces?

(A) 4

(B) 5

(D) 8

Answer:

Solution:

A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face.

We are given a pyramid with a square base. The base is a polygon with 4 sides.

The faces of a pyramid consist of:

1. One base: This is the square face at the bottom.

2. Lateral faces: These are the triangular faces that meet at the apex. Since the base is a square (4 sides), there are 4 lateral faces, one for each side of the square.

Total number of faces = Number of base faces + Number of lateral faces

Total number of faces = 1 + 4

Total number of faces = 5

A pyramid with a square base has 5 faces: 1 square base and 4 triangular lateral faces.

The correct option is (B) 5.

Question 35. If two line segments have the same length, they are called _____ line segments.

(A) Parallel

(B) Perpendicular

(D) Congruent

Answer:

Solution:

We are asked to complete the statement: "A line segment has a definite _____."

Let's consider the properties of a line segment:

A line segment is a part of a line that is bounded by two distinct endpoints. So it has endpoints, specifically two of them. The term "definite endpoint" (singular) isn't the most precise description of a defining definite property.
A line segment lies along a straight line, but the concept of a single definite direction is more applicable to a ray, which extends infinitely in one direction from an endpoint.
A line segment, being the part of a line between two fixed endpoints, has a specific, measurable distance between these endpoints. This distance is called its length. This length is a fixed and definite value for any given line segment.
In geometry, a line segment is considered one-dimensional; it has length but no width or thickness. Therefore, it does not have a definite width (it has zero width).

Among the given options, the property that is definite and uniquely associated with a specific line segment is its length.

Thus, a line segment has a definite length.

The correct option is (C) Length.

Short Answer Type Questions

Question 1. What are the three methods to compare line segments? Which method is the most accurate?

Answer:

Solution:

The three common methods to compare line segments are:

1. Comparison by Observation: This involves visually inspecting the two line segments and estimating which one is longer. This is the least accurate method as our perception can be easily deceived by the position or orientation of the segments.

2. Comparison by Tracing: This method involves using a tracing paper to trace one line segment and then placing the tracing directly on top of the other line segment to compare their lengths. This method is more accurate than observation, but inaccuracies can arise due to the thickness of the traced line or slight errors in tracing and placement.

3. Comparison using Instruments: This method involves using measuring instruments. There are two primary ways this is done:

Using a Ruler: Measure the length of each line segment using a ruler and then compare the obtained numerical measurements. Errors can occur in aligning the zero mark of the ruler with one endpoint or in reading the scale accurately, especially for lengths between markings.
Using a Divider: Place the two points of a divider on the endpoints of one line segment to set the distance. Then, without changing the opening of the divider, place its points on the endpoints of the second line segment. By observing whether the points match the endpoints, fall short, or go beyond, the lengths can be compared directly.

Among these methods, the comparison using a divider is considered the most accurate.

The divider allows for direct transfer and comparison of the lengths without relying on reading a scale (like a ruler) or dealing with the inaccuracies introduced by tracing (like the thickness of the line).

Thus, the three methods are comparison by observation, by tracing, and using instruments (ruler or divider). The most accurate method is comparison using a divider.

Question 2. If line segment AB has length $5$ cm and line segment CD has length $5$ cm, how do you compare their lengths?

Answer:

Solution:

Given:

Length of line segment AB = $5$ cm

Length of line segment CD = $5$ cm

To compare the lengths of the two line segments, we look at their measured lengths.

Length of AB $= 5$ cm

Length of CD $= 5$ cm

Since the numerical values of their lengths are the same, the lengths of the line segments AB and CD are equal.

Length of AB $=$ Length of CD

(Both are $5$ cm)

In geometry, if two line segments have the same length, they are said to be congruent.

Therefore, we compare their lengths by observing that their given measures are equal, which means the line segments AB and CD are congruent.

Question 3. What instrument is used to measure the length of a line segment accurately?

Answer:

To measure the length of a line segment accurately, an instrument called a ruler or a scale is used.

A ruler is a common tool used in geometry, drafting, and everyday life to measure distances or to draw straight lines.

It is typically a straight strip, often made of plastic, wood, or metal, with markings along its edge.

These markings represent standard units of length, such as centimeters (cm) and millimeters (mm) in the metric system, or inches in the imperial system.

To measure a line segment accurately using a ruler, one end of the line segment is placed precisely at the zero mark of the ruler. The reading on the ruler that aligns with the other end of the line segment gives the length of the segment.

For greater accuracy, rulers often have smaller divisions, such as millimeters, allowing for more precise readings than just centimeters or inches.

Factors affecting accuracy include the clarity and precision of the ruler's markings, the material of the ruler (which can expand or contract slightly with temperature), and the user's ability to align the ruler correctly and read the measurement without parallax error.

Question 4. What is the unit commonly used to measure angles?

Answer:

The unit commonly used to measure angles is the degree.

A degree is a unit of angular measure equal to $\frac{1}{360}$ of a full rotation. The symbol for degree is $^\circ$.

So, a full circle measures $360^\circ$.

Other units exist, such as radians, but degrees are the most commonly used unit in many practical applications and basic geometry.

Question 5. Define a right angle. What is its measure?

Answer:

A right angle is an angle that measures exactly $90^\circ$.

In geometry, a right angle is formed when two straight lines or rays intersect in a way that they create four equal angles at the point of intersection. Each of these four equal angles is a right angle.

Alternatively, a right angle can be described as an angle that is equal to one-quarter of a full rotation ($360^\circ / 4 = 90^\circ$) or one-half of a straight angle ($180^\circ / 2 = 90^\circ$).

The measure of a right angle is always $90^\circ$.

Question 6. What is an acute angle? Give an example of its possible measure.

Answer:

An acute angle is an angle whose measure is greater than $0^\circ$ and less than $90^\circ$.

In other words, an acute angle is any angle that is smaller than a right angle ($90^\circ$).

Examples of possible measures for an acute angle include:

$30^\circ$

$45^\circ$

$60^\circ$

$89^\circ$

Any angle $\theta$ such that $0^\circ < \theta < 90^\circ$ is an acute angle.

Question 7. What is an obtuse angle? Give an example of its possible measure.

Answer:

An obtuse angle is an angle whose measure is greater than $90^\circ$ and less than $180^\circ$.

In other words, an obtuse angle is any angle that is larger than a right angle ($90^\circ$) but smaller than a straight angle ($180^\circ$).

Examples of possible measures for an obtuse angle include:

$100^\circ$

$135^\circ$

$175^\circ$

Any angle $\theta$ such that $90^\circ < \theta < 180^\circ$ is an obtuse angle.

Question 8. What is a straight angle? What is its measure?

Answer:

A straight angle is an angle that measures exactly $180^\circ$.

A straight angle is formed by two rays or line segments that extend in opposite directions from a common endpoint (the vertex), forming a single straight line.

The measure of a straight angle is equal to half of a full rotation ($360^\circ / 2 = 180^\circ$).

It can also be thought of as the sum of two right angles ($90^\circ + 90^\circ = 180^\circ$) placed adjacent to each other along a straight line.

The measure of a straight angle is always $180^\circ$.

Question 9. How many right angles make a straight angle?

Answer:

Two right angles make a straight angle.

A right angle measures $90^\circ$.

A straight angle measures $180^\circ$.

To find out how many right angles make a straight angle, we divide the measure of a straight angle by the measure of a right angle:

Number of right angles = $\frac{\text{Measure of straight angle}}{\text{Measure of right angle}}$

Number of right angles = $\frac{180^\circ}{90^\circ} = 2$

Therefore, two right angles are needed to form a straight angle.

Question 10. What type of angle is formed by the hands of a clock at $9:00$ AM?

Answer:

At $9:00$ AM, the hands of a clock form a right angle.

At $9:00$ AM, the hour hand points exactly at the $9$ and the minute hand points exactly at the $12$.

The clock face is divided into $12$ hours, representing a full circle of $360^\circ$.

The angle between consecutive hour marks is $\frac{360^\circ}{12} = 30^\circ$.

At $9:00$ AM, the hands are separated by $3$ hour marks (from $9$ to $12$).

The angle between the hands is $3 \times 30^\circ = 90^\circ$.

An angle that measures exactly $90^\circ$ is defined as a right angle.

Question 11. What are perpendicular lines? Give a real-life example.

Answer:

Perpendicular lines are two lines that intersect each other at a right angle ($90^\circ$).

When two lines are perpendicular, they form four right angles at their point of intersection.

If line L1 is perpendicular to line L2, we can denote this as $L1 \perp L2$.

A real-life example of perpendicular lines is the intersection of the walls and the floor in a rectangular room.

The line where a wall meets the floor typically forms a right angle with the line where an adjacent wall meets the floor, as well as forming a right angle with the line of the floor itself.

Another common example is the intersection of the horizontal and vertical lines that form a plus sign ($+$) or the corners of a square or rectangle.

Question 12. Define a triangle based on the number of sides.

Answer:

A triangle is a polygon with three sides.

Based on the number of sides, a triangle is uniquely defined as having exactly three line segments as its sides.

These three sides connect at three points called vertices, forming three interior angles.

Question 13. A triangle has side lengths $3$ cm, $4$ cm, and $5$ cm. What type of triangle is it based on its sides?

Answer:

Based on its side lengths, the triangle is a scalene triangle.

The side lengths of the triangle are given as $3$ cm, $4$ cm, and $5$ cm.

In a scalene triangle, all three sides have different lengths.

Since $3 \neq 4$, $4 \neq 5$, and $3 \neq 5$, all three sides of the given triangle have different lengths.

Therefore, the triangle is classified as a scalene triangle.

Question 14. A triangle has angles measuring $40^\circ, 60^\circ,$ and $80^\circ$. What type of triangle is it based on its angles?

Answer:

Based on its angles, the triangle is an acute triangle.

The angles of the triangle are given as $40^\circ$, $60^\circ$, and $80^\circ$.

An acute triangle is a triangle in which all three interior angles are acute angles.

An acute angle is an angle that measures greater than $0^\circ$ and less than $90^\circ$.

Let's check the given angles:

$40^\circ$: $0^\circ < 40^\circ < 90^\circ$ (Acute angle)

$60^\circ$: $0^\circ < 60^\circ < 90^\circ$ (Acute angle)

$80^\circ$: $0^\circ < 80^\circ < 90^\circ$ (Acute angle)

Since all three angles ($40^\circ, 60^\circ, 80^\circ$) are acute angles, the triangle is classified as an acute triangle.

Question 15. What is a quadrilateral? How many sides and angles does it have?

Answer:

A quadrilateral is a polygon with four sides.

It is a closed two-dimensional figure made up of four line segments connected end-to-end.

A quadrilateral has:

Four sides

Four vertices (corner points)

Four interior angles

The sum of the interior angles of any convex quadrilateral is always $360^\circ$.

Question 16. Name a quadrilateral where all four sides are equal and all four angles are right angles.

Answer:

The quadrilateral where all four sides are equal and all four angles are right angles is a square.

A square is a special type of quadrilateral that possesses both the properties of a rhombus (all sides equal) and a rectangle (all angles are right angles).

Specifically, in a square:

All four sides have the same length.

All four interior angles are right angles, each measuring $90^\circ$.

Question 17. Name a quadrilateral where opposite sides are equal and parallel, and all four angles are right angles.

Answer:

The quadrilateral where opposite sides are equal and parallel, and all four angles are right angles is a rectangle.

A rectangle is a type of parallelogram with four right angles.

The properties given in the question match the definition of a rectangle:

Opposite sides are equal in length.

Opposite sides are parallel.

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

While a square also fits these properties, the question asks for "a quadrilateral", and rectangle is the broader term that perfectly describes the given conditions. A square is a special case of a rectangle where all four sides are equal, not just opposite sides.

Question 18. What is a cube? How many faces does it have?

Answer:

A cube is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex.

It is one of the five Platonic solids and is a special type of hexahedron.

In a cube:

All six faces are identical squares.

All edges are of equal length.

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

A cube has 6 faces.

Question 19. Give a real-life example of a sphere.

Answer:

A real-life example of a sphere is a football (soccer ball).

A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.

It is defined as the set of all points that are equally distant from a central point.

Common real-life objects that approximate the shape of a sphere include:

A football

A basketball

A tennis ball

An orange

The Earth (approximately)

Question 20. How many vertices does a cuboid have?

Answer:

A cuboid has 8 vertices.

A vertex (plural: vertices) is a corner point where three or more edges of a three-dimensional shape meet.

A cuboid has 6 rectangular faces, 12 edges, and 8 vertices.

You can visualize this by thinking of a box (a common example of a cuboid): it has 4 corners on the top and 4 corresponding corners on the bottom, totaling 8 vertices.

Question 21. What is a triangular prism? Describe its faces.

Answer:

A triangular prism is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.

It is a prism with a triangular base.

A triangular prism has 5 faces.

These faces are:

Two triangular faces: These are the bases of the prism. They are parallel and congruent triangles.

Three rectangular faces: These are the lateral faces that connect the corresponding sides of the two triangular bases. In a right triangular prism, these are rectangles; in an oblique triangular prism, they are parallelograms.

Question 22. How many edges does a square pyramid have?

Answer:

A square pyramid has 8 edges.

A square pyramid is a pyramid with a square base.

It consists of:

A square base (1 face)

Four triangular faces that meet at a point called the apex (4 faces)

To count the edges:

The square base has 4 edges.

There are 4 edges that connect each vertex of the square base to the apex.

Total number of edges = Edges on base + Slant edges = $4 + 4 = 8$ edges.

Question 23. What is the measure of a complete angle?

Answer:

A complete angle, also known as a full angle or full rotation, is an angle that measures exactly $360^\circ$.

It represents a full turn around a point, bringing the rotating ray back to its original position.

The measure of a complete angle is always $360^\circ$.

Question 24. What type of angle is $120^\circ$?

Answer:

An angle that measures $120^\circ$ is an obtuse angle.

An obtuse angle is defined as an angle whose measure is greater than $90^\circ$ and less than $180^\circ$.

The given angle is $120^\circ$.

We can see that $90^\circ < 120^\circ < 180^\circ$.

Since its measure falls between $90^\circ$ and $180^\circ$, it is classified as an obtuse angle.

Question 25. What is the difference between a square and a rectangle?

Answer:

The primary difference between a square and a rectangle lies in the lengths of their sides.

A rectangle is a quadrilateral with four right angles. In a rectangle, opposite sides are equal in length.

A square is also a quadrilateral with four right angles. However, in a square, all four sides are equal in length.

This means that a square is a special type of rectangle where the adjacent sides are also equal.

Every square is a rectangle, but not every rectangle is a square.

Long Answer Type Questions

Question 1. Describe the three methods of comparing line segments: by observation, by tracing, and by using a ruler and a divider. Explain the advantages and disadvantages of each method in terms of accuracy.

Answer:

Comparing line segments involves determining which segment is longer, shorter, or if they are of equal length. There are three common methods for this comparison:

1. Comparison by Observation:

In this method, we look at the two line segments and estimate which one appears longer just by sight.

Advantage: This is the quickest and simplest method. It requires no tools.

Disadvantage: This method is highly inaccurate, especially when the lengths of the line segments are very close to each other. Our visual perception can be easily tricked, leading to incorrect conclusions.

2. Comparison by Tracing:

In this method, we trace one of the line segments onto a piece of paper or tracing paper. Then, we place the traced copy over the other line segment to compare their lengths.

Process: Trace the first line segment (say AB) onto tracing paper. Place the tracing paper such that the endpoint A' of the traced segment aligns with the endpoint C of the second line segment (say CD). Observe where the endpoint B' of the traced segment falls relative to the endpoint D of the second segment.

If B' falls exactly on D, then AB = CD.

If B' falls beyond D, then AB > CD.

If B' falls before D, then AB < CD.

Advantage: This method is more accurate than comparison by observation.

Disadvantage: It requires tracing paper or a similar medium. The accuracy is limited by the thickness of the tracing lines and the precision of placing the traced segment. There can be errors introduced during the tracing process itself and during the overlaying process.

3. Comparison using a Ruler and a Divider:

This method uses measuring instruments for comparison.

Using a Ruler: Measure the length of each line segment using a ruler. Then compare the obtained numerical measures.

Process: Place the ruler along the first line segment (AB) such that the zero mark aligns with endpoint A. Read the marking on the ruler that aligns with endpoint B. This is the length of AB. Repeat the process for the second line segment (CD). Compare the two measured lengths.

Advantage: This method provides a numerical value for the length and is more objective than observation or tracing.

Disadvantage: Accuracy depends on the precision of the ruler markings and the ability to align the ruler correctly without parallax error (the apparent shift in the reading when viewed from different angles).

Using a Divider: A divider is a geometry tool with two arms that can be adjusted to span a certain distance. This is the most accurate method among the three.

Process: Adjust the opening of the divider such that the points of its two arms coincide exactly with the endpoints of the first line segment (AB). Without changing the opening, place the divider on the second line segment (CD) such that one point aligns with endpoint C. Observe where the second point of the divider falls relative to endpoint D.

If the second point falls exactly on D, then AB = CD.

If the second point falls beyond D, then AB > CD.

If the second point falls before D, then AB < CD.

To get the actual length, the span of the divider can then be measured using a ruler, placing one point on the zero mark and reading the position of the other point.

Advantage: This is the most accurate method because it avoids potential errors from tracing thickness and minimizes parallax errors associated with directly reading a ruler against the line segment. It captures the exact length between the endpoints.

Disadvantage: It requires specialized instruments (a divider and optionally a ruler). It can be tricky to handle the divider without accidentally changing its opening, especially for long segments or when comparing segments far apart.

In summary, while observation is the fastest but least accurate, tracing offers improved accuracy at the cost of using tracing paper and potential tracing errors. The method using a ruler and a divider, particularly the divider method, provides the highest accuracy for comparing line segments among these three techniques, but it requires specific tools.

Question 2. Explain how to measure an angle using a protractor. Draw an angle of measure $75^\circ$ using a protractor and label it correctly.

Answer:

Measuring an Angle using a Protractor:

A protractor is a semi-circular or circular instrument used to measure angles. A standard protractor is a semicircle marked with degrees from $0^\circ$ to $180^\circ$ along its curved edge.

Here are the steps to measure an angle (say $\angle \text{ABC}$) using a protractor:

1. Place the center point (origin or midpoint mark) of the protractor exactly on the vertex of the angle (point B).

2. Align the base line (the straight edge marked with $0^\circ$) of the protractor with one arm of the angle (say ray BA).

3. Make sure the $0^\circ$ mark is on the aligned arm. If the arm is to the right, use the $0^\circ$ on the right side of the base line and read the outer scale. If the arm is to the left, use the $0^\circ$ on the left side and read the inner scale.

4. Read the measure on the curved scale where the other arm of the angle (ray BC) crosses the scale. This reading gives the measure of the angle.

5. Ensure you read the correct scale (inner or outer) that starts from the $0^\circ$ aligned with the first arm.

Drawing an Angle of $75^\circ$ using a Protractor:

Here are the steps to draw an angle of measure $75^\circ$ (say $\angle \text{XYZ}$) using a protractor:

1. Draw a ray, which will be one arm of the angle. Let's call the starting point X and another point on the ray Y. This is ray XY.

2. Place the center point of the protractor exactly on the vertex of the angle, which is point X.

3. Align the base line of the protractor along the ray XY such that the $0^\circ$ mark on the protractor coincides with point Y (or lies on the ray XY).

4. Find the $75^\circ$ mark on the protractor's curved scale. If you aligned the ray XY with the $0^\circ$ mark on the right side, use the outer scale. If you aligned it with the $0^\circ$ mark on the left side, use the inner scale.

5. Make a small dot or mark on the paper at the $75^\circ$ mark on the protractor's scale. Let's call this point Z.

6. Remove the protractor. Draw a second ray from the vertex X through the point Z that you just marked.

7. The angle formed by the two rays XY and XZ, with vertex at X, is $\angle \text{YXZ}$ or $\angle \text{ZXY}$. The measure of this angle is $75^\circ$. Label the vertex X and points Y and Z on the arms. You can also indicate the angle measure inside the angle arc.

Question 3. Define and classify the different types of angles based on their measurement (Acute, Right, Obtuse, Straight, Reflex, Complete). Draw a diagram for each type of angle.

Answer:

Angles are classified based on their measure in degrees. Here are the different types of angles:

1. Acute Angle

Definition: An angle that measures less than $90^\circ$ but more than $0^\circ$.

Measure: $0^\circ < \theta < 90^\circ$

Diagram Description: Imagine two rays meeting at a point (vertex) and forming an opening that is smaller than the corner of a square. An arc connecting the two rays near the vertex shows the angle.

2. Right Angle

Definition: An angle that measures exactly $90^\circ$.

Measure: $\theta = 90^\circ$

Diagram Description: Imagine two rays meeting at a point (vertex) and forming a perfect 'L' shape. The two rays are perpendicular to each other. The angle is often marked with a small square symbol at the vertex.

3. Obtuse Angle

Definition: An angle that measures more than $90^\circ$ but less than $180^\circ$.

Measure: $90^\circ < \theta < 180^\circ$

Diagram Description: Imagine two rays meeting at a point (vertex) and forming an opening that is wider than a right angle but does not form a straight line. An arc connecting the two rays near the vertex shows the angle.

4. Straight Angle

Definition: An angle that measures exactly $180^\circ$.

Measure: $\theta = 180^\circ$

Diagram Description: Imagine two rays meeting at a point (vertex) and extending in exactly opposite directions, forming a straight line. The vertex lies on the line.

5. Reflex Angle

Definition: An angle that measures more than $180^\circ$ but less than $360^\circ$.

Measure: $180^\circ < \theta < 360^\circ$

Diagram Description: Imagine two rays meeting at a point (vertex). A reflex angle is the larger angle measured around the outside of the two rays, away from the interior angle (which would be acute, right, or obtuse). The arc indicating the angle goes around the vertex and extends beyond the straight angle.

6. Complete Angle (or Full Angle)

Definition: An angle that measures exactly $360^\circ$. It represents a full rotation.

Measure: $\theta = 360^\circ$

Diagram Description: Imagine a single ray starting from a vertex and rotating one full turn around the vertex back to its original position. This is usually represented by a circular arc drawn around the vertex, completing a full circle.

Note: As I cannot draw images directly in this format, I have provided textual descriptions of the diagrams.

Question 4. Explain the concept of perpendicular lines. Draw two perpendicular lines and label the angle formed by them. Give two examples of perpendicular lines you see around you in your school or home.

Answer:

Perpendicular lines are two lines that intersect each other at an angle of exactly $90^\circ$ (a right angle).

When two lines are perpendicular, they form four right angles at their point of intersection.

The symbol for perpendicularity is $\perp$. So, if line L1 is perpendicular to line L2, we write $L1 \perp L2$.

Diagram Description of Perpendicular Lines:

Draw two straight lines, say line AB and line CD, that cross each other at a point, say point O.

Ensure that the angle formed at the intersection point O is a right angle ($90^\circ$).

You can indicate the right angle by drawing a small square box at the vertex O in one of the four angles formed by the intersection. This square symbol signifies a $90^\circ$ angle.

Label the lines and the intersection point.

Real-Life Examples of Perpendicular Lines:

1. The edges of a wall meeting the floor in a room. The line formed by the bottom edge of the wall is perpendicular to the line formed by the edge of the floor (assuming they meet at a square corner).

2. The horizontal and vertical lines that form the grid on a piece of graph paper. These lines intersect at $90^\circ$ angles.

(Other examples include the corner of a door frame, the intersection of crosswalk lines on a road, the hands of a clock at 3:00 or 9:00).

Question 5. Classify triangles based on their sides (Scalene, Isosceles, Equilateral) and based on their angles (Acute-angled, Obtuse-angled, Right-angled). Draw a diagram for each type and mention its key properties.

Answer:

Triangles can be classified based on the lengths of their sides or the measures of their interior angles.

Classification Based on Sides:

1. Scalene Triangle

Definition: A triangle in which all three sides have different lengths.

Key Properties:

All three sides are of unequal length (a $\neq$ b $\neq$ c $\neq$ a).

All three angles are of unequal measure.

Diagram Description: Draw a triangle where visually, no two sides appear to be the same length. The angles will also look different.

2. Isosceles Triangle

Definition: A triangle in which at least two sides have equal lengths.

Key Properties:

At least two sides are of equal length (e.g., a = b $\neq$ c).

The angles opposite the two equal sides are also equal (Base angles are equal).

Diagram Description: Draw a triangle with two sides of the same length meeting at a vertex. The angles at the base (opposite the equal sides) should look equal. You can mark the two equal sides with small identical lines.

3. Equilateral Triangle

Definition: A triangle in which all three sides have equal lengths.

Key Properties:

All three sides are of equal length (a = b = c).

All three interior angles are equal, and each measures $60^\circ$.

An equilateral triangle is also an equiangular triangle.

Diagram Description: Draw a triangle where all three sides appear equal in length. All three angles should also look equal ($60^\circ$). Mark all three sides with the same small line symbol.

Classification Based on Angles:

1. Acute-angled Triangle

Definition: A triangle in which all three interior angles are acute angles (less than $90^\circ$).

Key Properties:

All three angles are acute ($\theta < 90^\circ$ for all angles).

The sum of the angles is $180^\circ$.

Diagram Description: Draw a triangle where all three angles appear to be less than $90^\circ$. None of the corners look like a square corner.

2. Right-angled Triangle

Definition: A triangle in which one of the interior angles is a right angle (exactly $90^\circ$).

Key Properties:

One angle measures exactly $90^\circ$.

The sum of the other two angles is $90^\circ$.

The side opposite the right angle is called the hypotenuse, which is the longest side.

Diagram Description: Draw a triangle with one angle clearly marked as a right angle using the square symbol. The other two angles will be acute.

3. Obtuse-angled Triangle

Definition: A triangle in which one of the interior angles is an obtuse angle (greater than $90^\circ$ but less than $180^\circ$).

Key Properties:

One angle measures greater than $90^\circ$ and less than $180^\circ$ ($90^\circ < \theta < 180^\circ$).

The other two angles are acute.

Diagram Description: Draw a triangle where one angle is noticeably larger than $90^\circ$. The other two angles will be small and acute.

Note: As I cannot draw images directly, I have provided textual descriptions of the diagrams.

Question 6. Describe the properties of a parallelogram and a rhombus. How are they similar, and how are they different? Draw a diagram for each shape.

Answer:

Both parallelograms and rhombuses are types of quadrilaterals (four-sided polygons).

Parallelogram:

Definition: A quadrilateral with two pairs of parallel sides.

Properties:

Opposite sides are parallel.

Opposite sides are equal in length.

Opposite angles are equal in measure.

Consecutive angles are supplementary (their sum is $180^\circ$).

The diagonals bisect each other (they cut each other in half at their intersection point).

Diagram Description: Draw a four-sided figure where the top and bottom sides are parallel and equal in length, and the left and right sides are parallel and equal in length. The angles at opposite vertices should look equal.

Rhombus:

Definition: A quadrilateral with all four sides equal in length.

Properties:

All four sides are equal in length.

Opposite sides are parallel (This means a rhombus is also a parallelogram).

Opposite angles are equal in measure.

Consecutive angles are supplementary.

The diagonals bisect each other at a right angle ($90^\circ$).

The diagonals bisect the angles at the vertices.

Diagram Description: Draw a four-sided figure where all four sides are the same length. It might look like a tilted square. Opposite angles should look equal. The diagonals drawn inside should meet at a right angle.

Similarities between a Parallelogram and a Rhombus:

Both are quadrilaterals with four sides.

Both have opposite sides that are parallel and equal in length.

Both have opposite angles that are equal in measure.

In both, consecutive angles are supplementary.

In both, the diagonals bisect each other.

Differences between a Parallelogram and a Rhombus:

In a rhombus, all four sides are equal in length, whereas in a general parallelogram, only opposite sides are guaranteed to be equal.

In a rhombus, the diagonals intersect at a right angle ($90^\circ$), whereas in a general parallelogram, the diagonals only bisect each other but do not necessarily meet at a right angle.

In a rhombus, the diagonals bisect the interior angles, whereas in a general parallelogram, this is not necessarily true (unless it is a rhombus or a square).

A rhombus is a special type of parallelogram, but a parallelogram is not necessarily a rhombus (unless all its sides are equal).

Question 7. Discuss the properties of a trapezium. Can a trapezium be a parallelogram? Explain why or why not. Draw a diagram of a trapezium.

Answer:

Properties of a Trapezium:

A trapezium (also known as a trapezoid in some regions) is a quadrilateral with at least one pair of parallel sides.

Key properties of a trapezium:

It has four sides.

It has exactly one pair of parallel sides (in the exclusive definition) or at least one pair of parallel sides (in the inclusive definition, commonly used in higher mathematics). We will use the inclusive definition here: at least one pair of parallel sides.

The parallel sides are called the bases.

The non-parallel sides are sometimes called legs.

The sum of consecutive angles between the parallel sides is $180^\circ$ (supplementary).

Can a Trapezium be a Parallelogram?

A parallelogram is a quadrilateral with two pairs of parallel sides.

Based on the inclusive definition of a trapezium (a quadrilateral with *at least one* pair of parallel sides), a parallelogram *can* be considered a special type of trapezium.

Since a parallelogram has two pairs of parallel sides, it certainly has "at least one pair" of parallel sides. Therefore, it satisfies the condition for being a trapezium under the inclusive definition.

However, if we use the exclusive definition of a trapezium (a quadrilateral with *exactly one* pair of parallel sides), then a parallelogram cannot be a trapezium because a parallelogram has two pairs of parallel sides.

In most modern mathematical contexts, the inclusive definition is preferred. So, generally, yes, a parallelogram can be a trapezium.

Diagram Description of a Trapezium:

Draw a quadrilateral. Draw two opposite sides such that they are parallel to each other (e.g., a top horizontal line and a bottom horizontal line). The other two sides should connect the endpoints of the parallel sides and should not be parallel to each other (unless it's an isosceles trapezium or a parallelogram, which is a special case).

Label the vertices, say A, B, C, D. If AB is parallel to DC, mark arrows on these two sides to indicate they are parallel.

Question 8. For the following 3D shapes: Cube, Cylinder, and Cone, list the number of faces, edges, and vertices for each. Give a real-life object that resembles each shape.

Answer:

Here are the number of faces, edges, and vertices for the given 3D shapes, along with real-life examples:

1. Cube

A cube is a polyhedron with six square faces.

Number of Faces: 6

Number of Edges: 12

Number of Vertices: 8

Real-life Example: A dice or a Rubik's Cube.

2. Cylinder

A cylinder is a 3D shape with two parallel circular bases and a curved lateral surface connecting them.

Number of Faces: 3 (2 flat circular faces + 1 curved face)

Number of Edges: 2 (the boundaries of the circular bases)

Number of Vertices: 0

Real-life Example: A can of soda or a battery.

3. Cone

A cone is a 3D shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

Number of Faces: 2 (1 flat circular base + 1 curved face)

Number of Edges: 1 (the boundary of the circular base)

Number of Vertices: 1 (the apex)

Real-life Example: An ice cream cone or a party hat.

Question 9. Explain the difference between a triangle, a quadrilateral, and a polygon. Can a circle be classified as a polygon? Justify your answer.

Answer:

Let's define each term to understand the differences:

Polygon:

A polygon is a closed two-dimensional shape made up of straight line segments joined end to end. The segments are called sides, and the points where the segments meet are called vertices. A polygon must have at least three sides.

Triangle:

A triangle is a type of polygon that has exactly three sides and three vertices.

Quadrilateral:

A quadrilateral is a type of polygon that has exactly four sides and four vertices.

The relationship is that triangles and quadrilaterals are specific types of polygons, distinguished by the number of sides they have.

A polygon is the general term for any closed figure made of straight sides. A triangle is a 3-sided polygon, and a quadrilateral is a 4-sided polygon.

Can a circle be classified as a polygon?

No, a circle cannot be classified as a polygon.

Justification: The definition of a polygon states that it must be made up of straight line segments. A circle is a curved shape; its boundary is a continuous curve, not a collection of straight line segments. Therefore, it does not meet the fundamental requirement of having straight sides, which is essential for a shape to be a polygon.

Question 10. Imagine you have a matchbox. What 3D shape does it represent? Describe the shape in terms of its faces, edges, and vertices. What is the shape of each face?

Answer:

A matchbox typically represents the 3D shape called a cuboid.

A cuboid is a three-dimensional solid object bounded by six rectangular faces.

Here is the description of a cuboid in terms of its faces, edges, and vertices:

Faces: A cuboid has 6 faces. These are the flat surfaces of the shape.

Edges: A cuboid has 12 edges. These are the lines where two faces meet.

Vertices: A cuboid has 8 vertices. These are the points (corners) where three edges meet.

The shape of each face of a typical matchbox (cuboid) is a rectangle.

In some matchboxes, two opposite faces might be squares, but the others will be rectangles. In a standard cuboid, all six faces are rectangles, though opposite faces are congruent.