CBSE Class 9 Maths Exemplar Chapter 7: Triangles NCERT Solutions

Reviewed by NCERT Help Editorial Team · Updated on 29 Jul 2026

NCERT Solutions PDF Class 9 PDF

CBSE Class 9 Maths Exemplar Chapter 7, Triangles, NCERT Solutions, offers a detailed exploration of triangle congruence. It covers the fundamental criteria for proving triangles congruent: SAS, ASA, and SSS, while also explaining why SSA is not a valid congruence rule. The solutions highlight key properties of isosceles triangles, such as the equality of angles opposite equal sides, and demonstrate the application of the angle sum property to determine unknown angles. Each problem is accompanied by a clear, step-by-step explanation, making complex concepts accessible. This resource aims to deepen students' understanding of the chapter's core principles and equip them with effective problem-solving strategies for exam preparation.

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Quick info

Board	CBSE
Class	Class 9
Subject	Maths Exemplar
Session	2026
Language	English
Type	NCERT Solutions
Chapter	Chapter 7

Chapter summary

Chapter 7 of the CBSE Class 9 Maths Exemplar focuses on the fundamental concepts of triangles, particularly congruence. This set of NCERT Solutions provides detailed explanations for multiple-choice questions that test the understanding of congruence criteria (SAS, ASA, SSS, and the invalidity of SSA) and the properties of isosceles triangles. It guides students through applying the angle sum property and the relationship between equal sides and opposite angles in triangles.

Learning outcomes

Identify valid criteria for triangle congruence (SAS, ASA, SSS).
Recognize that SSA is not a criterion for triangle congruence.
Apply the property that angles opposite equal sides in a triangle are equal.
Use the angle sum property of a triangle to find unknown angles.
Solve problems involving isosceles triangles.

Topics covered

Paper topics

Triangle Congruence
Congruence Criteria
SAS Congruence
ASA Congruence
SSS Congruence
SSA (Invalid Criterion)
Isosceles Triangles
Angles Opposite Equal Sides
Angle Sum Property of Triangles
Properties of Triangles

Important topics

Triangle Congruence Criteria (SSS, SAS, ASA)
Why SSA is not a congruence criterion
Properties of Isosceles Triangles
Angle Sum Property Application

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Questions and Solutions

Question 1

Which of the following is not a criterion for congruence of triangles?

A) SAS

B) ASA

C) SSA

D) SSS

Solution: The criteria for proving that two triangles are congruent are SSS, SAS, and ASA (and AAS, which can be derived). SSA (Side-Side-Angle) is not a valid congruence criterion because it is possible to construct two different triangles with the same SSA values. Therefore, SSA is not a criterion for congruence of triangles. The correct option is (C).

Question 2

If $AB = QR$ , $BC = PR$ and $CA = PQ$ , then which of the following congruence statements is correct?

A) $\triangle ABC \cong \triangle PQR$

B) $\triangle CBA \cong \triangle PRQ$

C) $\triangle BAC \cong \triangle RPQ$

D) $\triangle PQR \cong \triangle BCA$

Solution: We are given the following equalities between the sides of two triangles,

\triangle ABC

and

\triangle PQR

$AB = QR$

$BC = PR$

$CA = PQ$

To establish congruence, we need to find the correct correspondence between the vertices of the two triangles. Let's analyze the given side equalities:

The side AB in $\triangle ABC$ corresponds to QR in $\triangle PQR$ .
The side BC in $\triangle ABC$ corresponds to PR in $\triangle PQR$ .
The side CA in $\triangle ABC$ corresponds to PQ in $\triangle PQR$ .

From these equalities, we can deduce the vertex correspondence:

Since AB = QR, the vertex opposite to AB in $\triangle ABC$ (which is C) must correspond to the vertex opposite to QR in $\triangle PQR$ (which is P). So, $C \leftrightarrow P$ .
Since BC = PR, the vertex opposite to BC in $\triangle ABC$ (which is A) must correspond to the vertex opposite to PR in $\triangle PQR$ (which is Q). So, $A \leftrightarrow Q$ .
Since CA = PQ, the vertex opposite to CA in $\triangle ABC$ (which is B) must correspond to the vertex opposite to PQ in $\triangle PQR$ (which is R). So, $B \leftrightarrow R$ .

Therefore, the correct correspondence is

A \leftrightarrow Q, B \leftrightarrow R, C \leftrightarrow P

. This means

\triangle ABC \cong \triangle QRP

. Let's check the given options based on this correspondence:

A) $\triangle ABC \cong \triangle PQR$ : This implies A↔P, B↔Q, C↔R. This is incorrect.
B) $\triangle CBA \cong \triangle PRQ$ : This implies C↔P, B↔R, A↔Q. This matches our derived correspondence.
C) $\triangle BAC \cong \triangle RPQ$ : This implies B↔R, A↔P, C↔Q. This is incorrect.
D) $\triangle PQR \cong \triangle BCA$ : This implies P↔B, Q↔C, R↔A. This is incorrect.

Hence, the correct option is (B).

Question 3

In $\triangle ABC$ , $AB = AC$ and $\angle B = 50^{\circ}$ . Then $\angle C$ is equal to:

A) $40^{\circ}$

B) $50^{\circ}$

C) $80^{\circ}$

D) $130^{\circ}$

Solution: We are given a triangle

\triangle ABC

where

AB = AC

. This means that

\triangle ABC

is an isosceles triangle. A key property of isosceles triangles is that the angles opposite the equal sides are equal. In

\triangle ABC

, the angle opposite side AC is

\angle B

, and the angle opposite side AB is

\angle C

. Since

AB = AC

, it follows that

\angle C = \angle B

. We are given that

\angle B = 50^{\circ}

. Therefore,

\angle C

must also be

50^{\circ}

. To verify, we can find

\angle A

using the angle sum property:

\angle A + \angle B + \angle C = 180^{\circ}

. So,

\angle A + 50^{\circ} + 50^{\circ} = 180^{\circ}

, which gives

\angle A = 180^{\circ} - 100^{\circ} = 80^{\circ}

. Hence, the correct option is (B).

Question 4

In $\triangle ABC$ , $BC = AB$ and $\angle B = 80^{\circ}$ . Then $\angle A$ is equal to:

A) $80^{\circ}$

B) $40^{\circ}$

C) $50^{\circ}$

D) $100^{\circ}$

Solution: We are given a triangle

\triangle ABC

with

BC = AB

. This indicates that

\triangle ABC

is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal. The angle opposite side AB is

\angle C

, and the angle opposite side BC is

\angle A

. Since

BC = AB

, it follows that

\angle A = \angle C

. We are also given that

\angle B = 80^{\circ}

. Now, we use the angle sum property of a triangle, which states that the sum of all interior angles in a triangle is

180^{\circ}

$\angle A + \angle B + \angle C = 180^{\circ}$

Substitute the known values and the relationship

\angle C = \angle A

into the equation:

$\angle A + 80^{\circ} + \angle A = 180^{\circ}$

Combine the terms with

\angle A

$2\angle A + 80^{\circ} = 180^{\circ}$

Subtract

80^{\circ}

from both sides:

$2\angle A = 180^{\circ} - 80^{\circ}$

$2\angle A = 100^{\circ}$

Divide by 2 to find the value of

\angle A

$\angle A = \frac{100^{\circ}}{2}$

$\angle A = 50^{\circ}$

Therefore, the measure of angle A is

50^{\circ}

. The correct option is (C).

Common mistakes

Confusing SSA with valid congruence criteria.
Incorrectly applying the property of angles opposite equal sides.
Errors in algebraic manipulation when using the angle sum property.

Revision tips

Memorize the four standard congruence criteria: SSS, SAS, ASA, AAS.
Understand why SSA is not a congruence criterion by trying to construct triangles.
Practice identifying equal angles in isosceles triangles based on equal sides.
Review the angle sum property of triangles for every problem involving angles.

Practice MCQs

Q1. Which of the following is NOT a valid criterion for proving that two triangles are congruent?

Q2. If the sides of triangle ABC are related to triangle PQR such that AB = QR, BC = PR, and CA = PQ, which congruence statement is correct?

Q3. In triangle ABC, if AB = AC and angle B is 50 degrees, what is the measure of angle C?

Q4. For triangle ABC, if side BC is equal to side AB and angle B is 80 degrees, what is the measure of angle A?

Frequently asked questions

What are the main congruence criteria for triangles covered in this chapter?

This chapter covers the main congruence criteria for triangles: SSS (Side-Side-Side), SAS (Side-Angle-Side), and ASA (Angle-Side-Angle). It also highlights that SSA (Side-Side-Angle) is not a valid criterion.

How do these NCERT Solutions help with Class 9 Maths?

These solutions provide clear, rewritten explanations for problems in Chapter 7 (Triangles), helping students understand triangle congruence and properties of isosceles triangles, which are crucial for their CBSE exams.

What is the significance of equal sides in an isosceles triangle regarding its angles?

In an isosceles triangle, the angles opposite the equal sides are always equal. This principle is fundamental for solving problems involving isosceles triangles.

Can SSA be used to prove triangle congruence?

No, SSA (Side-Side-Angle) is not a valid criterion for proving triangle congruence. While three sides and two angles are given, the arrangement can sometimes lead to ambiguity, allowing for two different triangles.

How is the angle sum property used in these solutions?

The angle sum property, which states that the sum of angles in any triangle is 180 degrees, is used to find unknown angles when some angles or side-angle relationships are known.

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