📐 CBSE · Class 9 · Mathematics · Chapter 7

Chapter 7:
Triangles

Complete chapter resources for CBSE Class 9 Maths — congruence criteria, key theorems, sample questions, previous year board questions, and instant AI question paper generation.

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Key Concepts — Chapter 7
  • SAS: Two sides and included angle equal → congruent
  • ASA / AAS: Two angles + one side equal → congruent
  • SSS: All three sides equal → congruent
  • RHS: Right angle + hypotenuse + one side → congruent
  • Isosceles △: Angles opp. equal sides are equal
  • Inequality: Side opp. larger angle is longer
Chapter Overview

What this chapter covers

Chapter 7 of NCERT Class 9 Mathematics focuses on congruence of triangles — the conditions under which two triangles are identical in shape and size. The five congruence criteria studied are SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), SSS (Side-Side-Side), and RHS (Right angle-Hypotenuse-Side). Congruence is written using the symbol ≅, and the order of vertices in the correspondence must always be stated correctly.

A major application of congruence is in proving properties of isosceles triangles: if two sides of a triangle are equal, the angles opposite them are also equal, and vice versa. The chapter also establishes that the angle bisectors, medians, and altitudes drawn from the vertex angle of an isosceles triangle all coincide, making the triangle symmetric about that line.

The final section covers inequalities in triangles: the side opposite the greater angle is longer, and the sum of any two sides of a triangle is always greater than the third side (the Triangle Inequality). These results are used both in direct proofs and in comparison problems, and they form a common source of 4–5 mark questions in school and board examinations.

Topics

What's inside Chapter 7

As per NCERT Class 9 Mathematics (CBSE syllabus)

Topic 1
Congruence of Triangles
Introduction to the concept of congruence. Two triangles are congruent if all six corresponding parts (three sides and three angles) are equal. Notation: △ABC ≅ △DEF with vertex correspondence.
Topic 2
Criteria for Congruence — SAS, ASA, AAS
SAS: two sides and the included angle are equal. ASA: two angles and the included side are equal. AAS: two angles and a non-included side are equal. Proof and application of each criterion.
Topic 3
Criteria for Congruence — SSS and RHS
SSS: all three pairs of corresponding sides are equal. RHS: applicable only to right-angled triangles — right angle, hypotenuse, and one side are equal. Understanding when each criterion applies.
Topic 4
Properties of Isosceles Triangles
Theorem: angles opposite equal sides are equal. Converse: sides opposite equal angles are equal. The perpendicular bisector of the base, altitude from apex, median, and angle bisector all coincide in an isosceles triangle.
Topic 5
Inequalities in a Triangle
The side opposite the greater angle is longer. The sum of any two sides of a triangle is greater than the third side (Triangle Inequality). Application in comparing sides and angles given partial information about a triangle.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Ch 6 · Lines and Angles
Vertically opposite angles, alternate interior angles, linear pairs
Class 8 · Understanding Shapes
Basic angle-sum property, types of triangles, properties of quadrilaterals
Chapter 7 Triangles
Leads to
Ch 8 · Quadrilaterals
Proofs about parallelograms use triangle congruence extensively
Class 10 · Triangles (Ch 6)
Similarity criteria, Basic Proportionality Theorem, Pythagoras Theorem
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 9 school and board exam papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Objective 1 1–2 Identify the correct congruence rule, or apply triangle inequality
Very Short Answer 2 1 State a theorem, find missing angle/side using isosceles property
Short Answer 3 1 Prove two triangles congruent and deduce an equality
Long Answer / Proof 4–5 1 Multi-step proof using congruence + isosceles or inequality result
Total (approximate) 5–8 4–5 Weightage varies across school paper sets and exam terms
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 9 Maths Chapter 7. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
In △ABC and △PQR, AB = QR, BC = PR, and CA = PQ. Which congruence rule applies? (a) SAS (b) ASA (c) SSS (d) RHS
Q2 Easy 2 marks Short Answer
In an isosceles triangle ABC, AB = AC. The bisector of angle A meets BC at D. Prove that BD = DC.
Q3 Medium 2 marks Short Answer
In △ABC, angle B = 50° and angle C = 70°. Which is the longest side of the triangle? Give a reason.
Q4 Medium 3 marks Short Answer
Line segment AB is bisected at point O. Two line segments CD and EF bisect AB at O. Prove that C, O, F are collinear.
Q5 Medium 3 marks Proof
In the figure, PA ⊥ AB, QB ⊥ AB, and PA = QB. Prove that △OAP ≅ △OBQ and hence show that O is the midpoint of PQ.
Q6 Hard 4 marks Proof
In △ABC, the bisectors of angles B and C intersect at point O. Prove that angle BOC = 90° + (1/2) angle A.
Q7 Hard 4 marks Proof
AB is a line segment and P is its midpoint. D and E are points on the same side of AB such that angle BAD = angle ABE and angle EPA = angle DPB. Prove that △DAP ≅ △EBP and hence AD = BE.
Q8 Hard 5 marks Word Problem
In the figure, △ABC is an isosceles triangle with AB = AC. D is a point on BC such that AD ⊥ BC. (i) Prove that △ABD ≅ △ACD using the RHS criterion. (ii) Hence prove that angle BAD = angle CAD (i.e., AD bisects angle A). (iii) Is BD = DC? Justify your answer.
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 9 Maths exam papers — Triangles chapter.

Board 20223 marks
In △ABC, D is the midpoint of BC. DE ⊥ AB and DF ⊥ AC, where E is on AB and F is on AC. Prove that BE = CF. (CBSE SA-II 2022)
Board 20232 marks
In triangles ABC and PQR, angle A = angle Q and angle B = angle R. Which side of △PQR should be equal to side AB of △ABC so that the two triangles are congruent? Give reasons. (CBSE 2023)
Board 20204 marks
In a triangle ABC, E is the midpoint of median AD. Prove that ar(△BED) = (1/4) ar(△ABC). (Note: this question combines Chapter 7 congruence reasoning with Chapter 9 area concepts.) (CBSE 2020)
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FAQ

Questions teachers ask

How many marks does Triangles carry in the CBSE Class 9 Maths exam? +
Triangles (Chapter 7) typically carries 5–8 marks in CBSE Class 9 annual and term exams. Expect one 1-mark MCQ or assertion-reason, one 2-mark short answer on a congruence criterion or property, and one 4–5 mark proof-based question. Congruence criteria and the inequality theorems are the highest-frequency subtopics.
Which congruence criteria are most important for the CBSE Class 9 Maths exam? +
All five criteria — SAS, ASA, AAS, SSS, and RHS — are examinable. SAS and ASA appear most frequently in board and school exams because they lead naturally to proof-style questions. RHS is exclusively for right-angled triangles and is a common trap option in MCQs. Students should be able to state, apply, and justify each criterion.
What is the difference between AAS and ASA congruence in CBSE exams? +
In ASA (Angle-Side-Angle), the known side is between the two known angles. In AAS (Angle-Angle-Side), the known side is not between the two angles — it is adjacent to only one of them. Both are valid congruence criteria in the NCERT Class 9 syllabus. Board questions sometimes ask students to identify which rule applies given a diagram, so understanding the positional difference is essential.
How should students approach proof questions on Triangles in CBSE Class 9? +
Start by marking all given information on the diagram. Identify which congruence rule can be established (usually SAS, ASA, or SSS). Write the proof in a stepwise column format: State → Reason. Every step must cite a theorem, definition, or axiom. The final step must name the congruence rule used (e.g., "By SAS, triangle ABC ≅ triangle DEF"). Partial marks are given for each valid step, so always show full working.
How do I generate a custom question paper for Triangles using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 9 Mathematics, select Chapter 7 (Triangles), set your preferred question-type mix (MCQ, short answer, proof) and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.