Chapter 7 · Class IX Mathematics · NCERT Exercises

Triangles — Exercises

Congruence, Properties & Inequalities — All 3 Exercises Fully Solved

📂 3 Exercises 📝 21 Questions 🎓 High

▶ Start Exercises 📖 Revise Notes

Exercise Index

Congruence criteria: SAS, ASA, AAS, SSS, RHS; CPCT; prove two triangles congruent

Properties of isosceles triangle; angles opposite equal sides; angle bisector; altitude proofs

8 Q Moderate-Hard

Inequalities in triangles; side opposite larger angle; sum of two sides > third side

3 exercise files · 21 total questions

Chapter at a Glance

CBSE BoardsNTSEOlympiad

13 Concepts

9 Formulas

High Difficulty

12–14% Weightage

Before You Begin

Prerequisites

✓Basic triangle properties (Class VII)
✓Angle sum property
✓Congruence intuition

Have Ready

🔧Compass and ruler
🔧NCERT congruence criteria card
🔧Proof format template

Exercise Topic Map

Exercise 7.1 Identify which criterion applies (SAS/ASA/AAS/SSS/RHS); write ΔABC ≅ ΔPQR in correct vertex order; deduce CPCT

Exercise 7.2 Isosceles: AB=AC ⟹ ∠B=∠C (and converse); use median/altitude proofs via congruent triangles

Exercise 7.3 Larger side opposite larger angle; triangle inequality: a+b > c; use indirect proof (assume contradiction)

Key Formulae — Recall Before Solving

\(\triangle ABC \cong \triangle PQR \Rightarrow AB=PQ,\; BC=QR,\; CA=RP,\; \angle A=\angle P \text{ (CPCT)}\)

\(\text{SAS: two sides and included angle}\quad \text{ASA: two angles and included side}\)

\(\text{AAS: two angles and non-included side}\quad \text{SSS: three sides}\quad \text{RHS: right angle+hyp+side}\)

\(AB=AC \iff \angle B = \angle C \quad \text{(Isosceles Triangle Theorem)}\)

NCERT Solving Method

Step 1 — Proof format: always write Given, To Prove, Construction (if any), Proof with reasons. Step 2 — Choose criterion: look at what is GIVEN (sides/angles); match to SAS/ASA/AAS/SSS/RHS. Step 3 — Vertex correspondence: when writing ΔABC ≅ ΔPQR, ensure matching vertices are in the same position. Step 4 — CPCT: use it ONLY after proving congruence; state it explicitly in the proof. Step 5 — Inequality problems: remember the key principle — in a triangle, greater angle is opposite the greater side.

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