Chapter 7 · NCERT Class 9 Mathematics

Triangles

Two triangles are congruent when one fits exactly over the other. Learn the shortcuts — SSS, SAS, ASA, AAS, RHS — that prove congruence from just a few measurements, and what CPCT then lets you conclude. Tap each rule to see how it works.

📐 3 topics⏱ ~25 min📝 12-question quiz

Play with it

The congruence toolkit

You rarely need all six measurements to prove two triangles identical — the right three are enough. Tap each rule to see exactly which parts must match.

Explore · Proving congruencetap a rule

Learn

The three big ideas

Congruent — two figures with exactly the same shape and size; one can be placed over the other and cover it completely.
For triangles we write △ABC ≅ △PQR. The order matters: A ↔ P, B ↔ Q, C ↔ R show which parts correspond.
CPCT — Corresponding Parts of Congruent Triangles are Equal. Once two triangles are proved congruent, every matching side and angle is equal.

SSS — all three pairs of sides equal → congruent.
SAS — two sides and the included angle equal → congruent.
ASA — two angles and the included side equal → congruent.
AAS — two angles and a non-included side equal → congruent.
RHS — for right triangles: Right angle, Hypotenuse and one Side equal → congruent.

Worked example. Two triangles have all three pairs of sides equal. Which rule proves they are congruent?

Three pairs of equal sides matches the Side–Side–Side criterion → the rule is SSS.

Common mistake: thinking AAA (all three angles equal) proves congruence. It does not — equal angles only make the triangles similar (same shape, but possibly different size). You always need at least one pair of equal sides.

Isosceles triangle theorem — in a triangle with two equal sides, the angles opposite the equal sides are equal.
Converse — if two angles of a triangle are equal, the sides opposite them are equal, so the triangle is isosceles.
Combine these with the angle sum = 180° to find unknown angles in isosceles and equilateral triangles.

Where you'll meet it

Triangles at work

Rigid frames in bridges & towers

A triangle cannot be deformed without changing the length of a side, so triangulated trusses in bridges, cranes and transmission towers stay rigid under load — a four-sided frame would simply fold flat.

Surveying & finding distances

To measure a width they cannot reach — across a river or the height of a tower — surveyors build a congruent or similar triangle they CAN measure, then use CPCT (or similarity ratios) to read off the unknown distance.