Congruent figures
Two figures are congruent if they have exactly the same shape and size, so one can be placed on the other to cover it exactly. Two photocopies of the same picture, or two ₹10 coins, are congruent.
Two triangles are congruent if their three sides and three angles are equal in pairs. We write △ABC ≅ △PQR. The order of the letters matters: it tells us that A ↔ P, B ↔ Q and C ↔ R, so AB = PQ, ∠B = ∠Q and so on.
Once two triangles are known to be congruent, every pair of matching parts is equal. This is written CPCT: corresponding parts of congruent triangles.
The congruence rules
You don't need all six equalities. Any of these is enough:
Rule | What must be equal |
|---|---|
SAS (side-angle-side) | two sides and the angle between them |
ASA (angle-side-angle) | two angles and the side between them |
AAS (angle-angle-side) | two angles and any side in the same position |
SSS (side-side-side) | all three sides |
RHS (right angle-hypotenuse-side) | in right triangles, the hypotenuse and one other side |
SAS is taken as an axiom; the others are proved from it. AAS follows from ASA because if two angles are equal, the third angles are equal too (angle sum is 180°).
What doesn't work
SSA (two sides and an angle that is not between them) does not guarantee congruence. Two different triangles can share those measurements.
AAA (three equal angles) gives triangles of the same shape but possibly different sizes. They are similar, not necessarily congruent.
Isosceles triangles
A triangle with two equal sides is isosceles.
Theorem. Angles opposite equal sides of a triangle are equal.
Proof. In △ABC, let AB = AC. Draw the bisector of ∠A meeting BC at D.
In △BAD and △CAD: AB = AC (given), ∠BAD = ∠CAD (by construction), AD = AD (common).
So △BAD ≅ △CAD (SAS), and therefore ∠B = ∠C (CPCT). ∎
Converse. Sides opposite equal angles of a triangle are equal.
Consequences:
Each angle of an equilateral triangle is 60°.
In an isosceles triangle, the bisector of the angle between the equal sides is also the perpendicular bisector of the base.
A worked proof
Problem. In the figure, AB = AD and AC bisects ∠BAD. Prove that BC = DC.
Proof. In △ABC and △ADC:
AB = AD (given)
∠BAC = ∠DAC (AC bisects ∠BAD)
AC = AC (common side)
So △ABC ≅ △ADC by SAS, and BC = DC by CPCT. ∎
A worked RHS problem
Problem. AD is the altitude of an isosceles triangle ABC in which AB = AC. Show that AD bisects BC.
Proof. In right triangles △ADB and △ADC: hypotenuse AB = AC (given) and AD = AD (common). So △ADB ≅ △ADC by RHS, and BD = DC by CPCT. ∎
How to write a congruence proof
Name the two triangles you will compare, with vertices in matching order.
List three pairs of equal parts, giving a reason for each: given, common, vertically opposite, alternate angles, and so on.
State the rule (SAS, ASA, …) and conclude congruence.
Use CPCT for the final result.
Summary
△ABC ≅ △PQR means all matching sides and angles are equal; the vertex order shows the matching.
Valid rules: SAS, ASA, AAS, SSS, RHS. SSA and AAA are not valid.
Angles opposite equal sides are equal, and conversely.