Angles opposite equal sides are equal

The idea

In a triangle, the angles opposite to equal sides are equal, so the two angles opposite the equal sides of an isosceles triangle are equal.

A frame with two equal sides

The school gate has a triangular frame ABC with two equal sides, AB = AC, and the top angle ∠A = 80°. Meera wonders about the two angles at the bottom, ∠B and ∠C. What can we say about them?

How are ∠B and ∠C related?

• They are equal
• The bigger side has the smaller angle
• We cannot say

Fold the triangle

Swing the top angle A to three different sizes. AB and AC stay equal. Watch ∠B and ∠C. Then drop the line AD and fold the triangle.

Equal sides, equal angles

∠B and ∠C were equal every time. The line AD makes two right-angled triangles with equal hypotenuses (AB = AC) and the shared side AD, so they are congruent by RHS. Folding puts B exactly on C, so ∠B = ∠C. With ∠A = 80°, each of them is 50°.

In a triangle, the angles opposite to equal sides are equal. In an isosceles triangle with AB = AC, the angles ∠B and ∠C opposite them are equal.

• ∠A | ∠B | ∠C
• 80° | 50° | 50°
• 40° | 70° | 70°

Notes

In a triangle, the angles opposite to equal sides are equal. In an isosceles triangle with AB = AC, the angles ∠B and ∠C opposite them are equal.

Check yourself

∆ABC is isosceles with AB = AC and ∠A = 80°. What is ∠B, in degrees?

Answer: 50

∠B = ∠C and ∠A + ∠B + ∠C = 180°. So ∠B + ∠C = 100°, and each is 50°.

In ∆PQR, PQ = PR. Which two angles are equal?

In ∆LMN, LM = LN and ∠L = 40°. What is ∠M, in degrees?

Answer: 70

∠M + ∠N = 180° − 40° = 140°. They are equal, so ∠M = 70°.

In ∆ABC, AB = BC and ∠B = 50°. What is ∠A?

• ∠P and ∠Q. The equal angles are opposite the equal sides. PQ is opposite ∠R and PR is opposite ∠Q.
• ∠Q and ∠R — correct. Yes! PR is opposite ∠Q and PQ is opposite ∠R, so ∠Q = ∠R.
• ∠P and ∠R. The equal angles are opposite the equal sides. PQ is opposite ∠R and PR is opposite ∠Q.
• 65° — correct. Yes! AB = BC, so ∠C = ∠A. They share 180° − 50° = 130°, so each is 65°.
• 130°. 130° is 180° − 50°. That is ∠A and ∠C together.
• 50°. 50° is ∠B. The equal angles are opposite the equal sides AB and BC.