Angles of Isosceles and Equilateral Triangles

Class 7 Maths · Chapter 9: Geometric Twins · NCERT Ganita Prakash section P2 1.3

Learn

Take △ABC with AB = AC and ∠A = 80°. Draw the altitude AD to BC. Now △ADB and △ADC have a right angle at D, equal hypotenuses AB = AC, and the shared side AD: by RHS they are congruent. So ∠B = ∠C.

AB = AC → ∠B = ∠C

In an isosceles triangle, the angles opposite the equal sides are equal. With ∠A = 80°, ∠B + ∠C = 100°, so each is 50°. Going the other way, if the equal angles are 35° each, the third angle is 180° − 70° = 110°.

(180° − 80°) ÷ 2 = 50°

Equilateral triangles: AB = AC gives ∠B = ∠C, and AB = BC gives ∠A = ∠C. So all three angles are equal, and since they add to 180°, each is 60°. We found this just from congruence, without measuring!

3 × 60° = 180°

In a circle: if A is the centre and B, C are on the circle, AB and AC are both radii, so △ABC is isosceles. With ∠A = 120°, each of ∠B and ∠C is 30°. Congruent triangles appear everywhere: in bridges like Howrah's Rabindra Setu, in domes, pyramids and rangoli patterns.

(180° − 120°) ÷ 2 = 30°

Explore: isosceles angles.

1. AB = AC and ∠A = 80°. Open the angle to ∠B.
2. Open the angle to one angle of an equilateral triangle.
3. A is a circle's centre; B, C are on it; ∠A = 120°. Open to ∠B.

Remember

• Angles opposite equal sides are equal.
• Each angle of an equilateral triangle is 60°.

Quick check

In △PQR, PQ = PR and ∠Q = 70°. What is ∠P?

1. 40°
2. 70°
3. 55°
4. 110°

Think about it

Is the converse true: if ∠B = ∠C in a triangle, must AB = AC?

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.