# ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see Fig. 7.34). Show that ∠BCD is a right angle.

**Solution:**

We can use the property that angles opposite to equal sides are equal and then by the angle sum property in triangle BCD we can show the required result.

In isosceles triangle ABC,

AB = AC (Given)

∴ ∠ACB = ∠ABC (Angles opposite to equal sides of a triangle are equal)

Let ∠ACB = ∠ABC be x. ----------- (1)

In ΔACD,

AC = AD (Since, AB = AD)

∴ ∠ADC = ∠ACD (Angles opposite to equal sides of a triangle are equal)

Let ∠ADC = ∠ACD be y. ----------- (2)

Thus, ∠BCD = ∠ACB + ∠ACD = x + y ----------- (3)

In ΔBCD,

∠ABC + ∠BCD + ∠ADC = 180° (Angle sum property of a triangle)

Substituting the values we get,

x + (x + y) + y = 180° [From equation (1), (2) and (3)]

2 (x + y) = 180°

2(∠BCD) = 180° [From equation(3)]

∴ ∠BCD = 90°

**Video Solution:**

## ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see Fig. 7.34). Show that ∠BCD is a right angle.

### NCERT Maths Solutions Class 9 - Chapter 7 Exercise 7.2 Question 6:

**Summary:**

If ΔABC is an isosceles triangle in which AB = AC, side BA is produced to D such that AD = AB. We have proved that ∠BCD is a right angle.