# ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD

**Solution:**

A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle.

The sum of either pair of opposite angles of a cyclic quadrilateral is 180º. Opposite angles in a parallelogram are equal.

We can see that ABCE is a cyclic quadrilateral.

We know that in a cyclic quadrilateral, the sum of the opposite angles is 180°.

∠AEC + ∠CBA = 180°

∠AEC + ∠AED = 180° (Linear pair)

∠AED = ∠CBA....................(1)

We know that in a parallelogram, opposite angles are equal.

∠ADE = ∠CBA................... (2)

From (1) and (2),

∠AED = ∠ADE

AD = AE (sides opposite to equal angles in a triangle are equal).

Hence proved.

**Video Solution:**

## ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD

### Maths NCERT Solutions Class 9 - Chapter 10 Exercise 10.6 Question 6:

**Summary:**

ABCD is a parallelogram. The circle through A, B, and C intersect CD (produced if necessary) at E. We have proved that AE = AD.