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 the quadrilateral lie on a circle.

The sum of either pair of opposite angles of a cyclic quadrilateral is 180º. 

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)

Thus, ∠AED = ∠CBA....................(1)

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

∠ADE = ∠CBA................... (2)

From (1) and (2),

∠AED = ∠ADE

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

Hence proved.

☛ Check: NCERT Solutions Class 9 Maths Chapter 10

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.

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