Question

In the given figure, ABCD is a cyclic quadrilateral. AE is drawn parallel to CB and DA is produced. If ∠ADC = 92°, ∠FAE = 20°, determine ∠BCD.

Answer 1

Given:

ABCD is a cyclic quadrilateral.

AE ∥, CB, and DA are produced to F.

∠ADC = 92°, ∠FAE = 20°.

To find: ∠BCD

1. Use Cyclic Quadrilateral Property

Opposite angles of a cyclic quadrilateral are supplementary:

∠ADC + ∠ABC = 180°

92° + ∠ABC = 180°

∠ABC = 88°

2. Relate ∠FAE to ∠BAD

AE ∥ CB, so the angle made by AF (extension of AD) with AE equals the angle made by AD with CB:

∠FAE = angle between AD and CB.

∠BAD = interior angle between AD and AB.

Because CB is parallel to AE, and AF is the extension of AD:

Exterior angle ∠FAE equals the interior angle ∠BAD.

Therefore:

∠BAD = 20°

3. Use Property of Cyclic Quadrilateral (Opposite arcs and angles)

In the same quadrilateral ABCD:

∠BAD (20°) and ∠BCD are opposite angles of the cyclic quadrilateral.

∠BAD (20°) and ∠BCD are opposite angles of the cyclic quadrilateral.

Thus:

∠BAD + ∠BCD = 180°

20° + ∠BCD = 180°

∠BCD = 180° − 20°

= 108°