Question

In the quadrilateral ABCD, AB = CD and ∠CAD = 35°. Find ∠ACB and prove that ABCD is an isosceles trapezium.

Answer 1

Given:

• Quadrilateral ABCD is inscribed in a circle.
• AB = CD
• ∠CAD = 35°

To Prove:

• Find ∠ACB
• Prove that ABCD is an isosceles trapezium.

Proof:

1. Since AB = CD, chords AB and CD are equal.
2. Angles subtended by equal chords in the circle are equal. Therefore, ∠CAD = ∠ACB = 35°.
3. ∠CAD and ∠ACB are alternate interior angles of lines AD and BC. Hence, AD || BC.
4. Since AD || BC and ABCD is a cyclic quadrilateral, it forms a trapezium.
5. Given that AB = CD, the trapezium is isosceles.

So,

• ∠ACB = 35°
• ABCD is an isosceles trapezium because AB = CD and AD ∥ BC proved by alternate interior angles.