Question

In the given figure APB and CQD are two straight lines, then:

पर्याय

• AB // CD

• AC // PQ

• PQ // BD

• AC // BD

Answer 1

AC // BD

Explanation:

Let ∠BPQ = x and ∠DQP = y

We know that,

The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.

From figure,

In cyclic quadrilateral APQC,

∠A = ∠DQP = y and ∠C = ∠BPQ = x

From figure,

APB is a straight line.

∴ ∠APQ + ∠BPQ = 180°

⇒ ∠APQ + x = 180°

⇒ ∠APQ = 180° − x

CQD is a straight line.

∴ ∠CQP + ∠DQP = 180°

⇒ ∠CQP + y = 180°

⇒ ∠CQP = 180° − y

In cyclic quadrilateral PQDB,

∠B = ∠CQP = 180° − y and ∠D = ∠APQ = 180° − x

⇒ ∠A + ∠B = y + (180° − y) = 180°

⇒ ∠C + ∠D = x + (180° − x) = 180°

We know that,

Sum of adjacent angles in a trapezium is 180°

∴ ABDC is a trapezium.

∴ AC || BD