#### Question

If two non-parallel sides of a trapezium are equal, it is cyclic. Prove it. Or An isosceles trapezium is always cyclic. Prove it.

#### Solution

Given – ABCD is a trapezium in which AB ∥ CD and AD = BC To prove – ABCD is cyclic

Construction – draw DE ∥ BC

Proof –

DCBE is a parallelogram [by construction

∠DEB = ∠DCB [Opposite angles of parallelogram]

Also, ∠DEB = ∠EDA + ∠DAE [Exterior angle property]

In ∠ADE, ∠DAE = ∠DAE ……. (1) [since AD = BC = DE Or AD = DE]

Also, ∠DEB + ∠EDA =180° ………. (2)

From (1) and (2),

∠DEB + ∠DAE =180°

⇒ ∠DCB + ∠DAE = 180°

⇒ ∠C + ∠A = 180°

Hence ABCD is cyclic trapezium

Is there an error in this question or solution?

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If Two Non-parallel Sides of a Trapezium Are Equal, It is Cyclic. Prove It. Or an Isosceles Trapezium is Always Cyclic. Prove It. Concept: Cyclic Properties.

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