Question

3. ABCD is trapezium in which AB || CD. If AD = BC, show that angleA = angleB and
angleC = angleD.​

Answer 1

Answer:

Given:- ABC isi a trapezium where AB∥CD and AD=BC

To prove:- ∠A=∠B

Construction:- Extend AB and draw a line through C parallel to AD intersecting AB produced at E.

Proof:-

AD∥CE(Fro construction)

AE∥CD(As AB∥CD,&AB produced at E)

In AECD, both pairs of opposite sides are parallel.

∴AECD is a parallelogram.

∴AD=CE..(1)(∵Opposite sides of a parallelogram are equal)

AD=BC..(2)(Given)

From equation (1)&(2), we have

∴BC=CE

⇒∠CEB=∠CBE..3)(∵Angle opposite to equal sides are equal)

Now, for AD∥CE and AE is transversal,

∠A+∠CEB=180°

⇒∠A=180°−∠CEB...(4)

Also AE is a line,

∠B+∠CBE=180°(Linear pair)

⇒∠B+∠CEB=180°(From (3))

⇒∠B=180°−∠CEB..(5)

Now, from equation (4)&(5), we get

∠A=∠B

Hence proved.