Question

other. ABCD is trapezium in which AB || CD. If AD = BC, show that angle a is equal to angle b and angle c equal to angle d​

Answer 1

(i) <A=<B

[Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.

Given: ABCD is a trapezium where AB || CD and AD = BC

To prove: LA = ZB

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

Proof:

AD || CE

(From construction)

& AE || DC

(As AB || DC, & AB is extended)

In AECD, both pair of opposite sides are parallel,

AECD is a parallelogram

:. AD = CE

(Opposite sides of parallelogram are equal)

(Given)

But AD = BC

⇒BC = CE

So, ZCEB= 2CBE

(In A BCE, Angles opposite to equal sides are equal)

...(1)

For AD || CE,

& AE is the transversal

ZA + ZCEB = 180

(Interior angle on same side of transversal is supplementary)

...(2)

Also AE is a line,

So, ZB+ZCBE = 180

(Linear pair)

ZB+ZCEB = 180

(From (1))

ZB = 180°-2CEB

...(3)

ZA = 180° - ZCEB

From (2) & (3)

ZA = LB

Hence proved.

(ii) < C=2D

For AB || CD,

For AB || CD,

& AD is the transversal

ZA + ZD = 180°

(Interior angle on same side of transversal is supplementary)

ZD=180° - ZA

... (1)

& BC is the transversal

ZB + C = 180

(Interior angle on same side of transversal is supplementary)

ZC = 180° - ZB

ZC = 180° - LA

(As LA = LB proved in (i)) ...(2)

C

From (1) & (2)

ZD = ZC

Hence proved.