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Question

In given figure ABCD is a trapezium in which ABCD and AD=BC. Show that
(i) A=B
(ii) C=D
(iii) ABCBAD
(iv) diagonal AC= diagonal BD
463886.jpg

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Solution

Given that ABCD is a trapezium,

ABCD and AD=BC

To prove: (i) A=B
(ii) C=D
(iii) ABCBAD
(iv) AC=BD

Construction: Draw CEAD and extend AB to intersect CE at E.

Poof:

(i) As AECD is a parallelogram.

By construction, CE is parallel to AD and AE is parallel to CD,

AD=EC

But, AD=BC (Given)

BC=EC

3=4 (Angle opposite to equal side are equal)

Now, 1+4=1800 (consecutive interior angle of parallelogram )

And 2+3=1800 (linear pair )

1+4=2+3 1=23=4 (so get cancelled with each other )

A=B


(ii) ABCD

A+D=B+C=180o [Angles on the same side of transversal]

But, A=B

Hence, C=D

(iii) In ABC and BAD, we have

BC=AD (Given)
AB=BA (Common)

A=B (Proved)

Hence, by SAS congruence criterion,

ABCBAD

(iv) We have proved that, ABCBAD

AC=BD (CPCT)

Hence, all the four required results have been proved.

2105599_463886_ans_2d93e317545e48c0bc67d3da3736bc03.png

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