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Question

ABCD is a trapezium in which ABCD and AD=BC (see figure). Show that
(i) A=B
(ii) C=D
(iii) ABCBAD
(iv) diagonal AC= diagonal BD
1177783_2723aa13532049e39375d72552aba78f.png

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Solution

Given ABCD is trapezium where AD=BC.

(i) To prove: A=B

we can see that AECD is a parallelogram, so sum of adjacent angles =180o

A+E=180o

A+x=180o

A=180ox=B

Hence proved.

(ii) To prove: C=D

sum of adjacent angles in parallelogram is π, so

DC+180o2x=180o

C+D=2x

Now

B+C=180o

180ox+C=180o=0 C=x, so
D=x
And,

C=D

Hence proved.

(iii) ΔABC=ΔBAD

side AB is common.

AD=BC (given)

so the angle including both the sides is also same,

A=B. So

ΔABC=ΔBAD (By SAS congruent Rule)

Hence proved.

(iv) As ΔABC=ΔBAD

The third side of both triangles i.e. diagonals are equal AC=BD

1048998_1177783_ans_428555dce57a4ecfb4b89596a00f8eec.png

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