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Question

ABCD is quadrilateral in which ABCD. If AD=BC, show that A=B and C=D.


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Solution

Step 1: Drawing the diagram:

ABCD is quadrilateral in which ABCD and AD=BC.

ABCD is a trapezium in which AB || CD and AD = BC (see Fig. ? | Scholr™

Draw a line through C parallel to DA intersecting AB produced at E.

Step 2: Proving A=B and C=D:

We have, ABCD and

ADCE (By construction).

Therefore, AECD is a parallelogram.

Therefore, CE=AD (Opposite sides of a parallelogram)

AD=BC (Given)

Therefore, CE=BC

CBE=CEB (Angles opposite to equal sides are equal)

Also, A+CBE=180° (Interior angles on the same side of transversal is supplementary)

And B+CBE=180° (Linear pair axiom)

A=B

ABCD

A+D=180° [Sum of adjacent angles of parallelogram is supplementary ]

And B+C=180° (Interior angles on the same side of transversal is supplementary)

A+D=B+CD=C(A=B)

Hence proved.


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