Question

show that if each pair of opposite sides of a quadrilateral is equal then it is a parallelogram

Answer 1

Answer:

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Answer 2

Consider a random quadrilateral ABCD whose each pair of opposite sides are equal (AB = CD and BC = AD). Draw any one diagonal of the quadrilateral. I draw the diagonal AC.

We have two triangles now. ΔABC and ΔCDA.

AB = CD (given)

BC = DA (given)

AC = CA (same side)

So by SSS congruence rule, ΔABC ≅ ΔCDA

Hence,

∠BAC = ∠DCA. (corresponding angles of congruent triangles are equal)

Now consider the sides AB and CD. Diagonal AC acts as a transversal. Angles ∠BAC and ∠DCA are alternating interior angles. So we can say that

AB║CD

∠BCA = ∠DAC. (corresponding angles of congruent triangles are equal)

Now consider the sides AD and BC. Diagonal AC acts as a transversal. Angles ∠BCA and ∠DAC are alternating interior angles. So we can say that

AD║BC

Since opposite sides are parallel, quadrilateral ABCD is a parallelogram.