Question

Complete the following statement by means of one of those given in brackets against each:

If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a ...............

Options

• rectangle

• parallelogram

• rhombus

Answer 1

 If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a parallelogram.

Reason:

ABCD is a quadrilateral in which  AB = CD and BC = DA.

We need to show that ABCD is a parallelogram.

In Δ ACB and ΔCAD, we have

 AC = CA (Common)

CB = AD (Given)

 AB = CD (Given)

So, by SSS criterion of congruence, we have

 ΔACB ≅ ΔCAD

By corresponding parts of congruent triangles property.

∠CAB =∠ACD  …… (i)

And ∠ACB = ∠CAD

Now lines AC intersects AB and DC at A and C,such that

∠CAB = ∠ACD (From (i))

That is, alternate interior angles are equal.

Therefore, . AB || DC

Similarly,  AD || BC

Therefore, ABCD is a parallelogram.