Question

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

 If opposite angles of a quadrilateral are equal, then it is necessarily a ....................

Options

• parallelogram

•  rhombus

•  rectangle

Answer 1

 If opposite angles of a quadrilateral are equal, then it is necessarily a parallelogram.

Reason:

ABCD is a quadrilateral in which  ∠A =∠C and ∠B = ∠D.

We need to show that ABCD is a parallelogram.

In quadrilateral ABCD, we have

 ∠A = ∠C

∠B = ∠D

Therefore,

∠A + ∠B = ∠C + ∠D …… (i)

Since sum of angles of a quadrilateral is  360°

∠A + ∠B = ∠C + ∠D = 360°

From equation (i), we get:

(∠A + ∠B) + ( ∠A + ∠B) = 360°

                  2(∠A +∠B ) = 360°

                      ∠A + ∠B =  180°

Similarly,  ∠C + ∠D = 180°

Now, line AB intersects AD and BC at A and B respectively

Such that  ∠A +∠B = 180°

That is, sum of consecutive interior angles is supplementary.

Therefore, .AD || BC

Similarly, we get AB || DC.

Therefore, ABCD is a parallelogram.