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

#### Solution

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.