Question

Prove that if both pairs of opposite angles of a quadrilateral are equal, then it is a parallelogram. [3 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark
Proof : 1 Mark

Consider a parallelogram $ABCD$ with both pairs of opposite angles equal.

We know that, $2x+2y=360°$ [By angle sum property]

$\Rightarrow x+y=180°$

i.e., $\angle A+\angle B=x+y=180°$

So, $AD\parallel BC$.......(i) [Since the co interior angles are supplementary]

Similarly, $\angle A+\angle D=x+y=180°$

So, $AB\parallel DC$.......(ii) [Since the co interior angles are supplementary)]

From (i) and (ii) both opposite sides of $ABCD$ are parallel.

$\Rightarrow ABCD$ is a parallelogram.