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