Question

Prove Theorem 8.5 of Grade 9.
'If In a quadrilateral, each pair of opposite angles is equal then it is a parallelogram."

Answer 1

Given:

ABCD is a quadrilateral with equal opposite angles

∠A =∠C  

∠B = ∠D

To prove:

ABCD is a parallelogram

Solution:

Sum of angles of Quadrilateral = 360°

= ∠A+∠B∠C+D = 360°

Since ∠A =∠C and ∠B = ∠D,

Therefore,

∠A+∠D+∠A+∠D =360°

2∠A +2∠D = 360°

2∠A = 2∠D =360°

∠A+∠D = 180°

and AB II DC ( Interior angles on same side of transversal are supplementary )

Similarly,

∠A+∠B = 180°

and AD II BC

Thus, in ABCD both the pairs of opposite sides are parellel

= AB II DC and AD II BC

Hence ABCD is a parallelogram

Answer 2

hope the pinned image helps u