Question

Prove Theorem 8.5 with the diagram.
If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
DO NOT use the method by "Teachoo", I need a different method.
Thanx In Advance!

Answer 1

hey
hope it's help you

Answer 2

Given: ABCD is a quadrilateral where angleA = angleC and angleB = angleD.

To prove: ABCD is a parallelogram.

Proof:

In quadrilateral ABCD,

angleA = angleC (given)

angleB = angleD (given)

=> angleA + angleB = angleC + angleD

By Angle Sum Property of Parallelogram,

=> angleA + angleB + angleC + angleD = 360°

=> 2(angleA + angleB) = 360°

=> angleA + angleB = 360°/2

=> angleA + angleB = 180°

So,

angleA + angleB = angleC + angleD = 180°

* Lines AB intersects AD and BC at A and B respectively.

Such as,

angleA + angleB = 180°

So, AD || BC ________________________(1)

( Reason: sum of consecutive interior angle is 180°)

angleA + angleB = 180°

angleA + angleD = 180° ( angleB = angleD)

Hence, AB || CD _____________________(2)

From (1) and (2),

AB || CD and AD || BC

Therefore, ABCD is a parallelogram.