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

Answer:

Step-by-step explanation:

If opposite angles of a quadrilateral is equal then it is a parallelogram.

Given that ABCD is a quadrilateral whose opposite angles are equal

i.e. ∠A =∠C and ∠B = ∠D

To prove: ABCD is a parallelogram

Proof:

As we know that sum of angle of Quadrilateral is 360°

⇔∠A+∠B∠C+D = 360°

∵ ∠A =∠C and ∠B = ∠D , ⇔∠A+∠D+∠A+∠D =360°

⇔2∠A +2∠D = 360°

⇒2∠A = 2∠D =360°

⇔∠A+∠D = 180° ( Co-interior angle)

⇔AB║DC

Similarly

∠A+∠B+∠C+∠D = 360°

⇔∠A+∠B+∠A+∠B = 360° (∠A = ∠C and ∠B = ∠D)

⇔2∠A+2∠B = 360°

⇔∠A+∠B = 180°

⇔ AD║BC

therefore AB║DC and AD║BC

hence ABCD is a parallelogram..