Question

Prove theorem 8.5 of class 9 NCERT.

Answer 1

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Answer 2

Theorem 8.5 : If in quadrilateral, each pair of opposite angle are equal, the it’s a parallelogram.

Given Data : PQRS is a Quadrilateral, in which Opposite angles are equal.

∠1 = ∠3 ; ∠2 = ∠4

Solution (Proof) : We all know the sum of all four angles of Quadrilateral is 360.

∠1 + ∠2 + ∠3 + ∠4 = 360

(∠1 = ∠3 ; ∠2 = ∠4)

∠1 + ∠2 + ∠1 + ∠2 = 360

2 (∠1 + ∠2) = 360

(∠1 + ∠2) = 180

So,

∠1 + ∠4  and ∠2 + ∠3 is also equal to 180.

Line PS and QR:

Interior angle on same side of transverse are supplementary

So, PS || QR

Line PQ and SR :  

Interior angle on same side of transverse are supplementary

So, PQ || SR

If Both pairs of opposite sides are parallel, then PQRS is a Parallelogram.

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