Prove PQRS is a parallelogram.?

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Prove PQRS is a parallelogram.?

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Answer 1 · 2017-11-21T02:14:13

see explanation.

Explanation:

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As shown in the figure, a transversal passes through two parallel lines $L and M$ $L and M$ ,
As $L and M$ $L and M$ are parallel, the alternate interior angles are equal, $\Rightarrow ∠ A S Q = ∠ D Q S = 2 α, and ∠ B S Q = ∠ C Q S = 2 β$ $=> angleASQ=angleDQS=2alpha, and angleBSQ=angleCQS=2beta$ ,
$S P, S R, Q P and Q R$ $SP, SR, QP and QR$ are the bisectors of the interior angles,
As $2 α + 2 β = 180^{\circ}, \Rightarrow α + β = 90^{\circ}$ $2alpha+2beta=180^@, => alpha+beta=90^@$
$\Rightarrow ∠ S P Q = ∠ Q R S = 180 - (α + β) = 90^{\circ}$ $=> angleSPQ=angleQRS=180-(alpha+beta)=90^@$
Hence, $S, P, Q, and R$ $S,P,Q,and R$ form a rectangle.

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Prove PQRS is a parallelogram.?

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