# In the figure, PQRS is a trepezium in which PQ // SR and PS = QR, show that angle P = angle Q ? ( pls ans urgently and with detailed explanation)

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by vikasvini

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by vikasvini

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Let PQ is extended to T. Then, draw a line through R, which is parallel to PS, intersecting PT at point T. It is clear that PTRS is a parallelogram.

(i) PS = RT (Opposite sides of parallelogram PTRS are equal)

However, PS = QR (Given)

Therefore, QR = RT

∠RTQ = ∠RQT (Angle opposite to equal sides are also equal)

Consider the parallel lines PS and RT. PT is the transversal line for them..

∠P + ∠RTQ = 180º ( Sum of angles on the same side are equal to 180 )

∠P + ∠RQT = 180º (Using the relation ∠RTQ = ∠RQT ) ... (1)

However, ∠Q + ∠RQT = 180º (Linear pair angles) ... (2)

From equations (1) and (2), we obtain,

∠P = ∠Q.