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# ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. 8.21). Show that (ii) AP = CQ

Q: 10        ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. $\small 8.21$). Show that

(ii) $\small AP=CQ$

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Given: ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A  and C on diagonal BD.

To prove : $\small AP=CQ$

Proof: In$\small \Delta APB\, \, and\, \, \Delta CQD$,

$\angle$APB=$\angle$CQD             (Each $90 \degree$)

$\angle$ABP=$\angle$CDQ            (Alternate angles)

AB=CD                (Opposite sides of a parallelogram )

Thus, $\small \Delta APB\cong \Delta CQD$               (By SAS)

$\small AP=CQ$                 (CPCT)

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