Question

"Question 10 ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (See the given figure). Show that (i) ΔAPB ≅ ΔCQD (ii) AP = CQ

Class 9 - Math - Quadrilaterals Page 147"

Answer 1

Answer:

In the given quadrilateral ABCD,

AP and CQ are perpendiculars.

In the triangle APB and triangle CQD,

as,

AB is parallel to CD.

∠ABP = ∠CDQ (Alternate interior angles)

AB = CD (opposite sides of a parallelogram are equal)

and,

∠APB = ∠CQD = 90° (Right Angles)

Therefore,

ΔAPB ≅ ΔCQD (By ASA Congruence)

Hence, Proved.

(ii).

As,

ΔAPB ≅ ΔCQD. So, from the previous part we can say that,

The corresponding sides of the triangle are also equal.

i.e.

AP = CQ (By CPCT)

Hence, Proved.

Answer 2

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

To Find : Show that (i) ΔAPB ≅ ΔCQD (ii) AP = CQ

Solution:

Compare ΔAPB and  ΔCQD

AB = CD   opposite sides

∠APB = ∠CQD  = 90°

∠ABP = ∠CDQ    (∵ ∠ABD = ∠CDB  alternate interior angle  as AB || CD )

=> ΔAPB ≅  ΔCQD  ( RHS)

QED

=>  AP = CQ   ( Corresponding sides of congruent triangles are equal )

QED

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