Question

Please help me to solve question 9

Answer 1

Given data...

ABCD is a parallelogram in which BD is the diagonal on which P and Q are points taken on such that BP=BQ

To show

1.Triangle APD=~ Triangle CQB
proof...

From triangle APD and CQB we have

AB=BC (opposite side of a llgm are always equal)

angle ADP= angle QBC(alt. int. angle)
DP=BQ (gjven)

Therefore triangle APD congrueng triangle CQB by SAS congruency.

Therefore AP =CQ by CPCT...........(1)

2. AP =CQ

We have already proved AP=CQ.

AP=CQ(from euation 1)

3.Triangle AQB =~ Triangle CPD.
proof.

From triangle AQB and Triangle CPD we have

AB=CD (opposite side of a llgm always equal)

BQ=PD (given)

angle ABQ = angle CDP (alt. int. angle)

Therefore Triangle AQB =~ Triangle CPD by SAS congruency.

This implies AQ = CP by CPCT.......(2)

4.AQ=CP.

We have already proved above AQ=CP.

Therefore AQ=CP (from equation 2 )

5.APCQ is a parallelogram.

We know that parallelogram is a quadrilateral having opp sides equal .

Therefore adding equaltion 1 and 2 we get

AP=CQ

AQ=CP

Hence it is a llgm having opp sides equal.

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