Question

In the given figure, ABCD is a parallelogram and PD is parallel to QC.

S1 : $△$APD $\cong$ $△$BQC

S2 : ar ($△$APD) + ar (PBCD) = ar ($△$BQC) + ar (CDPB)

Choose the correct option and justify your choice.
(a) S1 is true but S2 is false

(b) S1 is false but S2 is true

(c) S1 and S2 are both true and S2 is the explanation for S1

(d) S1 and S2 are both true and S1 is the explanation for S2

[ 3 marks ]

Answer 1

In $△$APD and $△$BQC

$\angle PAD=\angle QBC$ (corresponding angles)

AD = BC ( opp. sides of parallelogram)

$\angle APD=\angle BQC$ (corresponding angles)

$△APD\cong △BQC$ by ASA criteria

And ar($△$APD) = ar($△$BQC)
(congruent triangles have equal areas)

[ 1.5 marks ]

Adding ar(PBCD) on both sides of the equation

ar($△$APD) + ar(PBCD) = ar($△$BQC) + ar(PBCD)

[ 1.5 marks ]

Therefore, both S1 and S2 are true and S1 is the explanation of S2.