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$\left(CDPB\right)$

• 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 are both true and S1 is the explanation for S2

Answer 1

The correct option is D

S1 and S2 are are both true and S1 is the explanation for S2

In $△APD$ and $△BQC$

$\angle PAD=\angle QBC$ [corresponding angles]

$AD=BC$ [opposite 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]

Adding ar$\left(PBCD\right)$ on both sides of the equation

ar($△APD)+$ ar$\left(PBCD\right)=$ ar($△BQC)+$ ar$\left(PBCD\right)$

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