Question

$\text{In the given figure, ABCD is a square and P is a point inside it such that PB = PD. Prove tht CPA is a straight line.}$

Answer 1

In △PAD and △PAB, we have:

AD=AB (Side of a square)

AP=AP (Common)

⇒∠APD=∠APB

PD =PB (Given)

△PAD≅PAB (SSS criterion)

In △CPD and △CPB, we have:

CD = CB (Sides of square)

CP=CP (Common)

PD=PB (Given)

△CPD≅△CPB (SSS test)

⇒∠CPD=∠CPB

∴∠APD+∠CPD=∠APB+∠CPB

But

∠APD+∠CPD+∠APB+∠CPB=360°

⇒∠APD+∠CPD=180°

So, CPA is a straight line.

Hence, proved.