Question

In figure, ABCD is a parallelogram and lines AQ and CP are the bisector of ∠A – ∠C respectively. Prove the APCQ is a parallelogram.

Answer 1

Given: ABCD is a parallelogram in which AQ and CP are the bisectors of ∠A and ∠C.

To prove: APCQ is a parallelogram.

Proof: In a parallelogram ABCD

∠A = ∠C

(opposite angles of a parallelogram)

\(\frac { 1 }{ 2 }\)∠A = \(\frac { 1 }{ 2 }\)∠C

∠PAQ = ∠PCQ

(as AQ bisect ∠A and PC bisect ∠C)

AB || CD

⇒ AP || CQ

APCQ is a parallelogram.

Hence proved.