Question

In parallelogram ABCD, PA and QC are bisectors of ∠A and ∠C respectively. Prove that APCQ is a parallelogram.

Answer 1

Consider the figure given below.

We know that opposite angles of a parallelogram are equal.

So ∠A = ∠C and ∠B = ∠D.

⇒ ∠A = ∠C

⇒ `1/2 ∠A = 1/2 ∠C`

⇒ ∠DAP = ∠BCQ

For ΔADP and ΔCBQ,

∠D = ∠B  ...[Properrty of parallelegram]

AD = CB  ...[Opposite sides of parallelegram]

∠DAP = ∠BCQ   ...[Proved above]

Therefore, by  congruency criterion, ΔADP ≅ ΔCBQ.

Now by corresponding parts of congruent triangles (CPCT),  ∠DPA = ∠CQB.

⇒ 180 – ∠CPA = 180 – ∠CQA (since DPA and ∠CPA form a linear pair and ∠CQB and ∠CQA form another linear pair)

⇒ ∠CPA = ∠CQA  ...(1)

Also as `1/2 ∠A = 1/2 ∠C`, we get ∠PAQ = ∠PCQ  ...(2)

From equations (1) and (2), we can say that opposite angles of quadrilateral AQCP are equal.

Therefore APCQ is a parallelogram. 

Hence, the given statement is proved.