In the given figure, AP is bisector of ∠A and CQ is bisector of ∠C of parallelogram ABCD. Prove that APCQ is a parallelogram.

Join AC.

Let AC intersect BD at point O.

As, AP is the bisector of ∠A and CQ is bisector of ∠C.

∴ ∠DAP = $\dfrac{∠A}{2}$ and ∠BCQ = $\dfrac{∠C}{2}$.

In || gm ABCD,

⇒ ∠A = ∠C (Opposite angles of || gm are equal)

⇒ $\dfrac{∠A}{2} = \dfrac{∠C}{2}$

⇒ ∠DAP = ∠BCQ.

In △ ADP and △ CBQ,

⇒ ∠DAP = ∠BCQ

⇒ AD = BC (Opposite sides of || gm are equal)

⇒ ∠ADP = ∠QBC (Alternate angles are equal)

∴ △ ADP ≅ △ CBQ (By A.S.A. axiom)

We know that,

Corresponding sides of congruent triangle are equal.

∴ DP = QB ……..(1)

We know that,

Diagonals of parallelogram bisect each other.

⇒ OD = OB …….(2)

⇒ OA = OC.

Subtracting equation (1) from (2), we get :

⇒ OD - DP = OB - QB

⇒ OP = OQ.

In quadrilateral APCQ,

⇒ OP = OQ and OA = OC.

Since, diagonals of quadrilateral APCQ bisect each other.

∴ APCQ is a parallelogram.

Hence, proved that APCQ is a parallelogram.