Assertion (A): ABCD is a parallelogram. AX is bisector of ∠A, CY is bisector of ∠C. Then quadrilateral AXCY is also a parallelogram.

Reason (R): If any one pair of opposite sides of a quadrilateral are equal and parallel then the quadrilateral is a parallelogram.

1. A is true, R is false.
2. A is false, R is true.
3. Both A and R are true.
4. Both A and R are false.

Both A and R are true.

Explanation

Given,

ABCD is a parallelogram.

The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y as shown in the figure below:

∠ A = ∠ C [∵ opposite angles in ||gm are equal]

Dividing by 2 on both sides,

$\dfrac{∠ A}{2}$ = $\dfrac{∠ C}{2}$

From figure,

∠1 = ∠2 ……….[1]

Since, AB || CD with CY as transversal,

∴ ∠2 = ∠3 [∵ Alternate angles]

∴ ∠1 = ∠3 [From 1]

As corresponding angles are equal,

∴ AX || YC

In parallelogram ABCD,

AB || DC

∴ AY || XC

As, AX || YC and AY || XC,

the opposite sides of the quadrilateral AXCY are parallel

∴ AXCY is a parallelogram [∵ Opposite sides are parallel]

∴ Assertion (A) is true.

Given : ABCD is a quadrilateral such that AB = DC and AB is parallel to DC.

To Prove : ABCD is a parallelogram.

Proof : Join A and C.

In Δ ABC and Δ CDA,

AB = CD (Given)

∠ BAC = ∠ DCA [∵ Alternate angles]

AC = AC [∵ Common side]

By SAS Congruency criterion,

Δ ABC ≅ Δ CDA

∴ ∠ BCA = ∠ DAC [∵ C.P.C.T.C.]

But, ∠ BCA and ∠ DAC are alternate angles

∴ AD || BC [∵ Alternate angles are equal]

∴ ABCD is a parallelogram [∵ Opposite sides are parallel]

∴ Reason (R) is true.

Hence, both Assertion (A) and Reason (R) are true.