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.
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

Question

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.

A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

KnowledgeBoat · Accepted Answer

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, = 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.

Mathematics

Rectilinear Figures

Related Questions

Assertion (A): In the given figure, ABCD is a rectangle and DCEF is a parallelogram then ABEF is a rectangle.
Reason (R):
AB = DC and DC = FE
⇒ AB = FE and so ABEF is a rectangle.
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

Mathematics

Rectilinear Figures

Related Questions

Assertion (A): In the given figure, ABCD is a rectangle and DCEF is a parallelogram then ABEF is a rectangle.Reason (R):AB = DC and DC = FE⇒ AB = FE and so ABEF is a rectangle.A is true, R is false.A is false, R is true.Both A and R are true.Both A and R are false.

Assertion (A): In the given figure, ABCD is a rectangle and DCEF is a parallelogram then ABEF is a rectangle.
Reason (R):
AB = DC and DC = FE
⇒ AB = FE and so ABEF is a rectangle.
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.