ABCD is a parallelogram and line segments AX . . .

Question : In given figure, AX and

CY are respectively the bisectors of the

opposite angles A and C of a parallelogram

ABCD. Show that AX || CY

OR

ABCD is a parallelogram and line segments AX, CY bisect the angles A and C respectively. Show that AX || CY.

Doubt by Saksham

Solution :

Given : ABCD is a parallelogram. AX and CY are the angle bisectors of ∠A and ∠C respectively.

To Prove : AX || CY

Proof : ABCD is ||gm (Given)
∠1=∠A/2 (Given)
∠A=2∠1 — (1)

∠2=∠C/2 (Given)
∠C=2∠2 — (2)

∠A=∠C (Opposite angles of a parallelogram are equal) 2∠1=2∠2 [From equation (1) and (2)] 
∠1=∠2 — (3) In Quadrilateral AXCY ∠1=∠2 [Proved above in equation (3)]

⇒ AXCY is a ||gm.

[If opposite angles of a quadrilateral are equal then it is a parallelogram]

Now, AX||CY [opposite sides of a parallelogram are parallel]

Hence proved.