ABCD is a parallelogram and line segments AX, CY bisect the angles A and C, respectively. Show that AX | | CY. D x C.
Since opposite angles are equal in a parallelogram . Therefore , in parallelogram ABCD , we have
∠A = ∠C
⇒ 1 / 2 ∠A = 1 / 2 ∠C
⇒ ∠1 = ∠2 —- i)
[∵ AX and CY are bisectors of ∠A and ∠C respectively]
Now, AB | | DC and the transversal CY intersects them.
∴ ∠2 and ∠3 —- ii) [∵ alternate interior angles are equal ]
From (i) and (ii) , we have
∠1 and ∠3
Thus , transversal AB intersects AX and YC at A and Y such that ∠1 = ∠3 i.e. corresponding angles are equal .
∴ AX | | CY .