Abcd Is Parallelogram And Line Segments Ax Cy Bisect Angles And C Respectively

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

Answers:

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 .