`A B C D` is a parallelogram andline segments `A X ,C Y` bisect the angles `A` and `C ,` respectively Figure. Show that `A X C Ydot` Figure

Explanation

Given: `ABCD` is a parallelogram and line segments `AX`, `CY` bisect the angles `A` and `C` respectively.
To Prove: `AX || CY`
Proof: `ABCD` is a parallelogram
`∴ /_A=/_C` [Opposite `∠s`]
`1/2/_A=1/2/_C` [ `∵` Halves of equals are equal]
`/_1=/_2` ....(1) [`∵ AX` is the bisector of `/_A` and `CY` is the bisector of `/_C`]
Now, `AB || DC` and `CY` intersects them
`:./_2=/_3` ...(2) [Alternate interior `/_s`]
From (1) and (2), we get:
`/_1=/_3`
But these form a pair of equal corresponding angles
`∴ AX || CY`
Hence proved.