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
| ∵ 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.
