Question 1021127: in a parallelogram ABCD,line AX and CY bisects angleA and angleC
prove that AX//CY
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! Without risking harm to the proof, I will assume that the point Y is the intersection of line CY and the side AB. Likewise with the point X, that it is the intersection of line AX and side CD.
Consider the parallelogram AYCX. Since the angles at vertices A and C are congruent, and lines Ax and CY bisect them, respectively, it follows that angle YAX and angle XCY are congruent. But sides AB and CD being parallel means that angle XCY and angle BYC are congruent.
==>angle BYC is congruent to angle YAX, by transitivity.
==> transversal AB cuts sides AX and CY at corresponding interior angles
==> Ax and CY are parallel by the transversal theorem.
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