Question

$AB$ is a line segment. $AX$ and $BY$ are two equal line segments drawn opposite sides of line $AB$such that $AX\left|\right|BY$. If line segments $AB$ and $XY$intersect each other at point $P$. Prove that
(a)$\Delta APX\cong \Delta BPY$
(b) line segments$AB$ and $XY$bisect each other at $P$.

Answer 1

Given$AX=BY$

$AX$ parallel to $BY$

$XY$cuts $AB$ at $P$.

In $∆AXP$ and $∆PYB$

$AX=BY$ (given)

$\angle AXP=\angle PYB$ (alternate interior angles)

$\angle XAP=\angle YBP$ (alternate interior angles)

So $\Delta APX\cong \Delta BPY$

Therefore $AP=BP$ and $XP=PY$

Hence the line segment $AB$ and $XY$ bisect each other.