Question

AB is a line segment AX and BY are two equal line segments drawn opposite sides of line AB such that AX ||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

In the given figure, segment $AX$ and $BY$ are equal and $AX||BY$

The line segment $AX||BY$ and line segment $AB$ join the parallel line at A and B.

And line segment $XY$ cut line $AB$ at point $P$

Then, in $\Delta AXP$ and $\Delta PYB$

$AX=BY$ ....... (Given)

$\angle XAP\left(3\right)=\angle YBP\left(4\right)$ (Adjacent angle )

$\angle AXP\left(1\right)=\angle PYB\left(2\right)$ (Adjacent angle )

Then $\Delta APX\cong \Delta BPY$

(B) The $\Delta APX\cong \Delta BPY$

Then $AP=BP$ and $XP=PY$

Thus, the line segment $AB$ and $XY$ bisect each other.