Question
AB is a line-segment. AX and BY are two equal line-segments drawn on opposite sides of line AB such that AX || BY. If AB and XY intersect each other at P. Prove that:
(i) ∆APX ≅ ∆BPY
(ii) AB and XY bisect each other at P.
Solution
(i) ∵ AX || BY and AB intersects them
∴ ∠PAX = ∠PBY ...(1)
| Alternate Angles
∵ AX || BY and XY intersects them
∴ ∠PXA = ∠PYB ...(2)
| Alternate Angles
In ∆APX and ∆BPY,
∠PAX = ∠PBY | From (1)
∠PXA = ∠PYB | From (2)
AX = BY | Given
∴ ∆APX = ∆BPY | ASA Axiom
(ii) ∵ AP = BP | C.P.C.T.
and PX = PY | C.P.C.T.
⇒ AB and XY bisect each other at P.
