Q10 of 13 Page 96

Two line segments PQ and RS intersect each other at X in such a way that XP=XR and ∠PSX =∠ROX. Let’s prove thatPXS ROX.


Given that two Two line segments PQ and RS intersect each other at X in such a way that XP=XR and ∠PSX =∠RQX


Now, ∠PSX=∠RQX and ∠RXQ=∠PXS (vertically opposite angles)……………(1)


In ΔRXQ,


∠RQX+∠RXQ+∠XRQ=180° ……………(2)


And in ΔPXS,


∠PXS+∠PSX+∠XPS=180° ………………(3)


Subtracting equation (2) from (3),


∠RQX+∠RXQ+∠XRQ-∠PXS-∠PSX-∠XPS=180° -180°


⇒ ∠RQX-∠PXS +∠RXQ-∠PSX +∠XRQ -∠XPS=0


⇒ ∠XRQ -∠XPS=0 (from (1))


⇒ ∠XRQ =∠XPS ……………..(4)


Now, in ΔPSX and ΔRQX,


XR=XP (given)


∠XRQ =∠XPS (from (4))


∠RXQ=∠PXS (vertically opposite angles)


∴ ΔPSX ≅ ΔRQX (by ASA rule)


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