# 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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