Q. 105.0( 3 Votes )

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.

Answer :


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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