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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