In Figure-7, $XY$ and $MN$ are two parallel tangents to a circle with centre $O$ and another tangent $AB$ with point of contact $C$ intersecting $XY$ at $A$ and $MN$ at $B$. Prove that $\angle AOB = 9

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In Figure-7, $XY$ and $MN$ are two parallel tangents to a circle with centre $O$ and another tangent $AB$ with point of contact $C$ intersecting $XY$ at $A$ and $MN$ at $B$. Prove that $\angle AOB = 90°$.

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Accepted Answer

Step 1: Join $OC$ (in given figure). $	riangle APO \cong 	riangle ACO$ [By RHS congruence]
Step 2: $	herefore \angle OAP = \angle OAC = x$ (let). Similarly, $	riangle OQB \cong 	riangle OCB$ $	herefore \angle OBC = \angle OBQ = y$ (let)
Step 3: $\because XY \parallel MN$. $	herefore \angle PAB + \angle ABQ = 180°$. $\Rightarrow 2x + 2y = 180° \Rightarrow x + y = 90°$. $	herefore \angle AOB = 180° - (x + y) = 90°$

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