`X Y`and `Xprime Yprime`are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting `X Y`at A and `XprimeYprime`at B. Prove that `/_A O B\ =\ 90^o`

Explanation

In quad. `APQB`, we have
`/_APO=90^(@)` and `/_BQO=90^(@)` [`:'` tangent at any point is perpendicular to the radius through the point of contact].
Now, `/_APO+/_BQO+/_QBC+/_PAC=360^(@)`
`implies/_PAC+/_QBC=360^(@)-(/_APO+/_BQO)=180^(@)`……`(i)`
We have
`/_CAO=(1)/(2)/_PAC` and `/_CBO=(1)/(2)/_QBC` [`:'`tangents from an external point are equally inclined to the line segment joining the centre to that point].
`:. /_CAO+/_CBO=(1)/(2)(/_PAC+/_QBC)=(1)/(2)xx180^(@)=90^(@)`.........`(ii)`
In `DeltaAOB`, we have
`/_CAO+/_AOB+/_CBO=180^(@)`
`implies/_AOB=180^(@)-(/_CAO+/_CBO)=90^(@)` [using `(i)`]