Show that tangent drawn at the end point of a diameter are parallel to

1.	Show that tangent drawn at the end point of a diameter are parallel to each other
Answer» Given: PQ is a diameter of a circle with centre O. The lines AB and CD are the tangents at P and Q respectively.\xa0To prove: AB\xa0{tex}\\parallel{/tex} CDProof: AB is a tangent to the circle at P and Op is the radius through the point of contact{tex}\\because{/tex}{tex}\\angle{/tex}OPA = 90o .......(1) [The tangent at any point of a circle is perpendicular to the radius through the point of contact]{tex}\\because{/tex}\xa0CD is a tangent to the circle at Q and OQ is the radius through the point of contact.{tex}\\therefore{/tex}\xa0{tex}\\angle{/tex}OQD = 90o ........(2) [The tangent at any point of a circle is perpendicular to the radius through the point of contact]From (1) and (2),{tex}\\angle{/tex}OPA = {tex}\\angle{/tex}OQDBut these form a pair of equal alternate angles.{tex}\\because{/tex}\xa0AB\xa0{tex}\\parallel{/tex} CD

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