Prove that tangents at the ends of a diameter of a circle are parallel

1.	Prove that tangents at the ends of a diameter of a circle are parallel.
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.To Prove: AB {tex}\\parallel{/tex} CDProof: Since AB is a tangent to the circle at P and OP is the radius through the point of contact.{tex}\\therefore{/tex}{tex}\\angle{/tex}OPA = 90o ........ (i)[The tangent at any point of a circle is\xa0{tex}\\perp{/tex} 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}{tex}\\angle{/tex}OQD = 90o ........ (ii)[The tangent at any point of a circle is {tex}\\perp{/tex} to the radius through the point of contact]From eq. (i) and (ii), {tex}\\angle{/tex}OPA = {tex}\\angle{/tex}OQDBut these form a pair of equal alternate angles also,{tex}\\therefore{/tex}\xa0AB {tex}\\parallel{/tex} CD

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