Prove that line segment between point …

Let PQ and RS be two parallel tangents of a circle where centre is O. Let their points of contact be A and B respectively. Join OA and OB.
 
Draw OC {tex}\parallel{/tex} PQ or RS
Now PQ {tex}\parallel{/tex} OC and transversal AO intersects them
{tex}\therefore{/tex} {tex}\angle{/tex}PAO + {tex}\angle{/tex}COA = 180^o  [{tex}\because{/tex} The sum of consecutive interior angles on the same side of the transverse is 180^o]
⇒ 90°+ {tex}\angle{/tex}COA = 180^o [{tex}\because{/tex} PA is a tangent and OA is the radius through the point of contact]
{tex}\therefore{/tex} {tex}\angle{/tex}PAO = 90^o [ the tangent of any point of a circle is perpendicular to the radius through the property contact]
{tex}\Rightarrow{/tex} {tex}\angle{/tex}COA = 90^o
Similarly,
{tex}\angle{/tex}COB = 90^o
{tex}\therefore{/tex} {tex}\angle{/tex}COA + {tex}\angle{/tex}COB  = 180^o
{tex}\Rightarrow{/tex}  AB is a straight line through O.