In the adjoining figure, PQ is a chord of a circle and PT is the tangent at P such that `angleQPT=60^(@).`Find `anglePRQ.`

Explanation

Join OP and OQ. Take any point S on the circumference in the alternate segment. Join SP and SQ.
Since,`" "OPbotPT`
(radius through point of contact is bot to the tangent)


`:." "angle2+angle1=90^(@)`
`implies" "angle2+60^(@)=90^(@)" "(given)`
`implies" "angle2=90^(@)-60^(@)=30^(@)" "...(1)`
But`" "OP=OQ " (each radii)"`
`:." "angle2=angle3`(angles opposite to equal sides are equal)
`" "...(2)`
Now in `DeltaPOQ,`
`angle2+angle3+angle4=180^(@)" "`(angle sum property)
`implies30^(@)+30^(@)+angle4=180^(@)" "["from" (1) and (2)]`
`implies" "angle4=120^(@)`
`:." "angle5=(1)/(2)xxangle4`
(`because` degree measure of an are is twice the angle subtended by it in alternate segment)
`implies" "angle5=(1)/(2)xx120^(@)`
`implies" "angle5-60^(@)`
Also,`" "angle5+angle6=180^(@)" "`(Sum of opposit angles of a cyclic quad.)
`implies" "60^(@)+angle6=180^(@)`
`implies" "angle6=anglePRQ=120^(@)`