Points A and B are on a circle with centre O. PAM and PBN are tangentsto the circle at A and B respectively from a point P outside the circle. Point Q is on the major arc AB such that ∠QAM = 58° and ∠QBN = 50°, then find the measure (in degrees) of ∠APB.

36

The angles of an alternate segment subtended by a tangent to a circle are equal.

\(\angle\)QAM = \(\angle\)ABQ = \({58}^\circ\)

\(\angle\)QBN = \(\angle\)BAQ = \({50}^\circ\)

So, \(\angle\) = \({72}^\circ\)

\(\angle\)BOA = 2\(\angle\)AQB = 2 x \({72}^\circ\) = \({144}^\circ\)

In quadrilateral PBOA,

\(\angle\)APB + \(\angle\)BOA = \({180}^\circ\)    (because OB is perpendicular to PN and PA is perpendicular to PM )

= \(\angle\)APB = 180 -144

= \(\angle\)APB = \({36}^\circ\)

Therefore, \(\angle\)APB is \({36}^\circ\).