Question

Line segment L N is tangent to circle O at point M and QM is a diameter.

Circle O is shown. Line segment Q M is a diameter. Line segments P M and Q R are secants. Line segment N L is a tangent and intersects the circle at point M. Angle P M O is 27 degrees and angle M Q R is 42 degrees.

Determine the measure of the following angles.

The measure of ∠QML is
degrees.

The measure of ∠PMN is
degrees.

Answer 1

∠QML = 90°

Reason: QM is a diameter so OM lies on QM; the tangent at M is perpendicular to the radius OM, hence MQ ⟂ ML.

∠PMN = 63°

Reason: the tangent at M is perpendicular to OM, so ∠PMN = 90° − ∠PMO = 90° − 27° = 63°.