Question

ln is tangent to circle o at point m and qm is a diameter. determine the measure of the following angles. the measure of ∠qml is degrees. the measure of ∠pmn is degrees.

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. Since $\overline{LN}$ is tangent to circle $O$ at $M$ and $\overline{QM}$ is a radius (as $\overline{QM}$ is a diameter), $\angle QML = 90^{\circ}$.

Step2: Use the property of angles in a circle

The angle between a tangent and a chord through the point of tangency is equal to the inscribed - angle subtended by the same chord. $\overline{PM}$ is a chord and $\overline{LN}$ is a tangent at $M$. The inscribed - angle $\angle MQP$ subtends the same arc as the angle between the tangent $\overline{LN}$ and the chord $\overline{PM}$. Given $\angle MQP = 27^{\circ}$, so $\angle PMN=27^{\circ}$.

Answer:

The measure of $\angle QML$ is 90 degrees.
The measure of $\angle PMN$ is 27 degrees.