Question

Fill in the blank with the correct angle measure.60°LK50°76°m2 QPM =M42°N

Answer 1

The measure of the inscribed angle = 1/2 the measure of its subtended arc

Inscribed angles subtended by the same arc are equal in measures

Let us use these 2 facts to solve the question

Since angle KLQ and angle KMQ subtended by the same arc KQ, then

$m\angle KLQ=m\angle KMQ=\mathring{50}$

Since the measure of arc LM = 76 degrees

Since angle LQM is an inscribed angle subtended by arc LM, then

$\begin{gathered} m\angle LQM=\frac{1}{2}(m\text{ arc LM)} \\ m\angle LQM=\frac{1}{2}(76)=38 \end{gathered}$

In triangle PQM

$\begin{gathered} m\angle PQM=\mathring{38} \\ m\angle PMQ=\mathring{50} \end{gathered}$

Then subtract them from 180 to find the measure of angle QPM

$\begin{gathered} m\angle QPM=180-38-50 \\ m\angle QPM=\mathring{92} \end{gathered}$