Question

type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar(s).
the measure of ∠mop is 219° as shown. what is the measure of ∠mnp?
m∠mnp = □°

Explanation:

Step1: Find the measure of the arc \( \overset{\frown}{MP} \)

The central angle \( \angle MOP = 219^{\circ} \). The measure of a full - circle is \( 360^{\circ} \). So, the measure of the arc \( \overset{\frown}{MP} \) (the minor arc) is \( 360^{\circ}-\angle MOP \).
\( m\overset{\frown}{MP}=360 - 219=141^{\circ} \)

Step2: Use the inscribed - angle theorem

The inscribed - angle theorem states that the measure of an inscribed angle \( \angle MNP \) is half the measure of its intercepted arc.
The inscribed angle \( \angle MNP \) intercepts the arc \( \overset{\frown}{MP} \). So, \( m\angle MNP=\frac{1}{2}m\overset{\frown}{MP} \)
Substitute \( m\overset{\frown}{MP} = 141^{\circ} \) into the formula: \( m\angle MNP=\frac{141}{2}=70.5^{\circ} \)

Answer:

\( 70.5 \)