Answer:
[tex]m\widehat{MP}=118^{\circ}[/tex]
Step-by-step explanation:
In any circle, the measure of an inscribed angle (an angle created when the intersection of two chords is on the circle) is equal to half of the arc it forms. In this case, [tex]\angle MPN[/tex] is an inscribed angle that forms [tex]\widehat{MN}[/tex]. Therefore, the measure of arc MN must be twice the measure of angle MPN:
[tex]m\widehat{MN}=2\cdot m\angle MPN,\\m\widehat{MN}=2\cdot 31^{\circ}=62^{\circ}[/tex]
Since we want to find the measure of arc MP, we need to know that there are 360 degrees in a circle. Since [tex]\overline{PN}[/tex] is a diameter of the circle (line that has two endpoints on the circle and that passes through the center of the circle), each side of [tex]\overline{PN}[/tex] must be 180 degrees.
Therefore, we have:
[tex]m\widehat{MN}+m\widehat{MP}=180^{\circ},\\62^{\circ}+m\widehat{MP}=180^{\circ},\\m\widehat{MP}=\boxed{118^{\circ}}[/tex]
