If the measure of the arc LN=80° and the measure of arc OP=50°, then m∠M=___.
To find the measure of angle M, we need to determine the measure of the intercepted arc LM.
First, let's understand the relationship between an inscribed angle and the arc it intercepts.
In a circle, an inscribed angle is formed by two chords (or a chord and a tangent) that intersect on the circle. The measure of the inscribed angle is half of the measure of the intercepted arc.
In this case, the measure of arc LN is given as 80°. Therefore, the measure of inscribed angle LNM is half of that, which is 40°.
Now, let's focus on the inscribed angle formed by chords NO and MO. Since the measure of the intercepted arc OP is given as 50°, the measure of the inscribed angle OMP is half of that, which is 25°.
To find the measure of angle M, we can subtract the angles LNM and OMP from 180° since angles LNM, M, and OMP form a straight line.
m∠M = 180° - m∠LNM - m∠OMP
m∠M = 180° - 40° - 25°
m∠M = 115°
Therefore, the measure of angle M is 115°.