Find the measure of angle LNM

m∠LNM = 54°

Here is why:

In this photo there are two important angles with very important features, a central angle and an inscribed angle.

A central angle is an angle that is in the center of the circle, so angle P is a central angle. The arc that is associated with this angle is going to be the same measure as the central angle. I have labeled this in the photo below as blue.

An inscribed angle is an angle that lies on the circle, so angle N is an inscribed angle. The arc that is associated with this angle will be double the amount of the inscribed angle, or the angle is half of the measure of the arc. I have labeled this in the photo below as red.

Since we know that the central angle is 108°, with what we know about central angles we know that arc LM is going to be 108° as well.

We also know that an inscribed angle is half the amount of the arc so...

108 ÷ 2 = 54

m∠LNM = 54°

2x + 8

step-by-step explanation: