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<em>m∠LNM = 54°</em>
<u><em>Here is why:</em></u>
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 <u>blue</u>.
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 <u>red</u>.
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
<em><u>m∠LNM = 54°</u></em>
