Use the Arc Addition Postulate. To find the measure of the minor arc corresponding to consecutive vertices, divide 360^(∘) by the number of sides of the polygon.
According to the Inscribed Angle Theorem, the measure of ∠ LRQ is half the measure of its intercepted arc, which is LNQ.
m ∠ LRQ = 1/2m LNQ
Notice that LNQ is divided into 5 smaller arcs. Additionally, since the octagon is regular, all these 5 arcs are congruent. The Arc Addition Postulate allows us to rewrite m LNQ.
m LNQ = 5mLM
Again, because the octagon is regular, the measure of the minor arc corresponding to consecutive vertices is equal to 360^(∘) divided by 8.
mLM = 360^(∘)/8 = 45^(∘)
Finally, we find the measure of the required angle by substituting the corresponding equations into the one written at the beginning.
