Question

Circle Q is shown. Secant L N and tangent P N intersect at point N outside of the circle. Secant L N intersects the circle at point M. Arc M P is y, arc L P is x, and arc M L is z.
Which equation is correct regarding the measure of ∠MNP?

m∠MNP = One-half(x – y)
m∠MNP = One-half(x + y)
m∠MNP = One-half(z + y)
m∠MNP = One-half(z – y)

Answer 1

m∠MNP = 1/2(x − y)

Reason: An angle formed outside a circle by a secant and a tangent equals half the difference of the measures of the intercepted arcs. Here that is 1/2(arc LP − arc MP) = 1/2(x − y).