Question

which equation is correct regarding the measure of ∠mnp?
○ (mangle mnp=\frac{1}{2}(x - y))
○ (mangle mnp=\frac{1}{2}(x + y))
○ (mangle mnp=\frac{1}{2}(z + y))
○ (mangle mnp=\frac{1}{2}(z - y))

Explanation:

Step1: Recall the tangent - secant angle formula

The measure of an angle formed by a tangent and a secant of a circle is half the difference of the measures of the intercepted arcs. Here, the tangent is $\overrightarrow{NP}$ and the secant is $\overrightarrow{NM}$, and the intercepted arcs are arc $x$ and arc $y$.

Step2: Apply the formula

The measure of $\angle MNP$ is given by the formula $m\angle MNP=\frac{1}{2}(x - y)$.

Answer:

$m\angle MNP=\frac{1}{2}(x - y)$ (the first option)