Question

which equation is correct regarding the measure of ∠mnp? 〇 $mangle mnp = \frac{1}{2}(x - y)$ 〇 $mangle mnp = \frac{1}{2}(x + y)$ 〇 $mangle mnp = \frac{1}{2}(z + y)$ 〇 $mangle mnp = \frac{1}{2}(z - y)$

Explanation:

Step1: Recall exterior angle theorem

The measure of an angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs.

Step2: Identify intercepted arcs

For $\angle MNP$, the intercepted arcs are the major arc $LPZ$ (measure $z$) and the minor arc $MP$ (measure $y$).

Step3: Apply the theorem

The formula is $m\angle MNP = \frac{1}{2}(\text{measure of larger arc} - \text{measure of smaller arc}) = \frac{1}{2}(z - y)$

Answer:

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