Question

quadrilateral wxyz is inscribed in circle t. complete the following.
(a) choose the correct expression to finish each statement.
$m\angle w$ = (choose one)
$m\angle y$ = (choose one)
(b) choose the correct expressions for the sum of $m\angle w$ and $m\angle y$.
$m\angle w + m\angle y$ = (choose one)
= (choose one)
(c) fill in the blank with the correct number.
$m\angle w + m\angle y$ = $\square^\circ$
(d) from the steps above, classify $\angle w$ and $\angle y$.
$\circ$ $\angle w$ and $\angle y$ are neither supplementary nor complementary.
$\circ$ $\angle w$ and $\angle y$ are complementary
$\circ$ $\angle w$ and $\angle y$ are supplementary.

Explanation:

Step1: Recall inscribed quadrilateral rule

For an inscribed quadrilateral, an inscribed angle is half the measure of its intercepted arc. So:
$m\angle W = \frac{1}{2}m\arc{ZYX}$
$m\angle Y = \frac{1}{2}m\arc{ZWX}$

Step2: Sum the two angle measures

Add the expressions for $m\angle W$ and $m\angle Y$:
$m\angle W + m\angle Y = \frac{1}{2}m\arc{ZYX} + \frac{1}{2}m\arc{ZWX}$

Step3: Factor and simplify the sum

Factor out $\frac{1}{2}$, and note $\arc{ZYX} + \arc{ZWX}$ is the full circle ($360^\circ$):
$m\angle W + m\angle Y = \frac{1}{2}(m\arc{ZYX} + m\arc{ZWX}) = \frac{1}{2}(360^\circ)$

Step4: Calculate the final sum

Compute the value:
$m\angle W + m\angle Y = 180^\circ$

Step5: Classify the angle pair

Angles that add to $180^\circ$ are supplementary.

Answer:

(a)
$m\angle W = \frac{1}{2}m\arc{ZYX}$
$m\angle Y = \frac{1}{2}m\arc{ZWX}$

(b)
$m\angle W + m\angle Y = \frac{1}{2}m\arc{ZYX} + \frac{1}{2}m\arc{ZWX}$
$= \frac{1}{2}(360^\circ)$

(c)
$m\angle W + m\angle Y = 180^\circ$

(d)
$\angle W$ and $\angle Y$ are supplementary.