Question

question 8 of 10 what is the measure of ∠xyz? a. 72° b. 116° c. 64° d. 108°

Explanation:

Step1: Recall circle - angle relationship

The sum of the measures of the arcs of a circle is $360^{\circ}$.

Step2: Find the measure of arc $VXW$

Given arc $VW = 152^{\circ}$ and arc $VZ = 64^{\circ}$, then arc $VXW=360^{\circ}-152^{\circ}-64^{\circ}=144^{\circ}$.

Step3: Use the property of vertical - angles and arc - angle relationship

$\angle XYZ$ is a vertical angle to an angle that intercepts arc $VXW$. The measure of an angle formed by two intersecting chords in a circle is half the sum of the measures of the intercepted arcs. But we can also note that $\angle XYZ$ and the angle intercepting arc $VZ$ are related. The angle $\angle XYZ$ and the angle with its vertex on the circle intercepting arc $VZ$ are supplementary (since they are related to the angles formed by the chords). The measure of the inscribed - angle intercepting arc $VZ$ is $\frac{1}{2}\times64^{\circ} = 32^{\circ}$. The angle $\angle XYZ$ and this inscribed - angle are supplementary. So $\angle XYZ=180^{\circ}- 64^{\circ}=116^{\circ}$.

Answer:

B. $116^{\circ}$