Question

question 4 of 10 what is the measure of (overparen{xy})?

Explanation:

Step1: Recall vertical - angle property

Vertical angles are equal. So, the angle opposite the 42° angle is also 42°.

Step2: Use the arc - angle relationship

The measure of an inscribed angle is half the measure of its intercepted arc. Let the measure of arc $\overparen{XY}$ be $x$. The inscribed angle $\angle UZV$ intercepts arc $\overparen{UV}$ with measure 38°, and the other inscribed - angle formed by the intersecting chords has two components. The sum of the measures of the angles around point $Z$ is 360°. But using the property that the measure of an angle formed by two intersecting chords is $\frac{1}{2}$ the sum of the measures of the intercepted arcs. Let the two intersecting chords be $XU$ and $YV$. The angle at $Z$ is given by $\frac{1}{2}(\text{measure of arc}\overparen{XY}+\text{measure of arc}\overparen{UV})$. We know one of the non - vertical angles at $Z$ is 42°. So, $42=\frac{1}{2}(x + 38)$.

Step3: Solve the equation for $x$

Multiply both sides of the equation $42=\frac{1}{2}(x + 38)$ by 2: $84=x + 38$. Then subtract 38 from both sides: $x=84 - 38=40$. So the measure of arc $\overparen{XY}$ is 40°.

Answer:

C. 40°