Question

question 8 of 10
what is the measure of shown in the diagram below?
a. 71°
b. 35.5°
c. 32°
d. 74.5°

Explanation:

Step1: Recall the formula for the measure of an inscribed angle formed by two chords

The measure of an inscribed angle formed by two chords is half the difference of the measures of the intercepted arcs. The formula is \(m\angle Z=\frac{1}{2}(m\widehat{VW}-m\widehat{XY})\). Here, we assume the circle has a total of \(360^{\circ}\), but in the case of two - arc relationship for the angle formed by two chords (the angle at \(Z\)), if we consider the major arc and minor arc. However, more accurately, for the angle formed outside the circle (by two secants), the formula is \(m\angle Z=\frac{1}{2}(m\widehat{VW}-m\widehat{XY})\). Wait, no, when two secants intersect outside the circle, the measure of the angle is \(\frac{1}{2}\) (measure of the intercepted major arc - measure of the intercepted minor arc). But in the standard formula for the angle formed by two secants outside the circle: \(\theta=\frac{1}{2}(m\widehat{VW}-m\widehat{XY})\). Given \(m\angle Z = 35.5^{\circ}\), if we use the formula \(\theta=\frac{1}{2}(m\widehat{VW}-m\widehat{XY})\), and assume \(m\widehat{VW} = 110^{\circ}\), then \(35.5^{\circ}=\frac{1}{2}(110^{\circ}-m\widehat{XY})\).

Step2: Solve the equation for \(m\widehat{XY}\)

Multiply both sides of the equation \(35.5^{\circ}=\frac{1}{2}(110^{\circ}-m\widehat{XY})\) by \(2\): \(2\times35.5^{\circ}=110^{\circ}-m\widehat{XY}\). So \(71^{\circ}=110^{\circ}-m\widehat{XY}\). Then, we can rewrite it as \(m\widehat{XY}=110^{\circ}- 71^{\circ}\).

Answer:

A. \(71^{\circ}\)