Question

question 7 of 10
what is the measure of ∠xyz?
a. 36°
b. 72°
c. 55°
d. 17°

Explanation:

Step1: Recall inscribed angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

Step2: Identify intercepted arc for ∠XYZ

∠XYZ intercepts arc XW. First calculate arc XW:
Arc XW = Arc VW - Arc XV = $55^\circ - 17^\circ = 38^\circ$

Step3: Calculate ∠XYZ

$\angle XYZ = \frac{1}{2} \times \text{Arc } XW$
$\angle XYZ = \frac{1}{2} \times 38^\circ = 19^\circ$
*Note: Correction - Re-identify the correct intercepted arc: ∠XYZ is formed by chords XZ and XY, with Y on the circle? No, Y is the center. Correct rule: Central angle equals its intercepted arc. Wait, no—∠XYZ has vertex Y (center), so it is a central angle intercepting arc XW? No, ∠XYZ has sides YX and YZ, intercepting arc XZ? No, arc XZ is $17^\circ$, that can't be. Wait, correct: ∠XYZ is an inscribed angle with vertex Z? No, vertex is Y. Wait, error: Arc VW is $55^\circ$, arc XZ is $17^\circ$. ∠XYZ is formed by chords ZY and XY, vertex Y (center) → no, ∠XYZ has vertex at Y, so it is a central angle intercepting arc XZ? No, that's $17^\circ$, not an option. Wait, correct: ∠XYZ is an inscribed angle with vertex at Z? No, label is ∠XYZ: vertex Y, sides YX and YZ. The inscribed angle at Z would be different. Wait, the correct approach: The angle formed by two chords intersecting on the circle (inscribed angle) intercepts an arc equal to twice the angle. Wait, ∠XYZ intercepts arc VW minus arc XZ? No, let's use the external angle theorem for circles: The measure of an angle formed by two secants intersecting outside the circle is half the difference of the intercepted arcs. Wait, no, the angle at Y: Y is inside the circle. The measure of an angle formed by two chords intersecting inside the circle is half the sum of the intercepted arcs. Wait, no, ∠XYZ: if Y is inside the circle, then ∠XYZ intercepts arc XW and arc VZ? No, arc VZ is $17^\circ$, arc XW is $55^\circ$. Then $\angle XYZ = \frac{1}{2}(55^\circ + 17^\circ) = 36^\circ$.

Step1: Correct rule for internal angle

Angle inside circle: half sum of intercepted arcs.

Step2: Identify intercepted arcs

∠XYZ intercepts arc XW ($55^\circ$) and arc XZ ($17^\circ$).

Step3: Calculate the angle

$\angle XYZ = \frac{1}{2}(55^\circ + 17^\circ) = \frac{1}{2}(72^\circ) = 36^\circ$

Answer:

A. $36^\circ$