Question

point o is the center of the given circle. what is the measure of widehat{yz}? a. 108° b. 27° c. 72° d. 54°

Explanation:

Step1: Recall inscribed angle theorem

An inscribed angle is half the measure of its intercepted central angle, and the central angle equals its intercepted arc. So $\angle XYZ$ intercepts arc $XZ$, and $\angle XOY$ is a central angle intercepting arc $XY$. First, note that $\angle OXY = 54^\circ$, and $OX=OY$ (radii of the circle), so $\triangle OXY$ is isosceles.

Step2: Calculate $\angle XOY$

In $\triangle OXY$, $\angle OXY = \angle OYX = 54^\circ$. Sum of angles in a triangle is $180^\circ$, so:
$\angle XOY = 180^\circ - 54^\circ - 54^\circ = 72^\circ$
This means arc $XY = 72^\circ$.

Step3: Find arc $XZ$

$XZ$ is a diameter, so arc $XZ = 180^\circ$. Arc $YZ = \text{arc }XZ - \text{arc }XY = 180^\circ - 72^\circ = 108^\circ$.

Step4: Calculate $\angle XYZ$

$\angle XYZ$ is an inscribed angle intercepting arc $XZ$? No, correction: $\angle XYZ$ intercepts arc $XZ$? No, $\angle YXZ = 54^\circ$ intercepts arc $YZ$. Wait, correct: Inscribed angle $\angle YXZ = 54^\circ$ intercepts arc $YZ$, so arc $YZ = 2\times54^\circ=108^\circ$.

Answer:

A. $108^\circ$