Question

in the diagram below, $overline{xy}$ and $overline{yz}$ are tangent to $odot o$. what is the measure of $angle y$? a. $48^{circ}$ b. $66^{circ}$ c. $96^{circ}$ d. $54^{circ}$

Explanation:

Step1: Recall tangent - radius property

The radius is perpendicular to the tangent at the point of tangency. So $\angle OXY = 90^{\circ}$ and $\angle OZY=90^{\circ}$.

Step2: Find the measure of the central angle

The measure of the arc intercepted by the non - tangent sides of the quadrilateral $OXYZ$ is $132^{\circ}$. The measure of the other arc is $360^{\circ}- 228^{\circ}=132^{\circ}$.

Step3: Use the sum of angles in a quadrilateral

In quadrilateral $OXYZ$, the sum of interior angles is $360^{\circ}$. Let $\angle Y = x$. Then $90^{\circ}+90^{\circ}+132^{\circ}+x = 360^{\circ}$.

Step4: Solve for $\angle Y$

$x=360^{\circ}-(90^{\circ}+90^{\circ}+132^{\circ})=48^{\circ}$. So the measure of $\angle Y$ is $48^{\circ}$.

Answer:

A. $48^{\circ}$