Question

question 6 of 10 in the diagram below, $overline{xy}$ and $overline{yz}$ are tangent to $odot o$. what is the measure of $angle xz$? a. 100° b. 180° c. 130° d. 90°

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: Consider the quadrilateral OXZY

The sum of the interior angles of a quadrilateral is $360^{\circ}$. In quadrilateral OXZY, we know $\angle{OXY} = 90^{\circ}$, $\angle{OZY}=90^{\circ}$, and $\angle{Y}=50^{\circ}$. Let $\angle{XOZ}=x$. Then $90^{\circ}+90^{\circ}+50^{\circ}+x = 360^{\circ}$.

Step3: Solve for $\angle{XOZ}$

$230^{\circ}+x=360^{\circ}$, so $x = 360^{\circ}- 230^{\circ}=130^{\circ}$.

Step4: Use the angle - arc relationship

The measure of an inscribed angle is half the measure of its intercepted arc. The arc $\overset{\frown}{XZ}$ has a central - angle measure of $130^{\circ}$, and the angle we want to find, $\angle{XZ}$ (assuming it's an inscribed angle subtended by arc $\overset{\frown}{XZ}$), and the measure of the non - intercepted arc is $230^{\circ}$. The measure of the angle $\angle{XZ}$ (the angle between the two tangents) and the central angle of the non - intercepted arc are related. The measure of the angle between two tangents is equal to half the difference of the measures of the intercepted arcs. Here, the measure of the angle between the two tangents $\angle{XZ}= \frac{1}{2}(230^{\circ}-130^{\circ})=\frac{1}{2}\times100^{\circ}=100^{\circ}$.

Answer:

A. 100°