Question

yz is tangent to the circle at z. find the measure of ∠xyz.
114°
172°
write your answer as a whole number or a decimal.
m∠xyz = ​°

Explanation:

Step1: Find the measure of the minor - arc

The sum of the measures of the major and minor arcs of a circle is 360°. Given the major - arc measure is 172°, the measure of the minor arc $\overset{\frown}{XZ}=360 - 172=188^{\circ}$.

Step2: Use the tangent - secant angle formula

The measure of an angle formed by a tangent and a secant to a circle is half the difference of the measures of the intercepted arcs. The formula is $m\angle XYZ=\frac{1}{2}(\text{major arc}-\text{minor arc})$.
Substitute the values of the major arc ($188^{\circ}$) and the minor arc ($114^{\circ}$) into the formula: $m\angle XYZ=\frac{1}{2}(188 - 114)$.

Step3: Calculate the angle measure

First, calculate the difference inside the parentheses: $188−114 = 74$. Then, multiply by $\frac{1}{2}$: $\frac{1}{2}\times74 = 37^{\circ}$.

Answer:

$37$