Question

question 5 of 10
in the diagram below, \\( \overline { x y } \\) and \\( \overline { y z } \\) are tangent to \\( \odot o \\). what is the measure of
\\( \angle z \\)?
\\( \bigcirc \\) a. \\( 74 ^ { \circ } \\)
\\( \bigcirc \\) b. \\( 90 ^ { \circ } \\)
\\( \bigcirc \\) c. \\( 106 ^ { \circ } \\)
\\( \bigcirc \\) d. \\( 148 ^ { \circ } \\)

Explanation:

Step1: Recall the property of the sum of angles in a quadrilateral

The sum of the interior angles of a quadrilateral is \(360^{\circ}\). In the quadrilateral \(OXY Z\), \(\angle OXY=\angle OZY = 90^{\circ}\) (tangent to a circle is perpendicular to the radius at the point of tangency). Let \(\angle XOZ\) be the central - angle and \(\angle XYZ\) be the given angle (\(\angle XYZ = 74^{\circ}\)).

Step2: Set up the angle - sum formula

We know that \(\angle OXY+\angle XYZ+\angle OZY+\angle XOZ=360^{\circ}\). Substituting \(\angle OXY = 90^{\circ}\), \(\angle XYZ=74^{\circ}\), \(\angle OZY = 90^{\circ}\) into the formula:

$$90^{\circ}+74^{\circ}+90^{\circ}+\angle XOZ=360^{\circ}$$

$$254^{\circ}+\angle XOZ=360^{\circ}$$

$$ \angle XOZ=360^{\circ}- 254^{\circ}=106^{\circ}$$

Answer:

C. \(106^{\circ}\)