Question

in the diagram below, \\(\overline{xy}\\) and \\(\overline{yz}\\) are tangent to \\(\odot o\\). what is the measure of \\(\widehat{xwz}\\)?

a. \\(240^\circ\\)
b. \\(180^\circ\\)
c. \\(150^\circ\\)
d. \\(210^\circ\\)

Explanation:

Identify the given values and target

Using the Angles Outside Circle Theorem and Arc Measure knowledge points

• Given: Tangents \(\overline{XY}\) and \(\overline{YZ}\) intersect at exterior point \(Y\) with angle \(m\angle Y = 60^\circ\).
• Minor arc measure \(m\widehat{XZ} = 120^\circ\).
• Target: Find the measure of major arc \(m\widehat{XWZ}\).

Apply the circle arc sum relationship

Using the Arc Measure knowledge point

$$ m\widehat{XWZ} + m\widehat{XZ} = 360^\circ $$

$$ m\widehat{XWZ} + 120^\circ = 360^\circ $$

$$ m\widehat{XWZ} = 240^\circ $$

Answer:

• (A) \(240^\circ\) (Correct answer)
• (B) \(180^\circ\)
• (C) \(150^\circ\)
• (D) \(210^\circ\)