Question

angle mon is a straight angle and \\( \overrightarrow { op } \\) bisects \\( \angle moq \\). what is the measure of \\( \angle mop? \\) \\( 29 ^ { \circ } \\) \\( 58 ^ { \circ } \\) \\( 61 ^ { \circ } \\) \\( 122 ^ { \circ } \\)

Explanation:

Step1: Find the measure of $\angle MOQ$

Since $\angle MON$ is a straight angle ($180^{\circ}$) and $\angle QON = 58^{\circ}$, then $\angle MOQ=180^{\circ}-\angle QON$.
$$\angle MOQ = 180^{\circ}- 58^{\circ}=122^{\circ}$$

Step2: Use the angle - bisector property

Because $\overrightarrow{OP}$ bisects $\angle MOQ$, then $\angle MOP=\frac{1}{2}\angle MOQ$.
Substitute $\angle MOQ = 122^{\circ}$ into the formula:
$$\angle MOP=\frac{1}{2}\times122^{\circ}=61^{\circ}$$

Answer:

$61^{\circ}$ (the third option)