Question

angle mon is a straight angle and overrightarrow{op} bisects angle moq. what is the measure of angle mop?
options: 29°, 58°, 61°, 122°

Explanation:

Step1: Determine ∠MOQ measure

Since ∠MON is a straight angle ($180^\circ$) and ∠NOQ is $58^\circ$, ∠MOQ = $180^\circ - 58^\circ = 122^\circ$.

Step2: Use angle bisector property

OP bisects ∠MOQ, so ∠MOP = $\frac{1}{2}$∠MOQ = $\frac{1}{2}×122^\circ = 61^\circ$? Wait, no, wait. Wait, the diagram: Wait, maybe I misread. Wait, the angle between OQ and ON is $58^\circ$, so ∠MOQ is $180 - 58 = 122$, but OP bisects ∠MOQ? Wait, no, maybe the angle between OP and OQ? Wait, no, the options: Wait, the correct answer? Wait, maybe I made a mistake. Wait, let's re-examine.

Wait, ∠MON is straight (180°). ∠NOQ is 58°, so ∠MOQ = 180 - 58 = 122°. If OP bisects ∠MOQ, then ∠MOP = 122 / 2 = 61°? But the options have 58, 61, etc. Wait, maybe the angle between OP and OQ is equal? Wait, no, maybe the diagram shows that ∠POQ is equal to ∠MOP? Wait, no, let's check the options. The options are 29, 58, 61, 122. Wait, maybe I misread the angle. Wait, the angle marked 58° is ∠NOQ. So ∠MOQ = 180 - 58 = 122. If OP bisects ∠MOQ, then ∠MOP = 122 / 2 = 61°. So the answer should be 61°? But the blue dot is on 58. Wait, maybe the problem is that OP bisects ∠MOQ, but ∠QON is 58, so ∠MOQ is 180 - 58 = 122, bisected by OP, so ∠MOP = 61. So the correct option is 61°, which is the third option: 61°.

Answer:

61° (the option with 61°)