Question

select the correct answer.
in the diagram, point m is the center of the circle. if ( mangle pmn = 134^circ ), what is ( mangle pom )?
diagram: circle with center m, points p, o, n on the circumference; angle at m between p and n is ( 134^circ )
a. ( 46^circ )
b. ( 67^circ )
c. ( 90^circ )
d. ( 134^circ )

Explanation:

Step1: Identify supplementary angles

Since \( \angle PMN = 134^\circ \) and \( \angle PMN \) and \( \angle PMO \) (or the angle adjacent to form a linear pair) are supplementary (they form a straight line, so their sum is \( 180^\circ \)). Wait, actually, \( \angle PMN \) and the central angle's adjacent angle? Wait, no, \( M \) is the center, so \( MP \), \( MO \), \( MN \) are radii. So \( \angle PMN \) and \( \angle PMO \) (wait, maybe \( \angle PMN \) and the angle \( \angle PMO \) are supplementary? Wait, no, let's correct. The angle \( \angle PMN = 134^\circ \), and the angle \( \angle PMO \) (the one forming a linear pair with \( \angle PMN \)) would be \( 180^\circ - 134^\circ = 46^\circ \)? Wait, no, maybe the central angle \( \angle POM \) and the inscribed angle? Wait, no, \( MP = MO = MN \) (radii). So triangle \( PMO \) is isoceles? Wait, no, maybe the key is that \( \angle PMN \) and \( \angle POM \) are related through the fact that \( \angle PMN \) is a reflex angle? Wait, no, the sum of a central angle and its adjacent angle (forming a linear pair) is \( 180^\circ \)? Wait, no, let's think again. The angle \( \angle PMN = 134^\circ \), and since \( M \) is the center, \( MP \) and \( MN \) are radii, and \( MO \) is also a radius. Wait, maybe the angle \( \angle PMN \) and the angle \( \angle POM \) are related such that \( \angle POM = 180^\circ - 134^\circ = 46^\circ \)? No, that doesn't seem right. Wait, no, maybe the inscribed angle theorem? Wait, no, \( \angle POM \) is a central angle, and maybe the angle at \( N \) or \( P \)? Wait, no, let's check the diagram. The diagram shows \( M \) as the center, with points \( P \), \( O \), \( N \) on the circle? Wait, \( O \) is on the circle? Wait, the diagram: \( O \), \( P \), \( N \) are on the circumference, \( M \) is the center. So \( MP \), \( MO \), \( MN \) are radii, so \( MP = MO = MN \). So triangle \( PMN \) has \( MP = MN \), so it's isoceles? Wait, no, the angle given is \( \angle PMN = 134^\circ \), and we need to find \( \angle POM \). Wait, maybe \( \angle PMN \) and \( \angle POM \) are related because \( \angle POM \) is the central angle, and \( \angle PMN \) is an angle at the center? No, \( \angle PMN \) is at \( M \) between \( P \) and \( N \), and \( \angle POM \) is at \( M \) between \( P \) and \( O \). Wait, maybe \( \angle PMN \) and \( \angle POM \) are supplementary? Wait, no, the sum of angles around a point is \( 360^\circ \), but maybe \( \angle PMN \) and \( \angle POM \) are related as \( \angle POM = 180^\circ - 134^\circ = 46^\circ \)? No, that's not. Wait, maybe I made a mistake. Wait, the correct approach: since \( M \) is the center, \( MP = MO = MN \) (radii). The angle \( \angle PMN = 134^\circ \), and we need to find \( \angle POM \). Wait, maybe \( \angle PMN \) and \( \angle POM \) are related such that \( \angle POM = 2 \times \) some inscribed angle, but no. Wait, actually, \( \angle PMN \) and \( \angle POM \) are adjacent angles forming a linear pair? No, the diagram shows a blue angle of \( 134^\circ \) at \( M \) between \( P \) and \( N \), and we need \( \angle POM \) at \( M \) between \( P \) and \( O \). So the sum of \( \angle PMN \) and \( \angle POM \) is \( 180^\circ \)? Wait, no, that would be if they are supplementary. Wait, \( 180 - 134 = 46 \), but option B is 67. Wait, maybe I messed up. Wait, no, maybe \( \angle PMN \) is the reflex angle, and the smaller angle at \( M \) between \( P \) and \( N \) is \( 180 - 134 = 46 \), but no. Wait, maybe the triangle \( POM \) is isoceles, and \( \angle POM \) is calculated as \( (180 - (180 - 134))/2 \)? No, that's confusing. Wait, let's start over.

Given \( M \) is the center, so \( MP = MO = MN \) (radii). The angle \( \angle PMN = 134^\circ \). We need to find \( \angle POM \).

Wait, the key is that \( \angle PMN \) and the angle \( \angle PMO \) (wait, \( O \) and \( N \) are on the circle, so \( MO \) and \( MN \) are radii, so \( \angle OMN \) is... Wait, maybe \( \angle PMN \) and \( \angle POM \) are related by the fact that \( \angle POM \) is the central angle, and \( \angle PMN \) is an angle at the center, but actually, the correct formula is that if two radii form an angle, and another angle is supplementary, but let's think about the linear pair. The angle \( \angle PMN = 134^\circ \), so the adjacent angle (forming a straight line) is \( 180^\circ - 134^\circ = 46^\circ \), but that's not the case. Wait, no, maybe \( \angle POM \) is half of \( (360^\circ - 2 \times 134^\circ) \)? No, that's not. Wait, I think I made a mistake. Let's check the options. The options are 46, 67, 90, 134.

Wait, maybe the angle \( \angle PMN = 134^\circ \), and since \( MP = MO \) (radii), triangle \( PMO \) is isoceles, but no. Wait, another approach: the sum of angles around point \( M \) is \( 360^\circ \), but that's not helpful. Wait, maybe the angle \( \angle PMN \) is the measure of the arc \( PN \), but no, central angle is equal to the arc. Wait, no, \( \angle PMN \) is a central angle? Wait, \( M \) is the center, so \( \angle PMN \) is a central angle, but it's \( 134^\circ \), so the arc \( PN \) is \( 134^\circ \). Then, what about \( \angle POM \)? Wait, \( O \) is another point on the circle, so maybe \( \angle POM \) is the central angle for arc \( PO \). Wait, maybe \( O \) and \( N \) are endpoints, and \( P \) is another point. Wait, the diagram: \( O \), \( P \), \( N \) on the circle, \( M \) center. So \( MO \), \( MP \), \( MN \) are radii. So \( \angle PMN = 134^\circ \) (central angle for arc \( PN \)), and we need \( \angle POM \) (central angle for arc \( PO \)). Wait, maybe \( O \) and \( N \) are such that \( \angle PMN \) and \( \angle POM \) are related by \( \angle POM = 180^\circ - 134^\circ = 46^\circ \)? No, that's option A. But wait, maybe I'm wrong. Wait, let's calculate:

If \( \angle PMN = 134^\circ \), then the angle supplementary to it (forming a linear pair) is \( 180 - 134 = 46^\circ \), but that's not. Wait, no, maybe \( \angle POM \) is half of \( (360 - 2 \times 134) \)? No, \( 360 - 2 \times 134 = 360 - 268 = 92 \), half of that is 46. Wait, that's option A. But the options include 67. Wait, maybe I misread the angle. Wait, the problem says \( m\angle PMN = 134^\circ \), maybe it's \( \angle PMO \)? No, the problem says \( \angle PMN \). Wait, maybe the triangle \( POM \) has \( MP = MO \), and \( \angle PMO = 180 - 134 = 46 \), so the base angles are \( (180 - 46)/2 = 67 \). Ah! That's it. So \( \angle PMN = 134^\circ \), so \( \angle PMO = 180^\circ - 134^\circ = 46^\circ \) (since they are supplementary, forming a linear pair). Then, in triangle \( PMO \), \( MP = MO \) (radii), so it's isoceles with \( \angle PMO = 46^\circ \). Therefore, the other two angles \( \angle MPO \) and \( \angle MOP \) (which is \( \angle POM \)) are equal. So \( \angle POM = (180^\circ - 46^\circ)/2 = 67^\circ \). Yes, that makes sense. So step by step:

Step1: Find supplementary angle to \( \angle PMN \)

\( \angle PMO = 180^\circ - 134^\circ = 46^\circ \) (linear pair)

Step2: Calculate \( \angle POM \) in isoceles triangle \( PMO \)

Since \( MP = MO \) (radii), \( \triangle PMO \) is isoceles with \( \angle PMO = 46^\circ \). Thus, \( \angle POM = \frac{180^\circ - 46^\circ}{2} = \frac{134^\circ}{2} = 67^\circ \)

Answer:

B. \( 67^\circ \)