Question

select the correct answer.
in the diagram, point m is the center of the circle. if ( mangle pmn = 134^circ ), what is ( mangle pon )?
a. ( 46^circ )
b. ( 67^circ )
c. ( 90^circ )
d. ( 134^circ )

Explanation:

Step1: Find supplementary angle of $\angle PMN$

$\angle PMO = 180^\circ - 134^\circ = 46^\circ$

Step2: Identify isosceles triangle $\triangle PMO$

Since $MP=MO$ (radii of the circle), $\angle MPO = \angle MOP$.

Step3: Calculate $\angle MOP$

Sum of angles in a triangle is $180^\circ$, so:
$\angle MOP = \frac{180^\circ - 46^\circ}{2} = 67^\circ$

Step4: Repeat for $\triangle MNO$

$\angle NMO = 46^\circ$, $MN=MO$, so $\angle MON = 67^\circ$

Step5: Calculate $\angle PON$

$\angle PON = \angle MOP + \angle MON = 67^\circ + 67^\circ = 134^\circ$

Answer:

D. $134^\circ$