We will first find the length of the segment MN and then the measures of the missing angles, m ∠ M and m ∠ N one at a time.
Finding MN
Note that we know an interior angle and the lengths of both of the adjacent sides. Therefore, we can use the Law of Cosines.
o^2=m^2+n^2 -2 m n cos O
Let's substitute m∠ O= 133, m= 21.5, and n= 31.5 to isolate o.
Now, we know the m ∠ O and the length of the side which is opposite to this angle. We can find m∠ M by using The Law of Sines!
sin O/o =sin M/m
Let's substitute o ≈ 48.8, m ∠ O= 133, and m= 21.5 to isolate sin M.
Finally, to find m ∠ N we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to 180.
133+ 19+ m ∠ N= 180 ⇔ m ∠ N≈ 28
Completing the Triangle
With all of the angle measures and side lengths, we can complete our diagram.
