Exercise 34 Page 676

Let's begin by color coding the opposite angles, sides, and the vertices in the given triangle. It will help us use the Law of Cosines and later the Law of Sines.

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.

Finding m∠ M

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.

Now we can use the inverse sine ratio to find m ∠ M.

Finding m ∠ N

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.