Let's first find the length of the missing segment PR and then the measures of the missing angles, m ∠ P and m ∠ R one at a time.
Finding PR
Note that we know an interior angle and the lengths of both of the adjacent sides. Therefore, we can use the Law of Cosines.
q^2=p^2+r^2 -2 p r cos Q
Let's substitute m∠ Q= 70, p= 9, and r= 15 to isolate q.
Now, we know the m ∠ Q and the length of the side which is opposite to this angle. We can find the measure of m∠ P by using the Law of Sines.
sin Q/q =sin P/p
Let's substitute q ≈ 14.6, m ∠ Q= 70, and p= 9 to isolate sin P.
Finally, to find m ∠ R we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to 180.
35 + 70 + m ∠ R = 180 ⇔ m ∠ R≈ 75
Completing the Triangle
With all of the angle measures and side lengths, we can complete our diagram.
