In triangle PQR, the measure of angle P is 30 more than

Given: \(\angle{P}=2*\angle{Q}+30\). Question: \(\angle{R}=?\).

Now, as the sum of the angles in a triangle equals to 180 degrees then: \(\angle{P}+\angle{Q}+\angle{R}=180\) --> \(\angle{R}=180-(\angle{P}+\angle{Q})=180-(2*\angle{Q}+30+\angle{Q})=150-3*\angle{Q}\) or \(\angle{R}=150-3*\frac{\angle{P}-30}{2}\). So, knowing the measure of \(\angle{P}\) ( or \(\angle{Q}\)) is sufficient to answer the question.

(1) PQ = QR --> triangle PQR is isosceles (base is PR) --> so \(\angle{R}=\angle{P}\). We also know that \(\angle{R}=150-3*\frac{\angle{P}-30}{2}\) so we have two distinct linear equations with two unknowns and we can solve for them. Sufficient.
(2) The measure of angle P is 78 --> so \(\angle{P}=78\) then we can solve \(\angle{R}=150-3*\frac{\angle{P}-30}{2}\) for \(\angle{R}\). Sufficient.

Answer: D.

For more on this issues check Triangles chapter of Math Book: math-triangles-87197.html

Hope it helps.