The trisectors of angles B and $C$ of scalene triangle ABC meet at points P and Q, as shown. Angle A measures 35 degrees and angle QBP measures 12 degrees. What is the measure of angle BPC?
Note that \(\angle B = 3 \times \angle QBP = 3 \times 12^\circ = 36^ \circ\)
This means that \(\angle C = 180 - 36 - 35 = 109\)
So, \(\angle PCB = {109^ \circ \over 3} = 33^ \circ\).
But, recall that \(\angle PBC = \angle QBP = 12\)
This means that \(\angle BPC = 180 - \angle PBC - \angle QBP = 180 - 33 - 12 = \color{brown}\boxed{135^\circ}\)