Full solution

Q. The trisectors of angles

B

and

C

of scalene triangle

A B C

meet at points

P

and

Q

, as shown. Angle

A

measures

39

degrees and angle QBP measures

14

degrees. What is the measure of angle BPC?

Find Angle B: Since angle A is $39$ degrees and the triangle ABC is scalene, we can find the measure of angle B by using the fact that the sum of angles in a triangle is $180$ degrees. $\newline$ Angle B + Angle C + $39$ degrees = $180$ degrees.
Calculate Angle C: We know that angle $QBP$ is $14$ degrees, and since $P$ is the trisector of angle $B$ , angle $QBP$ is one-third of angle $B$ . So, Angle $B = 3 \times 14$ degrees $= 42$ degrees.
Determine Angle BPQ: Now we can find angle $C$ by substituting the value of angle $B$ into the first equation. $\newline$ $42$ degrees + Angle $C$ + $39$ degrees = $180$ degrees. $\newline$ Angle $C$ = $180$ degrees - $42$ degrees - $39$ degrees. $\newline$ Angle $C$ = $B$ $1$ degrees.
Calculate Angle QPC: Since $P$ is the trisector of angle $B$ , angle $BPQ$ is also $14$ degrees (same as angle $QBP$ ).
Find Angle BPC: Angle $BPC$ is the sum of angle $BPQ$ and angle $QPC$ . But angle $QPC$ is also the trisector of angle $C$ , which is $99$ degrees. So, angle $QPC = \frac{99 \text{ degrees}}{3} = 33$ degrees.
Find Angle BPC: Angle $BPC$ is the sum of angle $BPQ$ and angle $QPC$ . But angle $QPC$ is also the trisector of angle $C$ , which is $99$ degrees. So, angle $QPC = \frac{99 \text{ degrees}}{3} = 33$ degrees. Now we can find angle $BPC$ . Angle $BPC = \text{angle } BPQ + \text{angle } QPC$ . Angle $BPC = 14$ degrees $BPQ$ $0$ degrees. Angle $BPQ$ $1$ degrees.