Question

Point $P$ is on side $\overline{AC}$ of triangle $ABC$ such that $\angle APB =\angle ABP$, and $\angle ABC - \angle ACB = 39^\circ$. Find $\angle PBC$ in degrees.

Answer 1

How about just typing:
Point P is on side AC of triangle ABC such that angle APB =angle ABP, and angle ABC - angle ACB = 39°. Find angle PBC in degrees.
let angle APB = angle ABP = x
let angle PBC = y, let angle BCP = k

given:
x+y - k = 39
k = x+y-39

In triangle PBC, the angles add up to 180°
y + 180-x + k = 180
y + 180 - x + x+y-39 = 180
2y = 39
y = 39/2 = 19.5°

check my arithmetic