In triangle ABC, point P is the concurrency point of the angle bisectors (the incenter). If m∠BCP = (4x − 2)° and m∠ACP = (3x + 4)°, find m∠BCP.

What the incenter tells you

Because P is the intersection of the angle bisectors, segment $CP$ bisects the angle at $C$. That means the two angles formed at $C$ are equal.

Set the two angle expressions equal

So we write: $$4x-2 = 3x+4$$

Solve for x

Subtract $3x$ from both sides: $$x-2 = 4$$ Add 2: $$x = 6$$

Compute $m\angle BCP$

Substitute $x=6$ into $m\angle BCP=4x-2$: $$m\angle BCP = 4(6)-2 = 24-2 = 22^\circ$$

So, $m\angle BCP=22^\circ$.