Here is a full solution.
Analyze the given information:
Point P is on side AC of triangle ABC.
Angles ACB and ABC are congruent (angle ACB = angle ABC).
The difference between angle ABC and angle ACB is 42 degrees (angle ABC - angle ACB = 42°).
Identify key properties of triangles:
The sum of angles in a triangle is always 180 degrees.
Solve for missing angle in triangle ABC:
Let x represent the unknown measure of angle ABC (and angle ACB by congruence).
From the triangle property: x + x + (x - 42) = 180°
Combine like terms: 2x - 42 = 180
Add 42 to both sides: 2x = 222
Divide both sides by 2 to find x: x = 111°
Find angle PBC:
Since angles ABC and ACB are congruent, each measures 111 degrees (as found previously).
Angle PBC is supplementary to angle ABC (they share a straight line).
Therefore, angle PBC = 180° - angle ABC = 180° - 111° = 69°.
Answer: Angle PBC = 69 degrees.
