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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.

bingboy Mar 26, 2024