Given: ABCD is a square and P is a point in its interior such that triangle PAB is equilateral.

To find: The measure of angle PCB.

Solution:
First, let us draw the diagram.



Step 1: Find the measure of angle PAB.

Since triangle PAB is equilateral, all angles are equal. Therefore, angle PAB = 60 degrees.

Step 2: Find the measure of angle APB.

Since triangle PAB is equilateral, angle APB is the exterior angle of triangle PAB at vertex P. Therefore, angle APB = 120 degrees.

Step 3: Find the measure of angle BPC.

Since ABCD is a square, angle BCD = 90 degrees.

Since triangle PAB is equilateral, angle PBA = 60 degrees.

Therefore, angle BPC = angle BCD - angle PBA - angle APB

= 90 degrees - 60 degrees - 120 degrees

= -90 degrees (Note: We get a negative answer because angle PBA and angle APB are both outside of triangle BPC)

However, angles cannot be negative, so we add 360 degrees to get the equivalent angle measurement.

Therefore, angle BPC = -90 degrees + 360 degrees

= 270 degrees.

Step 4: Find the measure of angle PCB.

Since angle BPC is 270 degrees, angle PCB = 360 degrees - angle BPC

= 360 degrees - 270 degrees

= 90 degrees.

Therefore, the measure of angle PCB is 90 degrees.

Answer: The measure of angle PCB is 90 degrees.