Given information:
- ABCQ is a square inscribed in an isosceles triangle PQR.
- A and C are the midpoints of PQ and QR respectively.

To find:
The measure of angle PBA.

Step-by-step solution:

1. Draw a diagram:
- Draw an isosceles triangle PQR.
- Mark midpoints A and C on PQ and QR respectively.
- Draw a square ABCQ inside the triangle, such that A and C are the midpoints of PQ and QR respectively.

2. Analyze the given information:
- Triangle PQR is isosceles, which means that PQ is equal to QR.
- Square ABCQ is inscribed in triangle PQR, which means that each vertex of the square lies on a side of the triangle.
- A is the midpoint of PQ, which means that PA is equal to AQ.
- C is the midpoint of QR, which means that CQ is equal to QR.

3. Identify the key angles:
- Angle P is the angle at vertex P of triangle PQR.
- Angle B is the angle at vertex B of square ABCQ.
- Angle A is the angle at vertex A of square ABCQ.

4. Apply angle properties:
- In a square, all angles are right angles, which means that angle B is equal to 90 degrees.
- In an isosceles triangle, the base angles are equal, which means that angle P is equal to angle Q.

5. Use the given information and angle properties:
- Since A is the midpoint of PQ and PA is equal to AQ, angle PAQ is a right angle (90 degrees).
- Since C is the midpoint of QR and CQ is equal to QR, angle CQR is a right angle (90 degrees).
- Since PQ is equal to QR, angle Q is equal to 180 - angle P - angle R.

6. Calculate angle P:
- Since triangle PQR is isosceles, angle P is equal to angle Q.
- Therefore, angle P + angle Q + angle R = 180 degrees.
- Substituting angle Q = angle P, we get: angle P + angle P + angle R = 180 degrees.
- Simplifying, we get: 2*angle P + angle R = 180 degrees.

7. Calculate angle R:
- Since triangle PQR is isosceles, angle P = angle Q and angle P + angle Q + angle R = 180 degrees.
- Substituting angle Q = angle P, we get: angle P + angle P + angle R = 180 degrees.
- Simplifying, we get: 2*angle P + angle R = 180 degrees.
- Rearranging the equation, we get: angle R = 180 degrees - 2*angle P.

8. Substitute angle R into the equation:
- Substituting angle R = 180 degrees - 2*angle P into the equation 2*angle P + angle R = 180 degrees, we get: 2*angle P + (180 degrees - 2*angle P) = 180 degrees.
- Simplifying, we get: 180 degrees = 180 degrees.
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