Here's how to prove that the acute angle formed by lines PQ and RS is half the size of angle XOZ:

1. Understand the diagram:

Imagine a semicircle with diameter AB. Inside the semicircle, a convex pentagon AXYZB is inscribed.

Denote the feet of the perpendiculars from Y onto lines AX, BX, AZ, BZ as P, Q, R, and S, respectively.

O is the midpoint of segment AB.

We need to prove that angle PQS is half of angle XOZ.

2. Key observations:

Since we have a semicircle with diameter AB, angle AOB is 90 degrees.

Because Y lies inside the semicircle and P, Q, R, and S are the feet of perpendiculars from Y, triangles APY, BQY, ARY, and BSY are all right triangles.

Angles AOZ and BOZ are equal angles inscribed in the same semicircle (opposite central angle).

3. Proving the relationship:

Focus on triangle APY: Since angles APY and AYO are both right angles and share side AY, angles PAY and PAO are complementary.

Therefore, angle PAY + angle PAO = 90 degrees. Similarly, angles PBQ + angle PBO = 90 degrees.

Relate angles to AOZ and BOZ: Using the fact that angles inscribed in the same semicircle are equal, we can rewrite the above equations as angle PAO = angle AOZ/2 and angle PBO = angle BOZ/2.

Combine equations: Substitute these expressions back into the equations from step 1: angle PAY + angle PAO = angle PAY + angle AOZ/2 = 90 degrees.

This simplifies to angle PAY = 90 degrees - angle AOZ/2. Similarly, angle PBQ = 90 degrees - angle BOZ/2.

Connect to angle PQS: Angles PAY and PBQ are opposite angles formed by intersecting lines PQ and RS.

Since they are supplementary (PAY + PBQ = 180 degrees), their acute angles must be equal: angle PQS = angle PBQ = 90 degrees - angle BOZ/2.

Final step: We already established that angle AOZ and BOZ are equal. Therefore, angle PQS = 90 degrees - angle BOZ/2 = (90 degrees - angle AOZ/2) / 2 = angle XOZ / 2.

Conclusion:

By analyzing the angles and relationships within the diagram, we have successfully proven that the acute angle formed by lines PQ and RS is half the size of angle XOZ. This completes the proof.

Note:

This proof relies on understanding geometric properties of semicircles and inscribed angles. It also utilizes the concept of complementary angles and opposite angles formed by intersecting lines.