Here are the short-form answers to the questions provided:

QIII: If two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are equal.
Proof: Equal arcs subtend equal central angles. Since the angles are equal and radii are equal (from congruent circles), the two corresponding chords must be equal by the Isosceles Triangle Theorem.

QIV: If the angle in a segment is greater than a semi-circle, it is less than a right angle.
Proof: The angle subtended by a chord at the center is 180°, and the angle at the circumference is half that. Since the angle in the segment is subtended by a chord greater than a semi-circle, it is less than 90°.

QV: A straight line drawn from the center of a circle to bisect a chord (which is not a diameter) is perpendicular.
Proof: The perpendicular from the center of a circle to a chord bisects the chord because it divides the segment into two equal parts. This follows from the property of a circle where the perpendicular from the center to any chord divides it into two equal lengths.

QVI: Construct a triangle with the given conditions: mAB = 5.2 cm, mBC = 5.6 cm, and ∠B = 45°.
Solution: Use the Law of Cosines to find the third side (AC) and then draw the triangle using the given lengths and angle. Construct a circumcircle around the triangle using a compass.

Let me know if you would like more details or clarifications.

Here are 5 related questions to further explore:

1. What is the Isosceles Triangle Theorem and how does it relate to this proof?
2. Can you explain why the angle in the segment is always less than 90°?
3. How do we prove that the perpendicular bisects the chord geometrically?
4. What is the Law of Cosines and how is it applied in triangle construction?
5. Can you explain how to construct a circumcircle for any triangle?

Tip: Always ensure you understand the underlying geometry concepts before jumping to formal proofs, as visualizing the shapes helps clarify the relationships between angles, chords, and radii.