Let's go through each question step-by-step.

Question 6

To find the angles ∠PBA and ∠PQA in the first circle diagram:

1. Identify known angles: There is an inscribed angle and other clues we’ll use to find these angles. Since no values are immediately provided, we can assume the use of circle theorems, like the angle in a semicircle being 90°, or the use of alternate segment theorem if appropriate.

I'll need to confirm some details for a more specific solution, such as which point is on the circumference opposite the diameter.

Question 7

This involves three parts:

• (a) Reflex angle AOB: The reflex angle is the larger angle between points $A$, $O$, and $B$, which spans outside of the angle at the center.
• (b) Angle ∠AQB: Since $Q$ is on the circumference, we use the circle theorem that an angle at the center is twice the angle at the circumference on the same arc. Thus, ∠AQB = 1/2 ∠AOB.
• (c) Angle ∠ARB: This angle also uses the same arc-based relationships and circle theorems. If any angles are marked (e.g., 34° at the center), we can solve for these precisely.

Question 8

Given:

• ∠BCQ = 55° and ∠BAP = 60°.

The question asks to find angles:

• (a) ∠OBA
• (b) ∠OBC
• (c) ∠AOC
• (d) ∠ATC

Each can be solved using circle theorems (like inscribed angles and angles subtended by the same chord).

If you provide me with any additional angle values or clarifications, I can give exact calculations for each question.

Would you like detailed solutions for each part?

Follow-up Questions:

1. What are circle theorems, and how do they apply to these problems?
2. How do we calculate reflex angles in circles?
3. What is the alternate segment theorem, and where can it be applied?
4. How can central and inscribed angles in circles be related?
5. What steps are involved in solving for unknown angles in cyclic quadrilaterals?

Tip: Always look for diameters and marked angles in circle problems; they often reveal useful properties.