https://file.aiquickdraw.com/mathbot/file/1727317368157e8pr6g7a.jpg

Let's analyze the given diagram and the corresponding questions:

1. **PS is a diameter of the circle**, and as a property of circles, any angle subtended by the diameter at the circumference is a right angle (90°).
2. **TRQ and TSP are straight lines**, meaning angles on a straight line add up to 180°.
3. **Given:**
   - Angle \( PQR = 108^\circ \)
   - Angle \( RTP = 24^\circ \)

### (a) We need to find:

#### 1. **Angle PRS**
   - Since \( PS \) is the diameter, angle \( PRS \) subtended by the diameter at \( R \) will be a right angle:
   \[
   \boxed{90^\circ} \quad \text{(Angle in a semicircle is } 90^\circ \text{)}
   \]

#### 2. **Angle QPT**
   - Consider the quadrilateral \( PQRT \) formed in the circle. The sum of angles in a cyclic quadrilateral is 180°.
   - In quadrilateral \( PQRT \), angles \( PQR \) and \( QPT \) are opposite.
   \[
   \text{Thus, } PQR + QPT = 180^\circ
   \]
   \[
   QPT = 180^\circ - 108^\circ = \boxed{72^\circ} \quad \text{(Opposite angles in a cyclic quadrilateral add up to } 180^\circ \text{)}
   \]

#### 3. **Angle PSR**
   - This angle can be found by considering the fact that the angle \( RTP = 24^\circ \), and angles on a straight line add up to 180°.
   \[
   PSR = 180^\circ - 24^\circ = \boxed{156^\circ} \quad \text{(Angles on a straight line add up to } 180^\circ\text{)}
   \]

#### 4. **Angle SRT**
   - Angle \( SRT \) can be found similarly. Since \( TRQ \) is a straight line and the angle \( RTP = 24^\circ \), the remaining angle is:
   \[
   SRT = 180^\circ - 156^\circ = \boxed{24^\circ} \quad \text{(Angles on a straight line add up to } 180^\circ\text{)}
   \]

### (b) Find **angle QPR**.
- Since angle \( PQR \) is 108° and \( PS \) is the diameter, \( QPR \) is the remaining angle in triangle \( PQR \). The sum of angles in a triangle is 180°:
\[
QPR = 180^\circ - (PQR + PRS) = 180^\circ - (108^\circ + 90^\circ) = \boxed{12^\circ}
\]

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Would you like further clarifications or a deeper explanation of any of these steps? Here are five related questions for further practice:

1. What is the sum of angles in any triangle?
2. How would you prove that angle PRS is 90° using circle theorems?
3. If \( PQR = 90^\circ \), what would the angle \( QPT \) be?
4. How would the answers change if \( PQR \) was less than 108°?
5. What is the relationship between angles subtended by the same arc in a circle?

**Tip**: In cyclic quadrilaterals, opposite angles always sum to 180°. This property can help solve many angle problems in circle geometry.

PS is a diameter of the circle. TRQ and TSP are straight lines. Angle PQR = 108° and angle RTP = 24°. Find angles PRS, QPT, PSR, SRT, and QPR, and give a reason for each statement.

Solve a circle geometry problem involving a diameter, cyclic quadrilaterals, and straight lines. Learn how to find angles PRS, QPT, PSR, SRT, and QPR, using key theorems and geometric properties.

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