`PQRST` is a cyclic pentagon and PT is a diameter, then `/_PQR + /_RST` is equal to the

Options

A. `180^@`
B. `270^@`
C. `216^@`
D. `144^@`

Answer

To solve the problem, we need to find the sum of the angles \( \angle PQR \) and \( \angle RST \) in the cyclic pentagon \( PQRST \) where \( PT \) is a diameter.

### Step-by-Step Solution:

1. **Understanding the Cyclic Pentagon**:
- A cyclic pentagon is a polygon whose vertices lie on the circumference of a circle. In this case, \( PQRST \) is a cyclic pentagon with \( PT \) as the diameter.

2. **Identifying Angles**:
- We need to find \( \angle PQR \) and \( \angle RST \).
- Let's denote \( \angle PQR \) as \( x \) and \( \angle RST \) as \( y \).

3. **Using the Inscribed Angle Theorem**:
- The inscribed angle theorem states that an angle inscribed in a semicircle is a right angle. Since \( PT \) is the diameter, both \( \angle PQR \) and \( \angle RST \) subtend the arc \( PR \) and \( RT \) respectively.

4. **Setting Up the Equations**:
- From the properties of cyclic quadrilaterals, we know that the opposite angles sum up to \( 180^\circ \). Thus, we can express this as:
\[ x + y = 180^\circ \]

5. **Finding the Values of Angles**:
- Since \( PT \) is the diameter, \( \angle PQR \) and \( \angle RST \) are both inscribed angles that subtend the same arc \( PR \) and \( RT \).
- Therefore, we can conclude that:
\[ x = y \]
- Substituting \( y \) with \( x \) in the equation gives:
\[ x + x = 180^\circ \implies 2x = 180^\circ \implies x = 90^\circ \]

6. **Calculating the Total**:
- Since both angles \( \angle PQR \) and \( \angle RST \) are equal:
\[ \angle PQR + \angle RST = 90^\circ + 90^\circ = 180^\circ \]

### Final Answer:
Thus, \( \angle PQR + \angle RST = 180^\circ \).