PQRS is a rhombus with `angle PQR = 54^(@)`. Determine `angle PRS`.

Answer

To solve the problem of finding angle PRS in rhombus PQRS where angle PQR = 54°, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Rhombus Properties**:
A rhombus has opposite angles that are equal and its diagonals bisect each other at right angles (90°).

2. **Identify Given Information**:
We know that angle PQR = 54°.

3. **Determine Angle QRS**:
Since opposite angles in a rhombus are equal, angle PSQ (which is opposite to angle PQR) is also 54°.

4. **Calculate Angle QRP**:
The diagonals bisect the angles. Therefore, angle QRP is half of angle PQR:
\[ \text{Angle QRP} = \frac{54°}{2} = 27° \]

5. **Identify Triangle SOR**:
Consider triangle SOR formed by the diagonals. Since the diagonals intersect at right angles, angle SOR = 90°.

6. **Use the Triangle Angle Sum Property**:
The sum of angles in triangle SOR is 180°:
\[ \text{Angle SOR} + \text{Angle QRP} + \text{Angle PRS} = 180° \]
Substituting the known values:
\[ 90° + 27° + \text{Angle PRS} = 180° \]

7. **Solve for Angle PRS**:
\[ 117° + \text{Angle PRS} = 180° \]
\[ \text{Angle PRS} = 180° - 117° = 63° \]

### Final Answer:
Angle PRS = 63°.