To calculate the area of the rhombus \(PQRS\) accurately, we need to determine the lengths of the diagonals \(QS\) and \(PR\) accurately.

### Given Data:
- \(PR = 10\)
- \(QR = 13\)
- \(\angle SRQ = 134^\circ\)

### Calculations:

1. **Using Law of Cosines in \(\triangle QRS\)**:
   - \(QR = RS = 13\)
   - \(\angle SRQ = 134^\circ\)

2. **Find the Length of \(QS\)**:
   \[
   QS^2 = QR^2 + RS^2 - 2 \cdot QR \cdot RS \cdot \cos(134^\circ)
   \]
   Since \(\cos(134^\circ) = -\cos(46^\circ)\):
   \[
   \cos(46^\circ) \approx 0.6947 \quad \text{(Using a calculator for precision)}
   \]
   Thus:
   \[
   \cos(134^\circ) = -0.6947
   \]
   Substitute into the formula:
   \[
   QS^2 = 13^2 + 13^2 - 2 \cdot 13 \cdot 13 \cdot (-0.6947)
   \]
   \[
   QS^2 = 169 + 169 + 2 \cdot 169 \cdot 0.6947
   \]
   \[
   QS^2 = 169 + 169 + 234.8686
   \]
   \[
   QS^2 = 572.8686
   \]
   \[
   QS \approx \sqrt{572.8686} \approx 23.93
   \]

3. **Calculate the Area of the Rhombus**:
   - The formula for the area of a rhombus is \(\text{Area} = \frac{1}{2} \times d_1 \times d_2\)
   - Where \(d_1 = PR = 10\) and \(d_2 = QS \approx 23.93\)

   \[
   \text{Area} = \frac{1}{2} \times 10 \times 23.93
   \]
   \[
   \text{Area} = 5 \times 23.93
   \]
   \[
   \text{Area} \approx 119.65
   \]

Therefore, the area of the rhombus \(PQRS\) is approximately \(119.65 \, \text{square units}\).

Would you like more details on any of the steps, or have any additional questions?

### Related Questions:

1. How do you verify the accuracy of trigonometric calculations in geometry problems?
2. What other properties of rhombus are useful in solving geometry problems?
3. Can you use other methods to find the area of a rhombus?
4. How does the concept of symmetry help in solving problems involving rhombus?
5. What are the properties of diagonals in a rhombus?
6. How can the Law of Cosines be applied in other types of quadrilaterals?
7. How do you find the angle measures using trigonometric functions?
8. What is the significance of using accurate trigonometric values in geometric calculations?

**Tip:** Always use precise trigonometric values from calculators or tables to ensure accuracy in geometric problems.

Learn how to calculate the area of a rhombus using the lengths of its diagonals and an included angle. This geometry problem involves applying the Law of Cosines and the formula for the area of a rhombus. Suitable for high school students.

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