In `trianglePQR , angleR = angle P,` `QR = 4 cm` and `PR = 5 cm` , Then `PQ` = ________

Options

A. 4 cm
B. 5 cm
C. 1 cm
D. 9 cm

Answer

To solve the problem, we will use the properties of triangles, specifically the properties of isosceles triangles since we know that two angles are equal.

### Step-by-Step Solution:

1. **Identify the Triangle and Given Information**:
We have triangle PQR where:
- Angle R = Angle P
- QR = 4 cm
- PR = 5 cm

2. **Recognize the Type of Triangle**:
Since angle R is equal to angle P, triangle PQR is an isosceles triangle. In an isosceles triangle, the sides opposite to equal angles are also equal.

3. **Determine the Sides Opposite to the Equal Angles**:
- The side opposite angle P is QR.
- The side opposite angle R is PQ.

4. **Set Up the Equation Based on the Isosceles Triangle Property**:
Since angle R = angle P, we have:
\[ QR = PQ \]

5. **Substitute the Known Length**:
We know that QR = 4 cm. Therefore:
\[ PQ = QR = 4 \text{ cm} \]

6. **Conclusion**:
Thus, the length of PQ is 4 cm.

### Final Answer:
PQ = 4 cm

---