In a `triangle PQR , angle P = 110^(@) `, PQ = PR . Find `angle Q and angle R `

Answer

To solve the problem, we need to find the angles Q and R in triangle PQR, where angle P is given as 110 degrees and sides PQ and PR are equal.

### Step-by-Step Solution:

1. **Identify Given Information**:
- Angle P = 110 degrees
- PQ = PR (which indicates that triangle PQR is isosceles)

2. **Use the Isosceles Triangle Property**:
- Since PQ = PR, the angles opposite these sides (angle Q and angle R) must be equal. Therefore, we can say:
\[ \text{Angle Q} = \text{Angle R} \]

3. **Apply the Angle Sum Property of a Triangle**:
- The sum of the angles in any triangle is 180 degrees. Therefore, we can write:
\[ \text{Angle P} + \text{Angle Q} + \text{Angle R} = 180^\circ \]
- Substituting the known values:
\[ 110^\circ + \text{Angle Q} + \text{Angle R} = 180^\circ \]

4. **Substitute Angle Q for Angle R**:
- Since Angle Q = Angle R, we can denote both as Q:
\[ 110^\circ + Q + Q = 180^\circ \]
\[ 110^\circ + 2Q = 180^\circ \]

5. **Solve for Q**:
- Rearranging the equation gives:
\[ 2Q = 180^\circ - 110^\circ \]
\[ 2Q = 70^\circ \]
- Dividing by 2:
\[ Q = \frac{70^\circ}{2} = 35^\circ \]

6. **Find Angle R**:
- Since Angle Q = Angle R, we have:
\[ \text{Angle R} = 35^\circ \]

### Final Answer:
- Angle Q = 35 degrees
- Angle R = 35 degrees