In `Delta PQR, angle P: angle Q : angle R = 1:3:5`. What is the value of (in degress) of `angle R - angle P`?

Options

A. 30
B. 80
C. 45
D. 60

Answer

To solve the problem, we need to find the value of \( \text{angle R} - \text{angle P} \) in triangle \( PQR \) where the angles are in the ratio \( 1:3:5 \).

### Step-by-Step Solution:

1. **Understand the Ratio of Angles**:
The angles of triangle \( PQR \) are given in the ratio \( 1:3:5 \). Let's denote:
- \( \text{angle P} = x \)
- \( \text{angle Q} = 3x \)
- \( \text{angle R} = 5x \)

2. **Set Up the Equation**:
The sum of the angles in any triangle is \( 180^\circ \). Therefore, we can write the equation:
\[ \text{angle P} + \text{angle Q} + \text{angle R} = 180^\circ \]
Substituting the expressions for the angles, we get:
\[ x + 3x + 5x = 180^\circ \]

3. **Combine Like Terms**:
Combine the terms on the left side:
\[ 9x = 180^\circ \]

4. **Solve for \( x \)**:
To find \( x \), divide both sides by 9:
\[ x = \frac{180^\circ}{9} = 20^\circ \]

5. **Calculate Each Angle**:
Now that we have \( x \), we can find the measures of each angle:
- \( \text{angle P} = x = 20^\circ \)
- \( \text{angle Q} = 3x = 3 \times 20^\circ = 60^\circ \)
- \( \text{angle R} = 5x = 5 \times 20^\circ = 100^\circ \)

6. **Find \( \text{angle R} - \text{angle P} \)**:
Now, we can calculate \( \text{angle R} - \text{angle P} \):
\[ \text{angle R} - \text{angle P} = 100^\circ - 20^\circ = 80^\circ \]

### Final Answer:
The value of \( \text{angle R} - \text{angle P} \) is \( 80^\circ \).