The angles of a quadrilateral are `r^(@), (x - 10)^(@), (x + 30)^(@) and (2x)^(@)` , the smallest angle is equal

Options

A. `68^(@)`
B. `52^(@)`
C. `58^(@)`
D. `47^(@)`

Answer

To solve the problem, we need to find the angles of the quadrilateral given by the expressions \( r^\circ, (x - 10)^\circ, (x + 30)^\circ, \) and \( (2x)^\circ \) and determine which one is the smallest.

### Step-by-step Solution:

1. **Understand the Sum of Angles in a Quadrilateral**:
The sum of all angles in a quadrilateral is \( 360^\circ \).

2. **Set Up the Equation**:
We can express the sum of the angles as follows:
\[ r + (x - 10) + (x + 30) + (2x) = 360 \]
Here, we will assume \( r = x \) for simplicity, as the problem does not provide a specific value for \( r \).

3. **Combine Like Terms**:
Combine the terms in the equation:
\[ x + (x - 10) + (x + 30) + (2x) = 360 \]
This simplifies to:
\[ 5x + 20 = 360 \]

4. **Solve for \( x \)**:
Subtract \( 20 \) from both sides:
\[ 5x = 360 - 20 \]
\[ 5x = 340 \]
Now, divide by \( 5 \):
\[ x = \frac{340}{5} = 68 \]

5. **Calculate Each Angle**:
Now that we have \( x = 68 \), we can find each angle:
- First angle: \( x = 68^\circ \)
- Second angle: \( x - 10 = 68 - 10 = 58^\circ \)
- Third angle: \( x + 30 = 68 + 30 = 98^\circ \)
- Fourth angle: \( 2x = 2 \times 68 = 136^\circ \)

6. **List the Angles**:
The angles are:
- \( 68^\circ \)
- \( 58^\circ \)
- \( 98^\circ \)
- \( 136^\circ \)

7. **Identify the Smallest Angle**:
Among the calculated angles, the smallest angle is \( 58^\circ \).

### Final Answer:
The smallest angle is \( 58^\circ \).