If the difference between the measures of the two smaller angles of a right angled triangle is `8^@`, then the smallest angle is :

Options

A. `37^@`
B. `41^@`
C. `42^@`
D. `49^@`

Answer

To solve the problem, we need to find the smallest angle of a right-angled triangle given that the difference between the two smaller angles is \(8^\circ\).

### Step-by-Step Solution:

1. **Understanding the Angles in a Right-Angled Triangle**:
In a right-angled triangle, one angle is \(90^\circ\). The sum of the other two angles must be \(90^\circ\) since the total sum of angles in a triangle is \(180^\circ\).

2. **Let the Angles be Defined**:
Let's denote the two smaller angles as \(x\) and \(y\). According to the problem, we know:
\[ y - x = 8^\circ \]
(where \(y\) is the larger angle).

3. **Sum of the Angles**:
Since the sum of the two smaller angles is \(90^\circ\), we can write:
\[ x + y = 90^\circ \]

4. **Substituting for y**:
From the first equation, we can express \(y\) in terms of \(x\):
\[ y = x + 8^\circ \]

5. **Substituting into the Sum Equation**:
Now substitute \(y\) into the sum equation:
\[ x + (x + 8^\circ) = 90^\circ \]
This simplifies to:
\[ 2x + 8^\circ = 90^\circ \]

6. **Solving for x**:
To isolate \(x\), subtract \(8^\circ\) from both sides:
\[ 2x = 90^\circ - 8^\circ \]
\[ 2x = 82^\circ \]
Now divide by 2:
\[ x = \frac{82^\circ}{2} = 41^\circ \]

7. **Finding y**:
Now, substitute \(x\) back to find \(y\):
\[ y = x + 8^\circ = 41^\circ + 8^\circ = 49^\circ \]

8. **Conclusion**:
The smallest angle \(x\) is \(41^\circ\).

### Final Answer:
The smallest angle is \(41^\circ\).