An exterior angle of a triangle is `105^(@)` and its two interior opposite angles are equal. Each of these equal angles is

Options

A. `37(1)/(2)@`
B. `52(1)/(2)@`
C. `72(1)/(2)@`
D. `75^(@)`

Explanation

To solve the problem step by step, we will follow the properties of triangles and exterior angles.

### Step-by-Step Solution:

1. **Understand the Problem**: We have a triangle where one of the exterior angles is given as \( 105^\circ \). The two opposite interior angles are equal. We need to find the measure of each of these equal angles.

2. **Draw the Triangle**: Let's denote the triangle as \( ABC \). Extend side \( BC \) to point \( D \) such that angle \( ACD \) is the exterior angle. Thus, \( \angle ACD = 105^\circ \).

3. **Identify the Interior Angles**: Let the two equal interior angles \( CAB \) and \( ABC \) be denoted as \( x \). Therefore, we have:
\[ \angle CAB = x \quad \text{and} \quad \angle ABC = x \]

4. **Apply the Exterior Angle Theorem**: According to the Exterior Angle Theorem, the exterior angle \( ACD \) is equal to the sum of the two opposite interior angles. Thus, we can write:
\[ \angle ACD = \angle CAB + \angle ABC \]
Substituting the known values, we get:
\[ 105^\circ = x + x \]

5. **Simplify the Equation**: This simplifies to:
\[ 105^\circ = 2x \]

6. **Solve for \( x \)**: To find \( x \), divide both sides by 2:
\[ x = \frac{105^\circ}{2} = 52.5^\circ \]

7. **Conclusion**: Each of the equal interior angles measures \( 52.5^\circ \).

### Final Answer:
Each of the equal angles is \( 52.5^\circ \).

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