The exterior angles obtained on producing the base BC of a tri angle ABC in both ways are `120^@` and `105^@`, then the vertical `/_A` of the triangle is of measure

Options

A. `36^@`
B. `40^@`
C. `45^@`
D. `55^@`

Answer

To find the measure of the vertical angle A of triangle ABC given the exterior angles at base BC, we can follow these steps:

### Step-by-Step Solution:

1. **Draw Triangle ABC**:
- Sketch triangle ABC with vertices A, B, and C. Extend the base BC in both directions to points D and E.

2. **Label the Exterior Angles**:
- The exterior angle at point D (produced from B) is given as 120°.
- The exterior angle at point E (produced from C) is given as 105°.

3. **Use the Exterior Angle Theorem**:
- According to the Exterior Angle Theorem, the exterior angle is equal to the sum of the two opposite interior angles.
- For angle D (120°):
\[ \text{Angle DBA} + \text{Angle ABC} = 180° \]
\[ 120° + \text{Angle ABC} = 180° \]
\[ \text{Angle ABC} = 180° - 120° = 60° \]

4. **Calculate Angle ACB**:
- For angle E (105°):
\[ \text{Angle ECA} + \text{Angle ACB} = 180° \]
\[ 105° + \text{Angle ACB} = 180° \]
\[ \text{Angle ACB} = 180° - 105° = 75° \]

5. **Sum of Angles in Triangle**:
- The sum of the angles in triangle ABC is 180°:
\[ \text{Angle A} + \text{Angle ABC} + \text{Angle ACB} = 180° \]
\[ \text{Angle A} + 60° + 75° = 180° \]
\[ \text{Angle A} + 135° = 180° \]

6. **Solve for Angle A**:
- Rearranging gives:
\[ \text{Angle A} = 180° - 135° = 45° \]

### Final Answer:
The measure of angle A in triangle ABC is **45°**.