If In `Delta ABC, /_ABC=5 /_ACB `and `/_BAC = 3 /_ACB`, then `/_ABC = ?`

Options

A. `130^@`
B. `80^@`
C. `100^@`
D. `120^@`

Answer

To solve the problem, we need to find the value of angle \( \angle ABC \) given the relationships between the angles in triangle \( ABC \).

### Step-by-step Solution:

1. **Define the Angles**:
- Let \( \angle ACB = x \).
- According to the problem, \( \angle ABC = 5 \times \angle ACB = 5x \).
- Also, \( \angle BAC = 3 \times \angle ACB = 3x \).

2. **Set Up the Equation**:
- The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can write the equation:
\[ \angle ABC + \angle ACB + \angle BAC = 180^\circ \]
- Substituting the expressions we defined:
\[ 5x + x + 3x = 180^\circ \]

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

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

5. **Find \( \angle ABC \)**:
- Now that we have \( x \), we can find \( \angle ABC \):
\[ \angle ABC = 5x = 5 \times 20^\circ = 100^\circ \]

### Final Answer:
Thus, \( \angle ABC = 100^\circ \).

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