If in a triangle ABC, the angles at B and Care 1.5 and 2.5 times of the angle at A respectively, then angle at B is

Options

A. `36^@`
B. `54^@`
C. `48^@`
D. `72^@`

Answer

To solve the problem, we need to find the angle at B in triangle ABC, given that the angles at B and C are 1.5 and 2.5 times the angle at A, respectively.

### Step-by-Step Solution:

1. **Define the angles**:
Let the angle at A be \( x \).
Then, the angle at B can be expressed as \( 1.5x \) and the angle at C as \( 2.5x \).

2. **Use the triangle angle sum property**:
The sum of the angles in a triangle is always 180 degrees. Therefore, we can write the equation:
\[ x + 1.5x + 2.5x = 180 \]

3. **Combine like terms**:
Adding the coefficients of \( x \):
\[ (1 + 1.5 + 2.5)x = 180 \]
This simplifies to:
\[ 5x = 180 \]

4. **Solve for \( x \)**:
To find \( x \), divide both sides of the equation by 5:
\[ x = \frac{180}{5} = 36 \]

5. **Find the angle at B**:
Now that we have \( x \), we can find the angle at B:
\[ \text{Angle at B} = 1.5x = 1.5 \times 36 = 54 \]

Thus, the angle at B is **54 degrees**.

### Summary:
- Angle A = 36 degrees
- Angle B = 54 degrees
- Angle C = 90 degrees (not needed for the answer but can be calculated)