If the three angles of a triangle are : `(x + 15^(@))`, `((6x)/5 + 6^(@))` and `((2x)/3 + 30^(@))`, then the triangle is :

Options

A. isosceles
B. right angled
C. equilateral
D. scalene

Answer

To solve the problem, we need to find the angles of the triangle and determine the type of triangle based on those angles.

### Step-by-Step Solution:

1. **Identify the angles of the triangle:**
- Let the three angles be:
- Angle A = \( x + 15^\circ \)
- Angle B = \( \frac{6x}{5} + 6^\circ \)
- Angle C = \( \frac{2x}{3} + 30^\circ \)

2. **Use the property of triangles:**
- The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can set up the equation:
\[ (x + 15) + \left(\frac{6x}{5} + 6\right) + \left(\frac{2x}{3} + 30\right) = 180 \]

3. **Combine like terms:**
- First, simplify the equation:
\[ x + 15 + \frac{6x}{5} + 6 + \frac{2x}{3} + 30 = 180 \]
- Combine the constant terms:
\[ x + \frac{6x}{5} + \frac{2x}{3} + 51 = 180 \]

4. **Eliminate the constants:**
- Subtract 51 from both sides:
\[ x + \frac{6x}{5} + \frac{2x}{3} = 129 \]

5. **Find a common denominator:**
- The common denominator for 1, 5, and 3 is 15. Rewrite the equation:
\[ \frac{15x}{15} + \frac{18x}{15} + \frac{10x}{15} = 129 \]
- Combine the fractions:
\[ \frac{15x + 18x + 10x}{15} = 129 \]
\[ \frac{43x}{15} = 129 \]

6. **Solve for x:**
- Multiply both sides by 15:
\[ 43x = 129 \times 15 \]
\[ 43x = 1935 \]
- Divide both sides by 43:
\[ x = \frac{1935}{43} = 45 \]

7. **Calculate the angles:**
- Angle A:
\[ A = x + 15 = 45 + 15 = 60^\circ \]
- Angle B:
\[ B = \frac{6x}{5} + 6 = \frac{6 \times 45}{5} + 6 = 54 + 6 = 60^\circ \]
- Angle C:
\[ C = \frac{2x}{3} + 30 = \frac{2 \times 45}{3} + 30 = 30 + 30 = 60^\circ \]

8. **Determine the type of triangle:**
- Since all three angles are equal (60°), the triangle is an **Equilateral Triangle**.

### Final Answer:
The triangle is an **Equilateral Triangle**.