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 given in terms of \( x \) and then determine the type of triangle based on those angles.

### Step-by-Step Solution:

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

2. **Combine the angles:**
Combine all the terms:
\[ x + 15 + \frac{6x}{5} + 6 + \frac{2x}{3} + 30 = 180 \]
Simplifying the constants:
\[ x + \frac{6x}{5} + \frac{2x}{3} + 51 = 180 \]

3. **Isolate the variable terms:**
Move the constant to the right side:
\[ x + \frac{6x}{5} + \frac{2x}{3} = 180 - 51 \]
\[ x + \frac{6x}{5} + \frac{2x}{3} = 129 \]

4. **Find a common denominator:**
The least common multiple (LCM) of the denominators 1, 5, and 3 is 15. Rewrite each term:
\[ \frac{15x}{15} + \frac{18x}{15} + \frac{10x}{15} = 129 \]
Combine the fractions:
\[ \frac{15x + 18x + 10x}{15} = 129 \]
\[ \frac{43x}{15} = 129 \]

5. **Solve for \( x \):**
Multiply both sides by 15:
\[ 43x = 129 \times 15 \]
Calculate \( 129 \times 15 \):
\[ 129 \times 15 = 1935 \]
Now divide by 43:
\[ x = \frac{1935}{43} \]
Calculate \( x \):
\[ x = 45 \]

6. **Calculate the angles:**
Now substitute \( x = 45 \) back into the expressions for the angles:
- First angle:
\[ x + 15 = 45 + 15 = 60^\circ \]
- Second angle:
\[ \frac{6x}{5} + 6 = \frac{6 \times 45}{5} + 6 = 54 + 6 = 60^\circ \]
- Third angle:
\[ \frac{2x}{3} + 30 = \frac{2 \times 45}{3} + 30 = 30 + 30 = 60^\circ \]

7. **Determine the type of triangle:**
All three angles are \( 60^\circ \), which means the triangle is an **equilateral triangle**.

### Final Answer:
The triangle is an **equilateral triangle**.