Two angles of a triangle are equal and the third angle is greater than each one of them by `18^(@)`. Find the angles.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the Angles
Let the two equal angles of the triangle be denoted as \( x \). Therefore, we have:
- Angle 1 = \( x \)
- Angle 2 = \( x \)

### Step 2: Define the Third Angle
According to the problem, the third angle is greater than each of the equal angles by \( 18^\circ \). Thus, we can express the third angle as:
- Angle 3 = \( x + 18^\circ \)

### Step 3: Use the Triangle Sum Property
The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can set up the equation:
\[ x + x + (x + 18) = 180 \]

### Step 4: Simplify the Equation
Combine like terms:
\[ 3x + 18 = 180 \]

### Step 5: Solve for \( x \)
Subtract \( 18 \) from both sides:
\[ 3x = 180 - 18 \]
\[ 3x = 162 \]
Now, divide both sides by \( 3 \):
\[ x = \frac{162}{3} = 54 \]

### Step 6: Find the Angles
Now that we have \( x \), we can find the three angles:
- Angle 1 = \( x = 54^\circ \)
- Angle 2 = \( x = 54^\circ \)
- Angle 3 = \( x + 18 = 54 + 18 = 72^\circ \)

### Conclusion
The angles of the triangle are:
- \( 54^\circ, 54^\circ, \) and \( 72^\circ \).

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