The angles of a triangle are in AP and the greatest angle is `75^(@)`. Find all the three angles in degrees :

Options

A. `55^(@), 55^(@), 70^(@)`
B. `45^(@), 60^(@), 75^(@)`
C. `40^(@), 65^(@), 75^(@)`
D. none of these

Answer

To solve the problem step by step, we will find the three angles of a triangle that are in arithmetic progression (AP) with the greatest angle being \(75^\circ\).

### Step 1: Define the angles
Let the three angles of the triangle be:
- First angle: \( a - d \)
- Second angle: \( a \)
- Third angle: \( a + d \)

Here, \(a\) is the middle angle, and \(d\) is the common difference.

### Step 2: Use the information given
According to the problem, the greatest angle is \(75^\circ\). Therefore, we can write:
\[ a + d = 75^\circ \quad \text{(1)} \]

### Step 3: Use the triangle angle sum property
The sum of the angles in a triangle is \(180^\circ\). Thus, we can write:
\[ (a - d) + a + (a + d) = 180^\circ \]
Simplifying this gives:
\[ 3a = 180^\circ \]
\[ a = \frac{180^\circ}{3} = 60^\circ \quad \text{(2)} \]

### Step 4: Substitute \(a\) back into equation (1)
Now, substitute \(a = 60^\circ\) into equation (1):
\[ 60^\circ + d = 75^\circ \]
Solving for \(d\):
\[ d = 75^\circ - 60^\circ = 15^\circ \quad \text{(3)} \]

### Step 5: Find the three angles
Now we can find the three angles:
1. First angle:
\[ a - d = 60^\circ - 15^\circ = 45^\circ \]
2. Second angle:
\[ a = 60^\circ \]
3. Third angle:
\[ a + d = 60^\circ + 15^\circ = 75^\circ \]

### Final Answer
The three angles of the triangle are:
- First angle: \(45^\circ\)
- Second angle: \(60^\circ\)
- Third angle: \(75^\circ\)