The smallest angle of a triangle is `40^@` less than the largest angle. If the largest angle is `80^@` then find the third angle of the triangle

Options

A. `40^@`
B. `90^@`
C. `80^@`
D. `60^@`

Answer

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

### Step 1: Identify the largest angle
We are given that the largest angle of the triangle is \(80^\circ\).

### Step 2: Determine the smallest angle
The smallest angle is \(40^\circ\) less than the largest angle. Therefore, we can calculate it as follows:
\[ \text{Smallest angle} = \text{Largest angle} - 40^\circ = 80^\circ - 40^\circ = 40^\circ \]

### Step 3: Use the triangle angle sum property
The sum of the angles in a triangle is always \(180^\circ\). Let the third angle be \(x\). We can set up the equation:
\[ \text{Largest angle} + \text{Smallest angle} + \text{Third angle} = 180^\circ \]
Substituting the known values:
\[ 80^\circ + 40^\circ + x = 180^\circ \]

### Step 4: Simplify the equation
Combine the known angles:
\[ 120^\circ + x = 180^\circ \]

### Step 5: Solve for the third angle
To find \(x\), we subtract \(120^\circ\) from both sides:
\[ x = 180^\circ - 120^\circ = 60^\circ \]

### Conclusion
The third angle of the triangle is \(60^\circ\).

### Summary of Angles
- Smallest angle: \(40^\circ\)
- Largest angle: \(80^\circ\)
- Third angle: \(60^\circ\)