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 find the third angle of the triangle, we can follow these steps:

### Step 1: Define the angles
Let the largest angle be denoted as \( A \), the smallest angle as \( B \), and the third angle as \( C \).

### Step 2: Assign the values
From the problem, we know:
- The largest angle \( A = 80^\circ \)
- The smallest angle \( B \) is \( 40^\circ \) less than the largest angle, so we can express it as:
\[ B = A - 40^\circ = 80^\circ - 40^\circ = 40^\circ \]

### Step 3: Use the triangle angle sum property
The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can write:
\[ A + B + C = 180^\circ \]
Substituting the known values:
\[ 80^\circ + 40^\circ + C = 180^\circ \]

### Step 4: Solve for the third angle \( C \)
Now, we can simplify the equation:
\[ 120^\circ + C = 180^\circ \]
To find \( C \), we subtract \( 120^\circ \) from both sides:
\[ C = 180^\circ - 120^\circ = 60^\circ \]

### Conclusion
Thus, the third angle \( C \) is \( 60^\circ \).

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