The sum of two angles of a triangle is `116^(@)` and their difference is `24^(@)`. Find the measure of each angle of the triangle.

Answer

To solve the problem, we need to find the measures of the two angles of a triangle given that their sum is \(116^\circ\) and their difference is \(24^\circ\).

Let's denote the two angles as \(a\) and \(b\).

### Step 1: Set up the equations
From the problem, we can write two equations based on the given information:
1. \(a + b = 116^\circ\) (Equation 1)
2. \(a - b = 24^\circ\) (Equation 2)

### Step 2: Solve the equations
To find the values of \(a\) and \(b\), we can add Equation 1 and Equation 2 together.

\[ (a + b) + (a - b) = 116^\circ + 24^\circ \]
This simplifies to:
\[ 2a = 140^\circ \]

### Step 3: Calculate \(a\)
Now, divide both sides by 2 to find \(a\):
\[ a = \frac{140^\circ}{2} = 70^\circ \]

### Step 4: Substitute \(a\) back to find \(b\)
Now that we have \(a\), we can substitute it back into Equation 1 to find \(b\):
\[ 70^\circ + b = 116^\circ \]
Subtract \(70^\circ\) from both sides:
\[ b = 116^\circ - 70^\circ = 46^\circ \]

### Step 5: Find the third angle \(c\)
In a triangle, the sum of all angles is \(180^\circ\). Therefore, we can find the third angle \(c\):
\[ c = 180^\circ - (a + b) = 180^\circ - (70^\circ + 46^\circ) \]
Calculating this gives:
\[ c = 180^\circ - 116^\circ = 64^\circ \]

### Final Answer
The measures of the angles in the triangle are:
- \(a = 70^\circ\)
- \(b = 46^\circ\)
- \(c = 64^\circ\)