Two angles of a hexagon are `120^@ and 160^@.` If the remaining four angles are equal, find each equal angle.

Options

A. `190^@`
B. `140^@`
C. `110^@`
D. `120^@`

Answer

To find each equal angle in the hexagon, we can follow these steps:

### Step 1: Calculate the sum of the interior angles of the hexagon.
The formula for the sum of the interior angles of a polygon is given by:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]
where \( n \) is the number of sides. For a hexagon, \( n = 6 \).

**Calculation:**
\[ \text{Sum of interior angles} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \]

### Step 2: Set up the equation for the angles.
We know two angles of the hexagon: \( 120^\circ \) and \( 160^\circ \). Let the remaining four equal angles be represented by \( x \).

**Equation:**
\[ 120^\circ + 160^\circ + 4x = 720^\circ \]

### Step 3: Simplify the equation.
First, add the known angles:
\[ 120^\circ + 160^\circ = 280^\circ \]
Now, substitute this back into the equation:
\[ 280^\circ + 4x = 720^\circ \]

### Step 4: Solve for \( x \).
Subtract \( 280^\circ \) from both sides:
\[ 4x = 720^\circ - 280^\circ \]
\[ 4x = 440^\circ \]
Now, divide both sides by 4:
\[ x = \frac{440^\circ}{4} = 110^\circ \]

### Conclusion:
Each of the four equal angles in the hexagon is \( 110^\circ \).

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