Two angles of a hexagon are 120 oand 100 o . If the remaining four angles are equal, then find its smallest angle.

1. Break Down the Problem

We need to find the smallest angle of a hexagon given two angles (120° and 100°) and that the remaining four angles are equal.

2. Relevant Concepts

The sum of the interior angles $S$ of a polygon can be calculated using the formula: $S = (n - 2) \times 180^\circ$ where $n$ is the number of sides in the polygon.

For a hexagon ($n = 6$): $S = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ$

3. Analysis and Detail

Now, let's denote the remaining four equal angles by $x$. The equation for the sum of the angles in the hexagon will be: $120^\circ + 100^\circ + 4x = 720^\circ$ Simplifying the equation: $220^\circ + 4x = 720^\circ$ Subtracting $220^\circ$ from both sides: $4x = 720^\circ - 220^\circ$ $4x = 500^\circ$ Now, divide by 4: $x = \frac{500^\circ}{4} = 125^\circ$

4. Verify and Summarize

All angles of the hexagon are:

• 120°
• 100°
• 125° (four times)

The sum: $120^\circ + 100^\circ + 4 \times 125^\circ = 120^\circ + 100^\circ + 500^\circ = 720^\circ$

Thus, the calculations verify the total as required.

Final Answer

The smallest angle of the hexagon is $125^\circ$.