In a octagon, four of the angles are equal and each of the others is 20° more than each of the first four. Find the angles of the octagon.

Answer

To find the angles of the octagon, we can follow these steps:

### Step 1: Understand the properties of an octagon
An octagon has 8 sides. The sum of the interior angles of a polygon can be calculated using the formula:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]
where \( n \) is the number of sides.

### Step 2: Calculate the sum of the interior angles of the octagon
For an octagon, \( n = 8 \):
\[ \text{Sum of interior angles} = (8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ \]

### Step 3: Define the angles
Let the measure of each of the four equal angles be \( x \). The other four angles are each \( 20^\circ \) more than the equal angles, so they can be expressed as \( x + 20^\circ \).

### Step 4: Set up the equation for the sum of the angles
The total sum of the angles can be expressed as:
\[ 4x + 4(x + 20^\circ) = 1080^\circ \]

### Step 5: Simplify the equation
Expanding the equation gives:
\[ 4x + 4x + 80^\circ = 1080^\circ \]
This simplifies to:
\[ 8x + 80^\circ = 1080^\circ \]

### Step 6: Solve for \( x \)
Subtract \( 80^\circ \) from both sides:
\[ 8x = 1080^\circ - 80^\circ \]
\[ 8x = 1000^\circ \]
Now, divide by 8:
\[ x = \frac{1000^\circ}{8} = 125^\circ \]

### Step 7: Find the other angles
The other four angles are:
\[ x + 20^\circ = 125^\circ + 20^\circ = 145^\circ \]

### Step 8: List all angles
Thus, the angles of the octagon are:
- Four angles of \( 125^\circ \)
- Four angles of \( 145^\circ \)

### Final Answer
The angles of the octagon are \( 125^\circ, 125^\circ, 125^\circ, 125^\circ, 145^\circ, 145^\circ, 145^\circ, 145^\circ \).

---