The interior angles of a pentagon are in the ratio 4:5:6:7:5. Find each angle of the pentagon.

Answer

To find each interior angle of the pentagon given the ratio 4:5:6:7:5, we can follow these steps:

### Step 1: Understand the formula for the sum of interior angles
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 of the polygon. For a pentagon, \( n = 5 \).

### Step 2: Calculate the sum of the interior angles of the pentagon
Substituting \( n = 5 \) into the formula:
\[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]

### Step 3: Set up the equations based on the given ratio
Let the angles of the pentagon be represented as:
- Angle 1 = \( 4x \)
- Angle 2 = \( 5x \)
- Angle 3 = \( 6x \)
- Angle 4 = \( 7x \)
- Angle 5 = \( 5x \)

### Step 4: Write the equation for the sum of the angles
Now, we can write the equation based on the sum of the angles:
\[ 4x + 5x + 6x + 7x + 5x = 540^\circ \]

### Step 5: Combine like terms
Combine the terms on the left side:
\[ (4x + 5x + 6x + 7x + 5x) = 27x \]
So, the equation becomes:
\[ 27x = 540^\circ \]

### Step 6: Solve for \( x \)
Now, divide both sides by 27:
\[ x = \frac{540^\circ}{27} = 20^\circ \]

### Step 7: Calculate each angle
Now that we have \( x \), we can find each angle:
- Angle 1: \( 4x = 4 \times 20^\circ = 80^\circ \)
- Angle 2: \( 5x = 5 \times 20^\circ = 100^\circ \)
- Angle 3: \( 6x = 6 \times 20^\circ = 120^\circ \)
- Angle 4: \( 7x = 7 \times 20^\circ = 140^\circ \)
- Angle 5: \( 5x = 5 \times 20^\circ = 100^\circ \)

### Step 8: List the angles
The interior angles of the pentagon are:
- \( 80^\circ \)
- \( 100^\circ \)
- \( 120^\circ \)
- \( 140^\circ \)
- \( 100^\circ \)