Which of the following is not true for an exterior angle of a regular polygon with n sides?

Options

A. Each exterior angle = `(360^(@))/n`
B. Exterior angle = 180° – interior angle
C. `n=(360^(@))/("exterior angle")`
D. Each exterior angle = `((n-2)xx180^(@))/n`

Answer

To determine which statement is not true for an exterior angle of a regular polygon with \( n \) sides, we can analyze each of the given conditions step by step.

### Step 1: Understand the properties of exterior angles
The sum of all exterior angles of any polygon is always \( 360^\circ \). For a regular polygon, each exterior angle can be calculated by dividing the total sum by the number of sides \( n \).

### Step 2: Evaluate the first condition
**Condition 1:** Each exterior angle is equal to \( \frac{360}{n} \).

- This is true because if the sum of the exterior angles is \( 360^\circ \) and there are \( n \) sides, then each exterior angle must be \( \frac{360}{n} \).

### Step 3: Evaluate the second condition
**Condition 2:** The exterior angle is equal to \( 180^\circ - \) interior angle.

- This is also true. The relationship between the exterior angle and the interior angle of a polygon is that they are supplementary, meaning they add up to \( 180^\circ \).

### Step 4: Evaluate the third condition
**Condition 3:** \( n = \frac{360}{\text{exterior angle}} \).

- This condition is true as well. If you know the measure of each exterior angle, you can find the number of sides \( n \) by rearranging the first condition.

### Step 5: Evaluate the fourth condition
**Condition 4:** Each exterior angle is equal to \( \frac{(n-2) \times 180}{n} \).

- This condition is **not true**. The formula \( \frac{(n-2) \times 180}{n} \) actually gives the measure of each interior angle, not the exterior angle. Therefore, this statement is incorrect.

### Conclusion
The statement that is not true for an exterior angle of a regular polygon with \( n \) sides is the fourth condition: **Each exterior angle is equal to \( \frac{(n-2) \times 180}{n} \)**.

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