Sum of the Measure of Exterior Angles

The sum of the measure of exterior angles in a polygon can be determined by using the formula:

Sum of Exterior Angles = 360°

Definition of Exterior Angle

An exterior angle of a polygon is an angle formed by one side of the polygon and the extension of its adjacent side. In other words, it is an angle that is outside the polygon.

Explanation

To find the sum of the measure of exterior angles, we need to consider each angle individually and then add them together.

Let's consider a polygon with n sides. Each exterior angle of this polygon can be determined by subtracting the measure of the corresponding interior angle from 180°. The sum of all these exterior angles will be equal to 360°.

Individual Exterior Angles

Let's assume the polygon has sides labeled as x, y, z, p, q, and r.

- Exterior angle for side x: Measure of exterior angle for side x = 180° - Measure of interior angle for side x
- Exterior angle for side y: Measure of exterior angle for side y = 180° - Measure of interior angle for side y
- Exterior angle for side z: Measure of exterior angle for side z = 180° - Measure of interior angle for side z
- Exterior angle for side p: Measure of exterior angle for side p = 180° - Measure of interior angle for side p
- Exterior angle for side q: Measure of exterior angle for side q = 180° - Measure of interior angle for side q
- Exterior angle for side r: Measure of exterior angle for side r = 180° - Measure of interior angle for side r

Sum of Exterior Angles

To find the sum of the measure of exterior angles, we need to add up all the individual exterior angles.

Sum of Exterior Angles = (180° - Measure of interior angle for side x) + (180° - Measure of interior angle for side y) + (180° - Measure of interior angle for side z) + (180° - Measure of interior angle for side p) + (180° - Measure of interior angle for side q) + (180° - Measure of interior angle for side r)

Finally, we can simplify the expression by combining like terms and applying the distributive property.

Sum of Exterior Angles = 6 * 180° - (Measure of interior angle for side x + Measure of interior angle for side y + Measure of interior angle for side z + Measure of interior angle for side p + Measure of interior angle for side q + Measure of interior angle for side r)

Since the sum of the interior angles of any polygon with n sides is given by the formula (n - 2) * 180°, we can substitute this value in the above equation.

Sum of Exterior Angles = 6 * 180° - ((n - 2) * 180°)

Therefore, the sum of the measure of the exterior angles x, y, z, p, q, and r is equal to 6 * 180° - ((n - 2) * 180°), where n represents the number of sides in the polygon.

360 degrees