6.2 Sum of Exterior Angles

1. Learning Objectives

Define an exterior angle of a polygon.
Understand the relationship between an interior angle and its adjacent exterior angle.
Apply the Exterior Angle Sum Theorem to solve geometric problems.

2. Key Notes

An exterior angle is formed by extending one side of a polygon. At each vertex, the interior and exterior angles are supplementary (they sum to 180°).

Sum of Exterior Angles = 360°

This rule applies to all convex polygons, regardless of the number of sides (n). Whether it is a triangle or a 100-sided polygon, the sum is always 360°.

1. What is the sum of exterior angles of a triangle?

By theorem, the sum for any convex polygon is 360°. Ans: 360°

2. What is the sum of exterior angles of a decagon (10 sides)?

The number of sides doesn't change the sum. Ans: 360°

3. If an interior angle is 120°, what is its adjacent exterior angle?

Interior + Exterior = 180°. 180° - 120° = 60°. Ans: 60°

4. A polygon has 50 sides. What is the sum of its exterior angles?

Still 360°. It is a constant property. Ans: 360°

5. If each exterior angle of a regular polygon is 45°, how many sides does it have?

n = 360 / exterior angle. 360 / 45 = 8. Ans: 8 sides (Octagon)

6. Find the number of sides if each exterior angle is 72°.

360 / 72 = 5. Ans: 5 sides (Pentagon)

7. In a regular hexagon, find the measure of one exterior angle.

Hexagon has 6 sides. 360 / 6 = 60°. Ans: 60°

8. If the sum of interior angles is 180°, what is the sum of exterior angles?

This describes a triangle. The exterior sum is always 360°. Ans: 360°

9. True or False: Exterior angles can be reflex angles (>180°).

False. For convex polygons, exterior angles are always < 180°. Ans: False

10. Find the exterior angle of a square.

360 / 4 = 90°. Also, 180 - 90 (interior) = 90°. Ans: 90°

11. A polygon has exterior angles of 100°, 80°, and 90°. Find the 4th angle.

360 - (100 + 80 + 90) = 360 - 270 = 90°. Ans: 90°

12. Find the measure of each exterior angle of a regular dodecagon (12 sides).

360 / 12 = 30°. Ans: 30°

13. If an exterior angle is 10°, how many sides are there?

360 / 10 = 36. Ans: 36 sides

14. What is the sum of exterior angles of a 1000-gon?

Constant property: 360°. Ans: 360°

15. If each exterior angle is 120°, what polygon is it?

360 / 120 = 3. Ans: Equilateral Triangle

16. Can a regular polygon have an exterior angle of 50°?

n = 360 / 50 = 7.2. Since 'n' must be a whole number, it is impossible. Ans: No

17. Find the interior angle if the exterior angle is 40°.

180 - 40 = 140°. Ans: 140°

18. The exterior angles of a triangle are in ratio 2:3:4. Find the smallest exterior angle.

2x + 3x + 4x = 360 → 9x = 360 → x = 40. Smallest is 2x = 80°. Ans: 80°

19. In a regular polygon, exterior angle = interior angle. Find n.

Both must be 90° (180/2). 360 / 90 = 4. Ans: 4 sides (Square)

20. As the number of sides increases, what happens to the size of each exterior angle?

As 'n' gets larger, 360/n gets smaller. Ans: Decreases