Lesson 6.1: Angle Sums

Learning Objectives

• Calculate the sum of interior angles in triangles, quadrilaterals, and n-gons.
• Understand the relationship between interior and exterior angles.
• Apply angle sum formulas in problem solving.

Topics Covered

• Triangle interior angle sum: 180°
• Quadrilateral interior angle sum: 360°
• Polygon interior angle sum formula: (n-2) × 180°
• Exterior angles and their sums
• Applications to regular polygons

Key Terms

• Interior Angle: An angle inside a polygon.
• Exterior Angle: An angle formed by extending one side of a polygon.
• Polygon: A closed figure with straight sides.
• Regular Polygon: All sides and angles are equal.

Lesson Examples

Example 1: Triangle Angle Sum

Triangle with angles 50°, 60°, 70° → 50 + 60 + 70 = 180° ✅

Example 2: Quadrilateral Angle Sum

Quadrilateral with angles 90°, 80°, 100°, 90° → Sum = 360° ✅

Example 3: Polygon Interior Angles

For a hexagon (n = 6): Sum = (6-2) × 180 = 720°

Example 4: Exterior Angles

Sum of exterior angles of any polygon = 360°

Practice Problems

1. Calculate the sum of interior angles of a pentagon.
2. Find one missing angle of a triangle with angles 50° and 60°.
3. Determine the measure of each exterior angle of a regular octagon.

Tips & Common Mistakes

• Remember the formula (n-2)×180° for polygons.
• Exterior and interior angles at the same vertex sum to 180°.
• Check your sums carefully to avoid arithmetic errors.

Summary

The sum of interior angles depends on the number of sides of a polygon. Triangles sum to 180°, quadrilaterals to 360°, and n-gons follow (n-2)×180°. Exterior angles always sum to 360°.

Challenge Problems

1. Calculate the sum of interior angles of a 12-sided polygon.
2. One interior angle of a regular pentagon is missing; find it.
3. Verify the sum of exterior angles for a 15-sided polygon.