Lesson 5.5: Exterior Angle Theorem & Triangle Sum Properties

Learning Objectives

• State and apply the Triangle Sum Property.
• Understand and prove the Exterior Angle Theorem.
• Use these properties to solve triangle problems and proofs.

Topics Covered

• Triangle Sum Property: \( \angle A + \angle B + \angle C = 180^\circ \)
• Exterior Angle Theorem: Exterior angle equals sum of opposite interior angles
• Applications in triangle proofs
• Diagram interpretation and angle calculations

Key Terms

• Interior Angle: An angle inside a triangle.
• Exterior Angle: An angle formed by extending a side of a triangle.
• Triangle Sum Property: Sum of interior angles of a triangle equals 180°.
• Exterior Angle Theorem: An exterior angle equals the sum of the two non-adjacent interior angles.

Lesson Examples

Example 1: Using the Triangle Sum Property

1. Given triangle ABC with angles A = 50°, B = 60°.
2. Find angle C.
3. Solution: \( \angle C = 180° - (50° + 60°) = 70° \)

Example 2: Using the Exterior Angle Theorem

1. Triangle XYZ with exterior angle at X = 110°.
2. Find sum of opposite interior angles Y + Z.
3. Solution: By the Exterior Angle Theorem, \( \angle Y + \angle Z = 110° \)

Practice Problems

1. Triangle DEF with angles 40° and 65°. Find the third angle.
2. Triangle GHI has an exterior angle of 120° at vertex G. Find sum of opposite interior angles.
3. Verify the Triangle Sum Property for triangle JKL with angles 55°, 75°, and 50°.

Tips & Common Mistakes

• Check that exterior angles are formed by extending a side of the triangle.
• Do not confuse the exterior angle with an interior angle.
• Ensure the sum of interior angles always equals 180°.

Summary

The Triangle Sum Property and Exterior Angle Theorem are fundamental tools for reasoning about angles in triangles. These principles are essential for proofs, solving unknown angles, and understanding triangle relationships.

Challenge Problems

1. Triangle MNO has interior angles 35° and 75°. Find the exterior angle at the third vertex.
2. Prove that an exterior angle of a triangle is always greater than either of the non-adjacent interior angles individually.
3. Given triangle PQR, extend side PQ and label exterior angle R. Verify the Exterior Angle Theorem numerically.