Overview
This lecture explains the two main properties of exterior angles in triangles, their proofs, and how to solve related problems using these concepts.
Properties of Exterior Angles in Triangles
- The sum of the exterior angles of any triangle is always 360 degrees.
- The sum of the exterior angles of any polygon is also 360 degrees.
- An exterior angle of a triangle is equal to the sum of the two remote interior angles.
Proofs
- Drawing a parallel line through a triangle demonstrates the exterior angles sum to a full circle (360°).
- Proof for the exterior angle theorem: X (exterior angle) = 180° - Y (interior angle); X = sum of two remote interior angles using properties of triangle and linear pair.
Example Problems and Solutions
- To find an exterior angle X, add the two remote interior angles.
- In a triangle with angles 40° and 50°, the exterior angle X = 40° + 50° = 90°.
- For an equilateral triangle, each angle is 60°, so exterior angle X = 180° - 60° = 120°.
- In an isosceles triangle with angle 20°, the other two angles are each 80°, so X = 80° + 20° = 100°.
- In a right triangle with an angle of 40°, the exterior angle X = 90° + 40° = 130°.
Additional Questions Demonstrating Key Rules
- Given two exterior angles (e.g., 150° and 80°), the third exterior angle X = 360° - 150° - 80° = 130°.
- For triangles with equal sides (isosceles), first find equal angles before calculating the exterior angle using the sum of remote interiors.
- For a triangle with unique configurations (right, isosceles), consistently use the two main rules for finding unknown angles.
Test Problem Example
- If in a triangle with three equal sides, let angles be A and B in the configuration, then the sum A + B = 90° as proven algebraically or by substituting values.
Key Terms & Definitions
- Exterior Angle — An angle formed by one side of a triangle and the extension of another side.
- Remote Interior Angles — The two interior angles of a triangle not adjacent to a given exterior angle.
- Linear Pair — Two adjacent angles that form a straight line, adding to 180°.
- Isosceles Triangle — A triangle with two sides of equal length.
- Equilateral Triangle — A triangle with all three sides and angles equal.
Action Items / Next Steps
- Practice problems: Find exterior angles for various triangle types using the discussed rules.
- Review definitions and properties for triangles and exterior angles.
- Attempt the test problem and verify your answer.