Complementary & Supplementary Angles

Overview

This lecture explains theorems involving complementary and supplementary angles, focusing on how congruence relationships arise when angles share the same complement or supplement, or are related to congruent angles. The theorems are useful for working with multiple angles, whether they overlap or come from different diagrams.

Complementary & Supplementary Angles: Definitions and Properties

• Complementary angles: Two angles whose measures add up to 90°.
• Supplementary angles: Two angles whose measures add up to 180°.
• These theorems apply only to angles, not to segments.
• The theorems are especially helpful when dealing with overlapping angles or angles from different diagrams.

Theorems Involving Shared Complements and Supplements

• There are four main versions of these theorems:
  • Complementary to the same angle: If two angles are each complementary to the same angle, then those two angles are congruent.
  • Complementary to congruent angles: If two angles are each complementary to two congruent angles, then those two angles are congruent.
  • Supplementary to the same angle: If two angles are each supplementary to the same angle, then those two angles are congruent.
  • Supplementary to congruent angles: If two angles are each supplementary to two congruent angles, then those two angles are congruent.
• These theorems are similar to the addition and subtraction properties discussed previously, but use the specific terms "complementary" and "supplementary."

Worked Examples & Applications

• Example 1: Complementary to the Same Angle
  • If ∠1 is complementary to ∠2, and ∠2 is complementary to ∠3, then ∠1 ≅ ∠3.
  • For instance, if ∠1 = 40°, then ∠2 = 50° (since 40° + 50° = 90°). If ∠2 is also complementary to ∠3, then ∠3 = 40°, so ∠1 ≅ ∠3.
  • Conclusion: If two angles are complementary to the same angle, they are congruent.
• Example 2: Supplementary to the Same Angle
  • If ∠A is supplementary to ∠B, and ∠C is supplementary to ∠B, then ∠A ≅ ∠C.
  • For example, if ∠A = 120°, then ∠B = 60° (since 120° + 60° = 180°). If ∠C is also supplementary to ∠B, then ∠C = 120°, so ∠A ≅ ∠C.
  • Conclusion: If two angles are supplementary to the same angle, they are congruent.
• Example 3: Complementary to Congruent Angles
  • If ∠1 is complementary to ∠3, ∠2 is complementary to ∠4, and ∠3 ≅ ∠4, then ∠1 ≅ ∠2.
  • For instance, if ∠1 = 30°, ∠3 = 60°, ∠4 = 60°, and ∠2 is complementary to ∠4, then ∠2 = 30°, so ∠1 ≅ ∠2.
  • Conclusion: If two angles are complementary to two congruent angles, they are congruent.
• Note: The same logic applies for supplementary angles and congruent supplements.

Key Terms & Definitions

• Complementary Angles: Two angles whose measures add up to 90°.
• Supplementary Angles: Two angles whose measures add up to 180°.
• Congruent Angles: Angles that have the same measure.

Action Items / Next Steps

• Add these theorems to your notes or table of contents under "Complementary and Supplementary Theorems."
• Practice applying these theorems to sample diagrams and problems, especially focusing on identifying when angles share the same complement or supplement, or are related to congruent angles.
• If you have questions, review the examples or ask for clarification.