Angle Relationships
Angles form special relationships! Complementary angles sum to 90°, supplementary to 180°, vertical angles are equal, and adjacent angles share a side. Master these relationships to solve geometric puzzles and real-world problems.
Concept
Angle relationships are powerful tools for finding unknown angle measures! **ANGLE BASICS:** **Measuring Angles:** - Unit: degrees (°) - Tool: protractor - Range: 0° to 360° **Angle Types:** - Acute: 0° < angle < 90° - Right: exactly 90° - Obtuse: 90° < angle < 180° - Straight: exactly 180° **1. COMPLEMENTARY ANGLES:** **Definition:** Two angles that add up to 90° **Formula:** ∠A + ∠B = 90° **Examples:** - 30° and 60° are complementary (30 + 60 = 90) - 45° and 45° are complementary (45 + 45 = 90) - 25° and 65° are complementary (25 + 65 = 90) **Finding Complement:** If one angle is x°, its complement is (90 - x)° Example: Complement of 35° = 90° - 35° = 55° **Key Point:** Complementary angles form a RIGHT angle (90°) together. **2. SUPPLEMENTARY ANGLES:** **Definition:** Two angles that add up to 180° **Formula:** ∠A + ∠B = 180° **Examples:** - 120° and 60° are supplementary (120 + 60 = 180) - 90° and 90° are supplementary (90 + 90 = 180) - 135° and 45° are supplementary (135 + 45 = 180) **Finding Supplement:** If one angle is x°, its supplement is (180 - x)° Example: Supplement of 110° = 180° - 110° = 70° **Key Point:** Supplementary angles form a STRAIGHT angle (180°) together. **3. VERTICAL ANGLES:** **Definition:** Opposite angles formed when two lines intersect **Property:** Vertical angles are ALWAYS EQUAL! **Diagram:** ∠1 | ∠2 ----+---- ∠3 | ∠4 ∠1 = ∠3 (vertical angles) ∠2 = ∠4 (vertical angles) Also: ∠1 + ∠2 = 180° (supplementary - linear pair) **Example:** If ∠1 = 65°, then ∠3 = 65° (vertical angles) And ∠2 = ∠4 = 115° (180° - 65°) **4. ADJACENT ANGLES:** **Definition:** Two angles that: - Share a common vertex (corner point) - Share a common side - Don't overlap **Diagram:** ray | | ∠1 -------+------- | ∠2 | ∠1 and ∠2 are adjacent **Key Point:** Adjacent angles can be ANY sum, but if they form a straight line, they're supplementary (180°). **LINEAR PAIR:** A special case of adjacent angles that form a straight line. - Always supplementary (sum to 180°) - Form a straight angle together **SOLVING ANGLE PROBLEMS:** **Strategy 1: Use Formulas** Example: Two complementary angles. One is 28°. Find the other. Solution: 90° - 28° = 62° **Strategy 2: Set Up Equations** Example: Two supplementary angles. One is 3x and the other is 2x. Find both. Equation: 3x + 2x = 180 5x = 180 x = 36 Angles: 3(36) = 108° and 2(36) = 72° Check: 108 + 72 = 180 ✓ **Strategy 3: Use Angle Relationships** Example: Two lines intersect. One angle is 55°. Find all four angles. - Vertical to 55°: 55° - Supplementary to 55°: 180° - 55° = 125° - Vertical to 125°: 125° Four angles: 55°, 125°, 55°, 125° **MULTI-STEP PROBLEMS:** Example: Angles A and B are complementary. Angle A is 20° more than angle B. Find both angles. Step 1: Define variables Let B = x Then A = x + 20 Step 2: Set up equation (complementary = 90°) x + (x + 20) = 90 Step 3: Solve 2x + 20 = 90 2x = 70 x = 35 Step 4: Find both angles B = 35° A = 35° + 20° = 55° Check: 35° + 55° = 90° ✓ **REAL-WORLD APPLICATIONS:** - Architecture: Roof angles, building corners - Navigation: Bearings and headings - Engineering: Bridge supports, trusses - Art: Perspective drawing - Sports: Ball trajectories, bank shots - Carpentry: Cutting angles for joints
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Master angle relationships through practice! **COMPLEMENTARY ANGLES:** 1. Find the complement of 40°. 2. Find the complement of 63°. 3. Two angles are complementary. One is 27°. Find the other. 4. Are 55° and 45° complementary? **SUPPLEMENTARY ANGLES:** 5. Find the supplement of 120°. 6. Find the supplement of 75°. 7. Two angles are supplementary. One is 135°. Find the other. 8. Are 100° and 80° supplementary? **VERTICAL ANGLES:** 9. Two lines intersect forming angles of 70°, x°, 70°, and y°. Find x and y. 10. Vertical angles are 5x and 3x + 40. Find x and the angle measures. **MIXED RELATIONSHIPS:** 11. ∠A and ∠B are complementary. ∠A = 2x and ∠B = x. Find both angles. 12. ∠C and ∠D are supplementary. ∠C = 4x and ∠D = 2x. Find both angles. **WORD PROBLEMS:** 13. Two complementary angles differ by 14°. Find both angles. (Hint: Let one be x, the other x + 14) 14. One supplementary angle is three times the other. Find both angles. 15. Two lines intersect. One angle is (2x + 10)° and its vertical angle is 50°. Find x. **DIAGRAM PROBLEMS:** Use the diagram where two lines intersect: ∠1 | ∠2 ----+---- ∠4 | ∠3 16. If ∠1 = 115°, find ∠2, ∠3, and ∠4. 17. If ∠2 = 3x - 15 and ∠4 = 2x + 10, find x and all four angles. **ADJACENT ANGLES:** 18. Adjacent angles forming a straight line measure 2x and 3x. Find both angles. 19. Three adjacent angles form a straight angle. They measure x, 2x, and 3x. Find each angle. **CHALLENGE:** 20. ∠A and ∠B are complementary. ∠B and ∠C are supplementary. If ∠A = 25°, find ∠C. 21. Two supplementary angles are in the ratio 2:3. Find both angles. (Hint: Let them be 2x and 3x) 22. Four angles are formed by two intersecting lines. Two adjacent angles are in the ratio 5:7. Find all four angles. **CRITICAL THINKING:** 23. Can two obtuse angles be complementary? Explain. 24. Can two obtuse angles be supplementary? Explain. 25. If two angles are congruent and complementary, what is their measure?