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.
Do this: Read the concept below, then try the quiz or activity.
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
Try it
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 | ∠316. 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?