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Complementary, supplementary, and vertical angles
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Angles are everywhere in geometry! Understanding how angles relate to each other helps you solve problems involving shapes, parallel lines, and intersections. Let's explore the key angle relationships you need to know!
Angle: Formed by two rays with a common endpoint (vertex)
Measuring angles:
Types of angles by measure:
Complementary angles are two angles that add up to 90°.
Examples:
Two angles are complementary. One angle measures 35°. What is the measure of the other angle?
Complementary angles sum to 90°.
Let x = the unknown angle 35° + x = 90° x = 90° - 35° x = 55°
Answer: 55°
Two angles are supplementary. One angle measures 128°. What is the measure of the other angle?
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Key: They don't have to be next to each other!
Example 1: What is the complement of 35°?
Solution: 90° - 35° = 55°
Example 2: Two complementary angles. One is 3x and the other is 2x. Find each angle.
Solution: 3x + 2x = 90 5x = 90 x = 18
Angles: 3(18) = 54° and 2(18) = 36°
Supplementary angles are two angles that add up to 180°.
Examples:
Key: They form a straight line when placed adjacent!
Example 1: What is the supplement of 110°?
Solution: 180° - 110° = 70°
Example 2: Two supplementary angles. One is twice the other. Find both angles.
Let x = smaller angle, then 2x = larger angle
Solution: x + 2x = 180 3x = 180 x = 60
Angles: 60° and 120°
Adjacent angles share a common vertex and a common side, but don't overlap.
Think: Side-by-side angles
Example: If you have a pizza slice, then cut it in half, you create two adjacent angles.
Note: Adjacent angles can be complementary, supplementary, or neither!
When two lines intersect, they form vertical angles - angles that are opposite each other.
Key Property: Vertical angles are ALWAYS EQUAL!
Example: If two lines intersect and one angle is 50°, what are the others?
When two lines cross, they form 4 angles. The angles opposite each other (vertical angles) are equal, and adjacent angles are supplementary.
Solution:
Two lines intersect. One angle is 3x and its vertical angle is 105°. Find x.
Solution: Vertical angles are equal! 3x = 105 x = 35
Answer: x = 35
A linear pair consists of two adjacent angles that form a straight line.
Key Property: Linear pairs are ALWAYS SUPPLEMENTARY (add to 180°)!
Example: Angles ABC and CBD form a linear pair. If angle ABC = 75°, find angle CBD.
Solution: 75° + CBD = 180° CBD = 105°
When a line (transversal) crosses two parallel lines, it creates 8 angles with special relationships!
Key angle pairs:
Location: Same position at each intersection
Property: Corresponding angles are EQUAL when lines are parallel
Example: If angle 1 = 70°, then angle 5 = 70° (corresponding)
Location: Between the parallel lines, on opposite sides of the transversal
Property: Alternate interior angles are EQUAL when lines are parallel
Example: If angle 3 = 110°, then angle 6 = 110° (alternate interior)
Location: Outside the parallel lines, on opposite sides of the transversal
Property: Alternate exterior angles are EQUAL when lines are parallel
Example: If angle 1 = 70°, then angle 8 = 70° (alternate exterior)
Location: Between the parallel lines, on the same side of the transversal
Property: Consecutive interior angles are SUPPLEMENTARY (add to 180°)
Example: If angle 3 = 110°, then angle 5 = 70° (consecutive interior: 110 + 70 = 180)
Example: Lines m and n are parallel. A transversal crosses them. If one angle is 65°, find all 8 angles.
Solution:
At first intersection: 65°, 115°, 65°, 115° (vertical and supplementary)
At second intersection (using corresponding angles):
All 8 angles: Four 65° angles and four 115° angles
Two vertical angles measure (2x + 10)° and (3x - 15)°. Find x.
Solution: Vertical angles are equal! 2x + 10 = 3x - 15 10 + 15 = 3x - 2x 25 = x
Answer: x = 25
Angles: 2(25) + 10 = 60° and 3(25) - 15 = 60° ✓
Two angles are supplementary. One measures (4x + 5)° and the other (3x + 10)°. Find both angles.
Solution: (4x + 5) + (3x + 10) = 180 7x + 15 = 180 7x = 165 x = 23.57... or about 23.6°
Angles: 4(23.6) + 5 = 99.4° and 3(23.6) + 10 = 80.6°
Check: 99.4 + 80.6 = 180° ✓
Lines are parallel. One angle is (5x - 20)° and its corresponding angle is (3x + 40)°. Find x.
Solution: Corresponding angles are equal! 5x - 20 = 3x + 40 5x - 3x = 40 + 20 2x = 60 x = 30
Answer: x = 30
Angles: 5(30) - 20 = 130° and 3(30) + 40 = 130° ✓
Problem: A roof truss forms an angle of 35° with the horizontal. What is the complement of this angle?
Solution: 90° - 35° = 55°
Problem: Two streets intersect. One corner angle is 72°. What are the other three angles?
Solution:
Angles: 72°, 108°, 72°, 108°
Problem: Parallel railroad tracks are crossed by a road. If one angle is 115°, what are the other angles?
Solution: All angles are either 115° or 65° (supplement) Using angle relationships: Four 115° angles and four 65° angles
| Relationship | Sum/Equality | Example |
|---|---|---|
| Complementary | Sum = 90° | 30° + 60° |
| Supplementary | Sum = 180° | 110° + 70° |
| Vertical | Equal | Both are 50° |
| Linear Pair | Sum = 180° | 75° + 105° |
| Corresponding (parallel) | Equal | Both are 65° |
| Alternate Interior (parallel) | Equal | Both are 120° |
| Alternate Exterior (parallel) | Equal | Both are 60° |
| Consecutive Interior (parallel) | Sum = 180° | 110° + 70° |
❌ Mistake 1: Confusing complementary and supplementary
❌ Mistake 2: Assuming all intersecting angles are 90°
❌ Mistake 3: Forgetting vertical angles are equal
❌ Mistake 4: Mixing up angle pairs with parallel lines
❌ Mistake 5: Not checking if lines are parallel
Tip 1: Draw a diagram
Tip 2: Look for key words
Tip 3: Use what you know
Tip 4: Set up equations
Tip 5: Always check
Finding complement: 90° - angle
Finding supplement: 180° - angle
With variables:
Are the angles:
Key Angle Relationships:
Complementary: Add to 90° Supplementary: Add to 180° Vertical: Opposite angles at intersection (always equal) Linear Pair: Adjacent angles forming straight line (supplementary)
With Parallel Lines: Corresponding: Equal (same position) Alternate Interior: Equal (between, opposite sides) Alternate Exterior: Equal (outside, opposite sides) Consecutive Interior: Supplementary (between, same side)
Remember:
Understanding angle relationships is essential for geometry proofs, construction, engineering, and solving real-world problems!
Supplementary angles sum to 180°.
Let x = the unknown angle 128° + x = 180° x = 180° - 128° x = 52°
Answer: 52°
Two vertical angles are formed by intersecting lines. If one vertical angle measures 65°, what is the measure of the other vertical angle?
Vertical angles are always equal.
If one vertical angle = 65° Then the other vertical angle = 65°
Answer: 65°
A linear pair is formed by two adjacent angles. If one angle measures 73°, what is the measure of the other angle?
A linear pair forms a straight line, so the angles are supplementary (sum to 180°).
Let x = the unknown angle 73° + x = 180° x = 180° - 73° x = 107°
Answer: 107°
Two parallel lines are cut by a transversal. If one corresponding angle measures 115°, what is the measure of its corresponding angle?
When parallel lines are cut by a transversal, corresponding angles are equal.
If one corresponding angle = 115° Then the other corresponding angle = 115°
Note: You could also find:
Answer: 115°