Angle Relationships

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!

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Review: Measuring Angles

Angle: Formed by two rays with a common endpoint (vertex)

Measuring angles:

• Use a protractor
• Measured in degrees (°)
• Full rotation = 360°

Types of angles by measure:

• Acute: Less than 90° (sharp)
• Right: Exactly 90° (square corner)
• Obtuse: Between 90° and 180° (wide)
• Straight: Exactly 180° (straight line)

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Complementary Angles

Complementary angles are two angles that add up to 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)

Key: They don't have to be next to each other!

Finding Complement

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°

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Supplementary Angles

Supplementary angles are two angles that add up to 180°.

Examples:

• 120° and 60° are supplementary (120 + 60 = 180)
• 90° and 90° are supplementary (90 + 90 = 180)
• 130° and 50° are supplementary (130 + 50 = 180)

Key: They form a straight line when placed adjacent!

Finding Supplement

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°

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Adjacent Angles

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!

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Vertical Angles

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:

• Vertical to 50° is also 50°
• Adjacent angles are supplementary: 180° - 50° = 130°
• All four angles: 50°, 130°, 50°, 130°

Example Problem

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

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Linear Pairs

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°

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Angles Formed by Parallel Lines and a Transversal

When a line (transversal) crosses two parallel lines, it creates 8 angles with special relationships!

Key angle pairs:

1. Corresponding Angles

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)

2. Alternate Interior Angles

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)

3. Alternate Exterior Angles

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)

4. Consecutive Interior Angles (Same-Side Interior)

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)

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Solving with Parallel Lines

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):

• Corresponding to 65° → 65°
• Corresponding to 115° → 115°
• Pattern repeats: 65°, 115°, 65°, 115°

All 8 angles: Four 65° angles and four 115° angles

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Using Variables with Angle Relationships

Example 1: Vertical 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° ✓

Example 2: Supplementary Angles

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° ✓

Example 3: Parallel Lines

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° ✓

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Real-World Applications

Architecture

Problem: A roof truss forms an angle of 35° with the horizontal. What is the complement of this angle?

Solution: 90° - 35° = 55°

Street Intersections

Problem: Two streets intersect. One corner angle is 72°. What are the other three angles?

Solution:

• Vertical: 72°
• Adjacent (supplementary): 180° - 72° = 108°
• Other vertical: 108°

Angles: 72°, 108°, 72°, 108°

Railroad Tracks

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

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Angle Relationship Summary Table

| 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° |

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Common Mistakes to Avoid

❌ Mistake 1: Confusing complementary and supplementary

• Complementary = 90° (Think: Corner = 90°)
• Supplementary = 180° (Think: Straight line = 180°)

❌ Mistake 2: Assuming all intersecting angles are 90°

• Only perpendicular lines form 90° angles!

❌ Mistake 3: Forgetting vertical angles are equal

• They're ALWAYS equal, no exceptions!

❌ Mistake 4: Mixing up angle pairs with parallel lines

• Corresponding: Same position
• Alternate interior: Between lines, opposite sides
• Consecutive interior: Between lines, same side

❌ Mistake 5: Not checking if lines are parallel

• Special angle relationships ONLY work when lines are parallel!

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Practice Tips

Tip 1: Draw a diagram

• Visual representation helps identify relationships
• Label angles clearly

Tip 2: Look for key words

• "Complement" → add to 90°
• "Supplement" → add to 180°
• "Vertical" → equal
• "Parallel lines" → use special relationships

Tip 3: Use what you know

• If you know one angle, you can find others!
• With parallel lines and one angle, you can find all 8!

Tip 4: Set up equations

• For variables, write an equation based on the relationship
• Solve algebraically

Tip 5: Always check

• Do complementary angles add to 90°?
• Do supplementary angles add to 180°?
• Does your answer make sense?

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Quick Formulas

Finding complement: 90° - angle

Finding supplement: 180° - angle

With variables:

• Complementary: x + y = 90
• Supplementary: x + y = 180
• Vertical: angle₁ = angle₂
• Linear pair: angle₁ + angle₂ = 180

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Identifying Angle Relationships Checklist

Are the angles:

1. Adding to 90°? → Complementary
2. Adding to 180°? → Supplementary
3. Opposite at an intersection? → Vertical (equal)
4. Adjacent and form a straight line? → Linear pair (supplementary)
5. Formed by parallel lines cut by a transversal?
   • Same position? → Corresponding (equal)
   • Between lines, opposite sides? → Alternate interior (equal)
   • Outside lines, opposite sides? → Alternate exterior (equal)
   • Between lines, same side? → Consecutive interior (supplementary)

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Summary

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:

• Vertical angles are ALWAYS equal
• Linear pairs are ALWAYS supplementary
• Parallel line relationships only work when lines are parallel!

Understanding angle relationships is essential for geometry proofs, construction, engineering, and solving real-world problems!