Angle Relationships

Review: Angle Measures

Angle: Formed by two rays sharing a common endpoint

Measured in degrees (°)

Types:

• Acute: Less than 90°
• Right: Exactly 90°
• Obtuse: Between 90° and 180°
• Straight: Exactly 180°

Complementary Angles

Complementary angles add up to 90°

Think: "C" for Corner (right angle)

Can be:

• Adjacent (next to each other)
• Non-adjacent (separate)

Example 1: Find Complement

Angle A = 35°

Complement = 90° − 35° = 55°

Answer: The complement is 55°

Check: 35° + 55° = 90° ✓

Example 2: Find Missing Angle

Two complementary angles. One is 28°. Find the other.

90° − 28° = 62°

Answer: 62°

Example 3: Set Up Equation

Angle A = x, Angle B = 2x

They are complementary.

Equation: x + 2x = 90°

Solve:

3x = 90°
x = 30°

Angle A: 30° Angle B: 2(30°) = 60°

Check: 30° + 60° = 90° ✓

Supplementary Angles

Supplementary angles add up to 180°

Think: "S" for Straight line

Often found:

• Linear pairs (adjacent angles on a straight line)
• Separate angles that sum to 180°

Example 1: Find Supplement

Angle C = 110°

Supplement = 180° − 110° = 70°

Answer: The supplement is 70°

Check: 110° + 70° = 180° ✓

Example 2: Linear Pair

Two angles form a linear pair. One is 65°. Find the other.

180° − 65° = 115°

Answer: 115°

Example 3: Algebraic

Angle X = 3x, Angle Y = x

They are supplementary.

Equation: 3x + x = 180°

Solve:

4x = 180°
x = 45°

Angle X: 3(45°) = 135° Angle Y: 45°

Check: 135° + 45° = 180° ✓

Vertical Angles

Vertical angles are formed when two lines intersect.

Key property: Vertical angles are always equal!

Formed as:

• Opposite angles at an intersection
• Non-adjacent

Example 1: Find Vertical Angle

Two lines intersect. One angle is 75°.

Vertical angle = 75° (equal)

Example 2: All Four Angles

Two lines intersect, forming angles 1, 2, 3, 4.

Given: Angle 1 = 120°

Find others:

• Angle 3 (vertical to 1) = 120°
• Angle 2 (supplementary to 1) = 60°
• Angle 4 (vertical to 2) = 60°

Check: All four sum to 360° ✓

Example 3: Solve for x

Vertical angles: 5x and 3x + 20

Set equal:

5x = 3x + 20
2x = 20
x = 10

Angle measure: 5(10) = 50°

Check: 3(10) + 20 = 50° ✓

Adjacent Angles

Adjacent angles:

• Share a common side
• Share a common vertex
• Don't overlap

May or may not add to 90° or 180°!

Example: Adjacent but Not Supplementary

Three angles around a point: 80°, 100°, 180°

Adjacent? Yes (share sides)

Supplementary? Only some pairs

All angles around point: Sum to 360°

Linear Pairs

Linear pair: Two adjacent angles that form a straight line

Always supplementary (add to 180°)

Example: Linear Pair

Angles A and B form a linear pair.

Angle A = 2x + 10 Angle B = 3x − 20

Equation: (2x + 10) + (3x − 20) = 180°

Solve:

5x − 10 = 180°
5x = 190°
x = 38°

Angle A: 2(38) + 10 = 86° Angle B: 3(38) − 20 = 94°

Check: 86° + 94° = 180° ✓

Angles Around a Point

All angles around a point sum to 360°

Example: Four Angles Around Point

Angles: 90°, 100°, x, 85°

Equation: 90 + 100 + x + 85 = 360

Solve:

275 + x = 360
x = 85°

Angle Bisector

An angle bisector divides an angle into two equal parts.

Example: Bisected Angle

Angle ABC = 80°. Ray BD bisects it.

Each part: 80° / 2 = 40°

Angle ABD = 40° Angle DBC = 40°

Parallel Lines and Transversals

When a line (transversal) crosses parallel lines:

Corresponding angles: Equal Alternate interior angles: Equal Alternate exterior angles: Equal Consecutive interior angles: Supplementary

Example: Parallel Lines

Lines l and m are parallel. Transversal crosses them.

One angle = 65°

Corresponding angle: 65° Alternate interior: 65° Consecutive interior: 115° (supplementary)

Real-World Applications

Construction: Ensuring right angles (complementary 45° angles)

Architecture: Designing roof angles (supplementary)

Navigation: Bearing angles and turns

Engineering: Structural supports with specific angles

Art: Creating perspective with angle relationships

Problem-Solving Strategy

Steps:

1. Draw a diagram if not given
2. Label known angles
3. Identify relationship (complementary, supplementary, vertical)
4. Set up equation
5. Solve for unknown
6. Check answer makes sense

Practice