Angle Relationships

TipLearning Objectives
  • Identify and use vertical, adjacent, complementary, and supplementary angles.
  • Write and solve equations based on angle relationships.
  • Understand how linear pairs and vertical angles help solve diagrams.

Key Ideas

Important angle relationships:

  • Vertical Angles: opposite angles formed by intersecting lines → equal
  • Adjacent Angles: share a vertex and side
  • Linear Pair: adjacent angles forming a straight line → sum to 180°
  • Supplementary: sum to 180°
  • Complementary: sum to 90°

Summary diagram showing vertical, adjacent, linear pair, supplementary, and complementary angles with dividing rays.

Common Problem Types

Vertical Angle Equations

Vertical angles are equal.

Example:
If ∠1 = 3x + 10 and ∠2 = x + 30 → set equal.

Linear Pairs

Sum to 180°.

Example:
If ∠A = 120°, ∠B = 60°.

Complementary Angle Problems

Sum to 90°.

Adjacent Angle Sums

Shared vertex → add/subtract if given total.

Summary Table of Angle Relationships

Relationship Rule
Vertical Equal
Linear Pair Sum to 180°
Supplementary Sum to 180°
Complementary Sum to 90°
Adjacent Share a side & vertex

Strategies

  • Look for intersecting lines → vertical & linear pairs.
  • Use equations when angles are algebraic.
  • Always identify whether the angle sum is 90° or 180°.

Worked Examples

Example 1 — Vertical

∠1 = 2x + 10
∠2 = 40

Vertical → 2x + 10 = 40 → x = 15.

Example 2 — Linear Pair

Angles on a line:

If ∠A = 4x and ∠B = 2x + 30:

\[ 4x + (2x+30) = 180 \]

WarningCommon Mistakes
  • Confusing vertical with adjacent angles.
  • Forgetting that linear pairs must form a straight line.
  • Using the wrong sum (90° vs 180°).
  • Solving equations without checking if the geometry fits.

Practice Problems

  1. Vertical angles: ∠1 = 50°. Find ∠2.
  2. Linear pair: angles are 5x and x + 30. Solve for x.
  3. Complementary: ∠A = 20°, find ∠B.
  4. If ∠B = 3x and ∠C = 180 - ∠B, what type of angles are B and C?
  1. 50°
  2. \(5x + x + 30 = 180 → x = 25\)
  3. 70°
  4. Supplementary

Summary

  • Vertical angles are equal.
  • Linear pairs, supplementary = 180°.
  • Complementary = 90°.
  • Adjacent angles share a vertex and side.
  • Opposite = vertical.
  • On a line = linear pair.
  • Look for right angles when complementary.