Lines and Angles - Basic Math

Complete Notes & Formulas

1. Lines, Line Segments, and Rays

Line

A line extends infinitely in BOTH directions

• Has NO endpoints

• Cannot be measured (infinite length)

• Shown with arrows on both ends ↔

• Named with two points: Line AB or AB̅ with line symbol

Line Segment

A line segment has TWO endpoints

• Has a definite start and end

• CAN be measured (has finite length)

• No arrows on ends —

• Named with two endpoints: Segment AB or AB̅

Ray

A ray has ONE endpoint and extends infinitely in ONE direction

• Starts at one point

• Extends forever in one direction

• Arrow on one end only →

• Named with starting point FIRST: Ray AB or AB⃗

Visual Comparison

2. What is an Angle?

An angle is formed when TWO RAYS share

a common endpoint called the VERTEX

Parts of an Angle

• Vertex: The common point where two rays meet

• Arms/Sides: The two rays that form the angle

• Measure: The amount of rotation between the two rays (in degrees °)

Measuring Angles

• Use a protractor to measure angles

• Place the center of protractor on the vertex

• Align one ray with the 0° line

• Read the degree measure where the other ray points

3. Types of Angles (Classification)

Angle Type	Measure	Description	Example
Acute Angle	0° < angle < 90°	Sharp, narrow angle	30°, 45°, 60°
Right Angle	angle = 90°	Forms an "L" shape	90° exactly
Obtuse Angle	90° < angle < 180°	Wide, open angle	120°, 135°, 150°
Straight Angle	angle = 180°	Forms a straight line	180° exactly
Reflex Angle	180° < angle < 360°	Greater than straight	210°, 270°, 300°
Complete Angle	angle = 360°	Full circle	360° exactly

Visual: Types of Angles

4. Naming Angles

Three Ways to Name an Angle

Method 1: Using Three Points

• ∠ABC or ∠CBA

• Vertex must be in the MIDDLE

• Order: Point-Vertex-Point

Method 2: Using the Vertex Only

• ∠B (if only ONE angle at vertex B)

• Cannot use if multiple angles at same vertex

Method 3: Using a Number

• ∠1, ∠2, ∠3, etc.

• Number is written inside the angle

Remember: When using 3 points, the VERTEX is ALWAYS in the middle!

5. Complementary Angles

∠A + ∠B = 90°

Complementary angles add up to 90°

They form a RIGHT angle together

Formula

If angle A + angle B = 90°

Then: Complement of angle A = 90° − A

Examples

Example 1: 30° and 60° are complementary (30 + 60 = 90)

Example 2: 45° and 45° are complementary (45 + 45 = 90)

Example 3: 25° and 65° are complementary (25 + 65 = 90)

Find the complement of 35°:

90° − 35° = 55°

Answer: 55°

6. Supplementary Angles

∠A + ∠B = 180°

Supplementary angles add up to 180°

They form a STRAIGHT angle together

Formula

If angle A + angle B = 180°

Then: Supplement of angle A = 180° − A

Examples

Example 1: 120° and 60° are supplementary (120 + 60 = 180)

Example 2: 90° and 90° are supplementary (90 + 90 = 180)

Example 3: 135° and 45° are supplementary (135 + 45 = 180)

Find the supplement of 75°:

180° − 75° = 105°

Answer: 105°

7. Vertical Angles

Vertical angles are formed when two lines intersect

They are OPPOSITE each other

Vertical angles are ALWAYS EQUAL!

Formula

If ∠1 and ∠3 are vertical angles

Then: ∠1 = ∠3

Visual: Vertical Angles

∠1 = ∠3 (Vertical angles)

∠2 = ∠4 (Vertical angles)

8. Adjacent Angles

Adjacent angles are angles that:

• Share a common VERTEX

• Share a common SIDE

• Are NEXT TO each other

• Do NOT overlap

Visual: Adjacent Angles

∠1 and ∠2 are adjacent (they share vertex and a common side)

9. Congruent Angles

Congruent angles have the SAME MEASURE

∠A ≅ ∠B means ∠A = ∠B

Examples:

• If ∠A = 45° and ∠B = 45°, then ∠A ≅ ∠B

• All right angles are congruent (all measure 90°)

• Vertical angles are always congruent

10. Finding Unknown Angle Measures

Using Complementary Angles

Problem: Two angles are complementary. One angle is 35°. Find the other angle.

Complementary angles add to 90°

35° + x = 90°

x = 90° − 35°

x = 55°

Answer: 55°

Using Supplementary Angles

Problem: Two angles are supplementary. One angle is 110°. Find the other angle.

Supplementary angles add to 180°

110° + x = 180°

x = 180° − 110°

x = 70°

Answer: 70°

Using Vertical Angles

Problem: ∠1 and ∠3 are vertical angles. If ∠1 = 65°, find ∠3.

Vertical angles are equal

∠3 = ∠1

∠3 = 65°

Answer: 65°

Quick Reference: Angle Relationships

Relationship	Definition	Formula/Rule
Complementary	Add to 90°	∠A + ∠B = 90°
Supplementary	Add to 180°	∠A + ∠B = 180°
Vertical	Opposite when lines cross	∠1 = ∠3, ∠2 = ∠4
Adjacent	Share vertex and side	Next to each other
Congruent	Equal measure	∠A ≅ ∠B means ∠A = ∠B