1. Identify Complementary, Supplementary, Vertical, Adjacent, and Congruent Angles

Complementary Angles:

Definition: Two angles whose measures add up to 90°

\( \angle A + \angle B = 90° \)

Example: If \( \angle A = 30° \), then its complement \( \angle B = 60° \)

Formula to find complement: \( 90° - x \)

Supplementary Angles:

Definition: Two angles whose measures add up to 180°

\( \angle A + \angle B = 180° \)

Example: If \( \angle A = 120° \), then its supplement \( \angle B = 60° \)

Formula to find supplement: \( 180° - x \)

Vertical Angles (Vertically Opposite Angles):

Definition: Angles opposite each other when two lines intersect

Key Property: Vertical angles are always EQUAL (congruent)

\( \angle 1 = \angle 3 \) and \( \angle 2 = \angle 4 \)

Characteristics:

• Opposite to each other
• Do NOT share a common side
• Always congruent (equal measure)

Adjacent Angles:

Definition: Angles that are next to each other

Characteristics:

• Share a common vertex
• Share a common side
• Do NOT overlap
• Can be complementary, supplementary, or neither

Linear Pair:

Definition: Two adjacent angles that form a straight line

Key Property: Linear pairs are always supplementary (sum = 180°)

\( \angle A + \angle B = 180° \)

Congruent Angles:

Definition: Angles that have the same measure

Notation: \( \angle A \cong \angle B \) means \( \angle A = \angle B \)

Example: If \( \angle A = 45° \) and \( \angle B = 45° \), then \( \angle A \cong \angle B \)

2. Find Measures of Complementary, Supplementary, Vertical, and Adjacent Angles

Finding Complementary Angles:

Example 1: One angle is 35°. Find its complement.

Complement = \( 90° - 35° = 55° \)

Example 2: Two complementary angles are in the ratio 2:3. Find both angles.

Let angles be \( 2x \) and \( 3x \)

\( 2x + 3x = 90° \)

\( 5x = 90° \) → \( x = 18° \)

Angles: \( 2(18°) = 36° \) and \( 3(18°) = 54° \)

Finding Supplementary Angles:

Example 1: One angle is 110°. Find its supplement.

Supplement = \( 180° - 110° = 70° \)

Example 2: Two supplementary angles differ by 40°. Find both angles.

Let angles be \( x \) and \( x + 40° \)

\( x + (x + 40°) = 180° \)

\( 2x + 40° = 180° \) → \( 2x = 140° \) → \( x = 70° \)

Angles: 70° and 110°

Finding Vertical Angles:

Example: If one vertical angle is 65°, what is the measure of its opposite angle?

Answer: 65° (vertical angles are equal)

Finding Adjacent Angles in a Linear Pair:

Example: Two adjacent angles form a linear pair. If one angle is 3 times the other, find both angles.

Let smaller angle = \( x \), larger angle = \( 3x \)

\( x + 3x = 180° \) → \( 4x = 180° \) → \( x = 45° \)

Angles: 45° and 135°

3. Write and Solve Equations Using Angle Relationships

Strategy:

1. Identify the angle relationship (complementary, supplementary, vertical, etc.)
2. Write an equation based on the relationship
3. Solve for the variable
4. Find the angle measures

Examples:

Example 1: Two complementary angles are \( (2x + 10)° \) and \( (3x - 5)° \). Find \( x \) and both angles.

Equation: \( (2x + 10) + (3x - 5) = 90 \)

\( 5x + 5 = 90 \)

\( 5x = 85 \) → \( x = 17 \)

Angles: \( 2(17) + 10 = 44° \) and \( 3(17) - 5 = 46° \)

Example 2: Two supplementary angles are \( (4x - 20)° \) and \( (2x + 50)° \). Find both angles.

Equation: \( (4x - 20) + (2x + 50) = 180 \)

\( 6x + 30 = 180 \)

\( 6x = 150 \) → \( x = 25 \)

Angles: \( 4(25) - 20 = 80° \) and \( 2(25) + 50 = 100° \)

Example 3: Vertical angles are \( (5x + 15)° \) and \( (7x - 5)° \). Find \( x \).

Equation: \( 5x + 15 = 7x - 5 \) (vertical angles are equal)

\( 15 + 5 = 7x - 5x \)

\( 20 = 2x \) → \( x = 10 \)

Each angle: \( 5(10) + 15 = 65° \)

Example 4: Two angles in a linear pair are \( (3x + 10)° \) and \( (5x - 2)° \). Find both angles.

Equation: \( (3x + 10) + (5x - 2) = 180 \)

\( 8x + 8 = 180 \)

\( 8x = 172 \) → \( x = 21.5 \)

Angles: \( 3(21.5) + 10 = 74.5° \) and \( 5(21.5) - 2 = 105.5° \)

4. Identify Alternate Interior and Alternate Exterior Angles

Transversal: A line that intersects two or more lines at different points.

Alternate Interior Angles:

Definition: Angles on opposite sides of the transversal, between (interior to) the two lines.

Location:

• On opposite sides of the transversal
• Between the two lines (interior)
• Do NOT share a vertex

Property: If lines are parallel, alternate interior angles are EQUAL

If lines are parallel: \( \angle 3 = \angle 6 \) and \( \angle 4 = \angle 5 \)

Alternate Exterior Angles:

Definition: Angles on opposite sides of the transversal, outside (exterior to) the two lines.

Location:

• On opposite sides of the transversal
• Outside the two lines (exterior)
• Do NOT share a vertex

Property: If lines are parallel, alternate exterior angles are EQUAL

If lines are parallel: \( \angle 1 = \angle 8 \) and \( \angle 2 = \angle 7 \)

Converse:

If alternate interior angles are equal OR alternate exterior angles are equal, then the lines are parallel.

5. Transversals of Parallel Lines: Name Angle Pairs

When a transversal intersects two parallel lines, it creates 8 angles. These angles form special pairs:

1. Corresponding Angles:

Definition: Angles in the same position at each intersection

Property: If lines are parallel, corresponding angles are EQUAL

Pairs: \( \angle 1 = \angle 5 \), \( \angle 2 = \angle 6 \), \( \angle 3 = \angle 7 \), \( \angle 4 = \angle 8 \)

2. Alternate Interior Angles:

Pairs: \( \angle 3 = \angle 6 \), \( \angle 4 = \angle 5 \)

3. Alternate Exterior Angles:

Pairs: \( \angle 1 = \angle 8 \), \( \angle 2 = \angle 7 \)

4. Consecutive Interior Angles (Co-Interior/Same-Side Interior):

Definition: Angles on the same side of the transversal, between the two lines

Property: If lines are parallel, consecutive interior angles are SUPPLEMENTARY (sum = 180°)

\( \angle 3 + \angle 5 = 180° \) and \( \angle 4 + \angle 6 = 180° \)

Summary Table:

6. Transversals of Parallel Lines: Find Angle Measures

Strategy:

1. Identify the given angle
2. Determine the relationship between angles
3. Use properties: equal angles or supplementary angles
4. Calculate unknown angles

Examples:

Example 1: If \( \angle 1 = 65° \), find all other angles when two parallel lines are cut by a transversal.

Step 1: Vertical angles: \( \angle 3 = 65° \) (opposite to \( \angle 1 \))

Step 2: Linear pair: \( \angle 2 = 180° - 65° = 115° \)

Step 3: Vertical to \( \angle 2 \): \( \angle 4 = 115° \)

Step 4: Corresponding angles: \( \angle 5 = \angle 1 = 65° \)

Continue: \( \angle 6 = 115° \), \( \angle 7 = 65° \), \( \angle 8 = 115° \)

Example 2: Two parallel lines are cut by a transversal. One angle is 40°. Find its alternate interior angle.

Answer: 40° (alternate interior angles are equal when lines are parallel)

Example 3: Two parallel lines are cut by a transversal. If one consecutive interior angle is 110°, find the other.

\( x + 110° = 180° \)

\( x = 70° \)

7. Transversals of Parallel Lines: Solve for x

Examples:

Example 1: Two parallel lines are cut by a transversal. If corresponding angles are \( (3x + 20)° \) and \( (5x - 10)° \), find \( x \).

Equation: \( 3x + 20 = 5x - 10 \) (corresponding angles are equal)

\( 20 + 10 = 5x - 3x \)

\( 30 = 2x \) → \( x = 15 \)

Example 2: Alternate interior angles are \( (4x + 5)° \) and \( (6x - 15)° \). Find \( x \) and the angle measures.

Equation: \( 4x + 5 = 6x - 15 \)

\( 5 + 15 = 6x - 4x \)

\( 20 = 2x \) → \( x = 10 \)

Angles: \( 4(10) + 5 = 45° \)

Example 3: Consecutive interior angles are \( (2x + 30)° \) and \( (3x + 20)° \). Find \( x \).

Equation: \( (2x + 30) + (3x + 20) = 180 \) (consecutive interior angles are supplementary)

\( 5x + 50 = 180 \)

\( 5x = 130 \) → \( x = 26 \)

Example 4: Two alternate exterior angles are \( (7x - 10)° \) and \( (5x + 30)° \). Find \( x \) and the angle measures.

Equation: \( 7x - 10 = 5x + 30 \)

\( 7x - 5x = 30 + 10 \)

\( 2x = 40 \) → \( x = 20 \)

Angles: \( 7(20) - 10 = 130° \)

8. Find Lengths and Measures of Bisected Line Segments and Angles

Segment Bisector:

Definition: A line, ray, or segment that divides a line segment into two equal parts.

Midpoint: The point where the bisector intersects the segment

If segment AB is bisected at point M, then: \( AM = MB = \frac{AB}{2} \)

Segment Bisector Examples:

Example 1: A segment AB is 24 cm long. Point M bisects AB. Find AM and MB.

\( AM = MB = \frac{24}{2} = 12 \) cm

Example 2: Point M bisects segment PQ. If PM = 3x + 5 and MQ = 5x - 7, find x and the length of PQ.

Equation: \( 3x + 5 = 5x - 7 \) (bisected parts are equal)

\( 5 + 7 = 5x - 3x \) → \( 12 = 2x \) → \( x = 6 \)

\( PM = 3(6) + 5 = 23 \) units

\( PQ = 2 \times 23 = 46 \) units

Angle Bisector:

Definition: A ray that divides an angle into two equal parts.

If ray BD bisects \( \angle ABC \), then: \( \angle ABD = \angle DBC = \frac{\angle ABC}{2} \)

Angle Bisector Examples:

Example 1: Ray BD bisects \( \angle ABC \) which measures 80°. Find \( \angle ABD \).

\( \angle ABD = \frac{80°}{2} = 40° \)

Example 2: Ray QS bisects \( \angle PQR \). If \( \angle PQS = 35° \), find \( \angle PQR \).

\( \angle PQR = 2 \times 35° = 70° \)

Example 3: Ray BD bisects \( \angle ABC \). If \( \angle ABD = (2x + 10)° \) and \( \angle DBC = (3x - 5)° \), find x and \( \angle ABC \).

Equation: \( 2x + 10 = 3x - 5 \) (bisected angles are equal)

\( 10 + 5 = 3x - 2x \) → \( x = 15 \)

\( \angle ABD = 2(15) + 10 = 40° \)

\( \angle ABC = 2 \times 40° = 80° \)

Key Formulas:

For Bisected Segments:

• Each part = \( \frac{\text{Total length}}{2} \)
• Total length = \( 2 \times \text{One part} \)

For Bisected Angles:

• Each angle = \( \frac{\text{Total angle}}{2} \)
• Total angle = \( 2 \times \text{One angle} \)

Quick Reference Guide

💡 Key Tips for Lines and Angles

• ✓ Complementary adds to 90°, Supplementary adds to 180°
• ✓ Vertical angles are always equal (no exceptions!)
• ✓ Linear pair = supplementary + adjacent
• ✓ Adjacent angles share a vertex and a side
• ✓ When lines are parallel: corresponding, alternate interior, and alternate exterior angles are equal
• ✓ Consecutive interior angles are supplementary when lines are parallel
• ✓ Transversal creates 8 angles when crossing two lines
• ✓ Bisector divides into 2 equal parts (segments or angles)
• ✓ Always write equations based on angle relationships
• ✓ Check your answers: Do complementary angles sum to 90°? Do supplementary sum to 180°?