Parallel and Perpendicular Lines - Tenth Grade Geometry
Introduction to Parallel and Perpendicular Lines
Parallel Lines: Lines in the same plane that never intersect
Perpendicular Lines: Lines that intersect at 90° (right angle)
Transversal: A line that intersects two or more other lines
Coplanar: Lying in the same plane
Perpendicular Lines: Lines that intersect at 90° (right angle)
Transversal: A line that intersects two or more other lines
Coplanar: Lying in the same plane
1. Identify Parallel, Intersecting, and Skew Lines and Planes
Parallel Lines
Parallel Lines: Two lines in the same plane that never intersect
Symbol: $\ell \parallel m$ (read as "line $\ell$ is parallel to line $m$")
Distance: Parallel lines are always the same distance apart
Slopes: In coordinate plane, parallel lines have equal slopes
Symbol: $\ell \parallel m$ (read as "line $\ell$ is parallel to line $m$")
Distance: Parallel lines are always the same distance apart
Slopes: In coordinate plane, parallel lines have equal slopes
Parallel Lines Properties:
• Must be coplanar (in same plane)
• Never intersect, no matter how far extended
• Equidistant at all points
• If $\ell \parallel m$ and $m \parallel n$, then $\ell \parallel n$ (transitive property)
Notation: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$
• Must be coplanar (in same plane)
• Never intersect, no matter how far extended
• Equidistant at all points
• If $\ell \parallel m$ and $m \parallel n$, then $\ell \parallel n$ (transitive property)
Notation: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$
Intersecting Lines
Intersecting Lines: Two lines that meet at exactly one point
Point of Intersection: The common point where lines meet
Must be: Coplanar (in the same plane)
Point of Intersection: The common point where lines meet
Must be: Coplanar (in the same plane)
Types of Intersecting Lines:
1. Perpendicular Lines: Intersect at 90°
• Symbol: $\ell \perp m$
• Form four right angles
2. Oblique Lines: Intersect at any angle other than 90°
• Form angles that are not right angles
1. Perpendicular Lines: Intersect at 90°
• Symbol: $\ell \perp m$
• Form four right angles
2. Oblique Lines: Intersect at any angle other than 90°
• Form angles that are not right angles
Skew Lines
Skew Lines: Lines that are NOT parallel and do NOT intersect
Key Property: Non-coplanar (lie in different planes)
Exist in: Three-dimensional space only
Cannot exist: In 2D (on a flat plane)
Key Property: Non-coplanar (lie in different planes)
Exist in: Three-dimensional space only
Cannot exist: In 2D (on a flat plane)
Skew Lines Definition:
Lines $\ell$ and $m$ are skew if and only if:
1. They do NOT intersect
2. They are NOT parallel
3. They are NOT coplanar
Example: The edges of a rectangular box that don't share a face
Lines $\ell$ and $m$ are skew if and only if:
1. They do NOT intersect
2. They are NOT parallel
3. They are NOT coplanar
Example: The edges of a rectangular box that don't share a face
Example 1: Identify line relationships
In a rectangular prism (box):
• Top edge AB and bottom edge CD: Parallel (same plane when viewed from front)
• Top edge AB and side edge AE: Intersecting (meet at point A)
• Top edge AB and opposite bottom edge (not below AB): Skew (different planes, don't intersect)
In a rectangular prism (box):
• Top edge AB and bottom edge CD: Parallel (same plane when viewed from front)
• Top edge AB and side edge AE: Intersecting (meet at point A)
• Top edge AB and opposite bottom edge (not below AB): Skew (different planes, don't intersect)
Parallel and Intersecting Planes
Parallel Planes: Two planes that never intersect
Intersecting Planes: Two planes that meet along a line
Line of Intersection: Where two planes meet
Intersecting Planes: Two planes that meet along a line
Line of Intersection: Where two planes meet
Plane Relationships:
Parallel Planes:
• No points in common
• Always same distance apart
• Example: Floor and ceiling of a room
Intersecting Planes:
• Meet along a line
• Share infinite points (all points on intersection line)
• Example: Two walls meeting at a corner
Parallel Planes:
• No points in common
• Always same distance apart
• Example: Floor and ceiling of a room
Intersecting Planes:
• Meet along a line
• Share infinite points (all points on intersection line)
• Example: Two walls meeting at a corner
2. Construct a Perpendicular Line
Perpendicular: Forms 90° angle
Construction: Using compass and straightedge
Two types: Through a point on the line, or through a point not on the line
Construction: Using compass and straightedge
Two types: Through a point on the line, or through a point not on the line
Construction 1: Perpendicular Through Point on Line
Given: Line $\ell$ and point P on line $\ell$
Construct: Line through P perpendicular to $\ell$
Step 1: Place compass on P
Step 2: Draw arcs on both sides of P intersecting $\ell$ at points A and B
Step 3: Open compass wider (more than half of AB)
Step 4: Place compass on A, draw arc above line
Step 5: Keep same width, place compass on B, draw arc intersecting first arc at C
Step 6: Draw line through P and C
Result: Line PC is perpendicular to $\ell$ at P
Construct: Line through P perpendicular to $\ell$
Step 1: Place compass on P
Step 2: Draw arcs on both sides of P intersecting $\ell$ at points A and B
Step 3: Open compass wider (more than half of AB)
Step 4: Place compass on A, draw arc above line
Step 5: Keep same width, place compass on B, draw arc intersecting first arc at C
Step 6: Draw line through P and C
Result: Line PC is perpendicular to $\ell$ at P
Construction 2: Perpendicular From External Point
Given: Line $\ell$ and point P NOT on line $\ell$
Construct: Line through P perpendicular to $\ell$
Step 1: Place compass on P
Step 2: Draw arc intersecting $\ell$ at two points A and B
Step 3: Place compass on A, set width more than half of AB
Step 4: Draw arc below line
Step 5: Keep same width, place compass on B
Step 6: Draw arc intersecting first arc at point Q
Step 7: Draw line through P and Q
Result: Line PQ is perpendicular to $\ell$
Construct: Line through P perpendicular to $\ell$
Step 1: Place compass on P
Step 2: Draw arc intersecting $\ell$ at two points A and B
Step 3: Place compass on A, set width more than half of AB
Step 4: Draw arc below line
Step 5: Keep same width, place compass on B
Step 6: Draw arc intersecting first arc at point Q
Step 7: Draw line through P and Q
Result: Line PQ is perpendicular to $\ell$
3. Transversals: Name Angle Pairs
Transversal: A line that intersects two or more lines at different points
Creates: 8 angles when crossing two lines
Angle pairs: Special relationships between these angles
Creates: 8 angles when crossing two lines
Angle pairs: Special relationships between these angles
Standard Angle Numbering:
When transversal $t$ crosses lines $\ell$ and $m$:
At first intersection: angles 1, 2, 3, 4
At second intersection: angles 5, 6, 7, 8
Typical arrangement:
• Angles 1 and 2 above line $\ell$
• Angles 3 and 4 below line $\ell$
• Angles 5 and 6 above line $m$
• Angles 7 and 8 below line $m$
When transversal $t$ crosses lines $\ell$ and $m$:
At first intersection: angles 1, 2, 3, 4
At second intersection: angles 5, 6, 7, 8
Typical arrangement:
• Angles 1 and 2 above line $\ell$
• Angles 3 and 4 below line $\ell$
• Angles 5 and 6 above line $m$
• Angles 7 and 8 below line $m$
Types of Angle Pairs
1. Corresponding Angles:
• Same position at each intersection
• Same side of transversal
• One interior, one exterior (or both exterior, both interior)
• Pairs: ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8
• Pattern: "F" shape or "backwards F"
2. Alternate Interior Angles:
• Between the two lines (interior)
• Opposite sides of transversal
• Pairs: ∠3 & ∠6, ∠4 & ∠5
• Pattern: "Z" shape or "backwards Z"
3. Alternate Exterior Angles:
• Outside the two lines (exterior)
• Opposite sides of transversal
• Pairs: ∠1 & ∠8, ∠2 & ∠7
• Pattern: Outside "Z" shape
4. Consecutive Interior Angles (Same-Side Interior):
• Between the two lines (interior)
• Same side of transversal
• Pairs: ∠3 & ∠5, ∠4 & ∠6
• Pattern: "C" shape or "backwards C"
• Also called: Co-interior angles
• Same position at each intersection
• Same side of transversal
• One interior, one exterior (or both exterior, both interior)
• Pairs: ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8
• Pattern: "F" shape or "backwards F"
2. Alternate Interior Angles:
• Between the two lines (interior)
• Opposite sides of transversal
• Pairs: ∠3 & ∠6, ∠4 & ∠5
• Pattern: "Z" shape or "backwards Z"
3. Alternate Exterior Angles:
• Outside the two lines (exterior)
• Opposite sides of transversal
• Pairs: ∠1 & ∠8, ∠2 & ∠7
• Pattern: Outside "Z" shape
4. Consecutive Interior Angles (Same-Side Interior):
• Between the two lines (interior)
• Same side of transversal
• Pairs: ∠3 & ∠5, ∠4 & ∠6
• Pattern: "C" shape or "backwards C"
• Also called: Co-interior angles
Example 1: Identify angle pairs
Given transversal crossing two lines with angles labeled 1-8:
Name all corresponding angle pairs:
∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8
Name all alternate interior angle pairs:
∠3 and ∠6, ∠4 and ∠5
Name all alternate exterior angle pairs:
∠1 and ∠8, ∠2 and ∠7
Name all consecutive interior angle pairs:
∠3 and ∠5, ∠4 and ∠6
Given transversal crossing two lines with angles labeled 1-8:
Name all corresponding angle pairs:
∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8
Name all alternate interior angle pairs:
∠3 and ∠6, ∠4 and ∠5
Name all alternate exterior angle pairs:
∠1 and ∠8, ∠2 and ∠7
Name all consecutive interior angle pairs:
∠3 and ∠5, ∠4 and ∠6
4. Transversals of Parallel Lines: Find Angle Measures
Key Concept: When a transversal crosses PARALLEL lines, special angle relationships exist
These relationships: Only work when lines are parallel
Used for: Finding unknown angle measures
These relationships: Only work when lines are parallel
Used for: Finding unknown angle measures
Parallel Lines Angle Theorems:
When $\ell \parallel m$ and transversal $t$ crosses both:
1. Corresponding Angles Postulate:
$$\text{Corresponding angles are } \textbf{congruent}$$
Example: If ∠1 = 70°, then ∠5 = 70°
2. Alternate Interior Angles Theorem:
$$\text{Alternate interior angles are } \textbf{congruent}$$
Example: If ∠3 = 110°, then ∠6 = 110°
3. Alternate Exterior Angles Theorem:
$$\text{Alternate exterior angles are } \textbf{congruent}$$
Example: If ∠1 = 70°, then ∠8 = 70°
4. Consecutive Interior Angles Theorem:
$$\text{Consecutive interior angles are } \textbf{supplementary}$$
Example: If ∠3 = 110°, then ∠5 = 70° (sum = 180°)
When $\ell \parallel m$ and transversal $t$ crosses both:
1. Corresponding Angles Postulate:
$$\text{Corresponding angles are } \textbf{congruent}$$
Example: If ∠1 = 70°, then ∠5 = 70°
2. Alternate Interior Angles Theorem:
$$\text{Alternate interior angles are } \textbf{congruent}$$
Example: If ∠3 = 110°, then ∠6 = 110°
3. Alternate Exterior Angles Theorem:
$$\text{Alternate exterior angles are } \textbf{congruent}$$
Example: If ∠1 = 70°, then ∠8 = 70°
4. Consecutive Interior Angles Theorem:
$$\text{Consecutive interior angles are } \textbf{supplementary}$$
Example: If ∠3 = 110°, then ∠5 = 70° (sum = 180°)
Example 1: Find all angles
Given: Lines $\ell \parallel m$, transversal $t$, ∠1 = 65°
Find: All other angles
Using vertical angles:
∠3 = ∠1 = 65° (vertical)
Using linear pairs (supplementary):
∠2 = 180° - 65° = 115°
∠4 = 115° (vertical to ∠2)
Using corresponding angles (parallel lines):
∠5 = ∠1 = 65° (corresponding)
∠6 = ∠2 = 115° (corresponding)
∠7 = ∠3 = 65° (corresponding)
∠8 = ∠4 = 115° (corresponding)
Summary: Four angles are 65°, four angles are 115°
Given: Lines $\ell \parallel m$, transversal $t$, ∠1 = 65°
Find: All other angles
Using vertical angles:
∠3 = ∠1 = 65° (vertical)
Using linear pairs (supplementary):
∠2 = 180° - 65° = 115°
∠4 = 115° (vertical to ∠2)
Using corresponding angles (parallel lines):
∠5 = ∠1 = 65° (corresponding)
∠6 = ∠2 = 115° (corresponding)
∠7 = ∠3 = 65° (corresponding)
∠8 = ∠4 = 115° (corresponding)
Summary: Four angles are 65°, four angles are 115°
Example 2: Use alternate interior angles
Given: $\ell \parallel m$, ∠4 = 3x + 15, ∠5 = 2x + 35
Find: x and both angle measures
∠4 and ∠5 are alternate interior angles
So they are congruent:
$3x + 15 = 2x + 35$
$3x - 2x = 35 - 15$
$x = 20$
∠4 = 3(20) + 15 = 75°
∠5 = 2(20) + 35 = 75° ✓
Answer: x = 20, both angles = 75°
Given: $\ell \parallel m$, ∠4 = 3x + 15, ∠5 = 2x + 35
Find: x and both angle measures
∠4 and ∠5 are alternate interior angles
So they are congruent:
$3x + 15 = 2x + 35$
$3x - 2x = 35 - 15$
$x = 20$
∠4 = 3(20) + 15 = 75°
∠5 = 2(20) + 35 = 75° ✓
Answer: x = 20, both angles = 75°
Example 3: Use consecutive interior angles
Given: $\ell \parallel m$, ∠3 = 4x, ∠5 = 2x + 30
Find: x and both angles
∠3 and ∠5 are consecutive interior angles
So they are supplementary:
$4x + (2x + 30) = 180$
$6x + 30 = 180$
$6x = 150$
$x = 25$
∠3 = 4(25) = 100°
∠5 = 2(25) + 30 = 80°
Check: 100° + 80° = 180° ✓
Answer: x = 25, ∠3 = 100°, ∠5 = 80°
Given: $\ell \parallel m$, ∠3 = 4x, ∠5 = 2x + 30
Find: x and both angles
∠3 and ∠5 are consecutive interior angles
So they are supplementary:
$4x + (2x + 30) = 180$
$6x + 30 = 180$
$6x = 150$
$x = 25$
∠3 = 4(25) = 100°
∠5 = 2(25) + 30 = 80°
Check: 100° + 80° = 180° ✓
Answer: x = 25, ∠3 = 100°, ∠5 = 80°
5. Transversals of Parallel Lines: Solve for x
Strategy for Solving:
Step 1: Identify the angle pair relationship
Step 2: Determine if angles are congruent or supplementary
Step 3: Set up an equation
Step 4: Solve for the variable
Step 5: Check your answer by substituting back
Step 1: Identify the angle pair relationship
Step 2: Determine if angles are congruent or supplementary
Step 3: Set up an equation
Step 4: Solve for the variable
Step 5: Check your answer by substituting back
Example 1: Multiple angles
Given: $\ell \parallel m$, ∠1 = (5x - 10)°, ∠2 = (3x + 30)°
∠1 and ∠2 are supplementary (linear pair)
$(5x - 10) + (3x + 30) = 180$
$8x + 20 = 180$
$8x = 160$
$x = 20$
Verify:
∠1 = 5(20) - 10 = 90°
∠2 = 3(20) + 30 = 90°
90° + 90° = 180° ✓
Answer: x = 20
Given: $\ell \parallel m$, ∠1 = (5x - 10)°, ∠2 = (3x + 30)°
∠1 and ∠2 are supplementary (linear pair)
$(5x - 10) + (3x + 30) = 180$
$8x + 20 = 180$
$8x = 160$
$x = 20$
Verify:
∠1 = 5(20) - 10 = 90°
∠2 = 3(20) + 30 = 90°
90° + 90° = 180° ✓
Answer: x = 20
Example 2: Corresponding angles
Given: Lines parallel, ∠2 = (7x + 5)°, ∠6 = (9x - 15)°
These are corresponding angles
$7x + 5 = 9x - 15$
$5 + 15 = 9x - 7x$
$20 = 2x$
$x = 10$
Answer: x = 10
Given: Lines parallel, ∠2 = (7x + 5)°, ∠6 = (9x - 15)°
These are corresponding angles
$7x + 5 = 9x - 15$
$5 + 15 = 9x - 7x$
$20 = 2x$
$x = 10$
Answer: x = 10
6. Construct Parallel Lines
Method: Copy a corresponding angle
Principle: If corresponding angles are congruent, lines are parallel
Tools: Compass and straightedge
Principle: If corresponding angles are congruent, lines are parallel
Tools: Compass and straightedge
Construction: Parallel Line Through Point P
Given: Line $\ell$ and point P not on $\ell$
Construct: Line through P parallel to $\ell$
Step 1: Draw any line through P that intersects $\ell$ (this creates a transversal)
• Call intersection point Q
Step 2: Copy ∠PQ$\ell$ to create corresponding angle at P
a) Place compass on Q, draw arc intersecting both rays
b) Keep same width, place compass on P
c) Draw arc intersecting line through P
d) Measure the arc at Q with compass
e) Transfer that measurement to arc at P
f) Draw ray from P through new point
Result: New line through P is parallel to $\ell$
Why: Corresponding angles are congruent
Given: Line $\ell$ and point P not on $\ell$
Construct: Line through P parallel to $\ell$
Step 1: Draw any line through P that intersects $\ell$ (this creates a transversal)
• Call intersection point Q
Step 2: Copy ∠PQ$\ell$ to create corresponding angle at P
a) Place compass on Q, draw arc intersecting both rays
b) Keep same width, place compass on P
c) Draw arc intersecting line through P
d) Measure the arc at Q with compass
e) Transfer that measurement to arc at P
f) Draw ray from P through new point
Result: New line through P is parallel to $\ell$
Why: Corresponding angles are congruent
7-8. Proofs Involving Parallel Lines
Two types of proofs:
1. Prove lines are parallel (converses)
2. Prove angle relationships given parallel lines
Key theorems: Both directions (original and converse)
1. Prove lines are parallel (converses)
2. Prove angle relationships given parallel lines
Key theorems: Both directions (original and converse)
Theorems and Their Converses
Original Theorems (Given: Lines are parallel):
1. If lines are parallel, corresponding angles are congruent
2. If lines are parallel, alternate interior angles are congruent
3. If lines are parallel, alternate exterior angles are congruent
4. If lines are parallel, consecutive interior angles are supplementary
Converse Theorems (To prove: Lines are parallel):
1. If corresponding angles are congruent, then lines are parallel
2. If alternate interior angles are congruent, then lines are parallel
3. If alternate exterior angles are congruent, then lines are parallel
4. If consecutive interior angles are supplementary, then lines are parallel
1. If lines are parallel, corresponding angles are congruent
2. If lines are parallel, alternate interior angles are congruent
3. If lines are parallel, alternate exterior angles are congruent
4. If lines are parallel, consecutive interior angles are supplementary
Converse Theorems (To prove: Lines are parallel):
1. If corresponding angles are congruent, then lines are parallel
2. If alternate interior angles are congruent, then lines are parallel
3. If alternate exterior angles are congruent, then lines are parallel
4. If consecutive interior angles are supplementary, then lines are parallel
Example 1: Prove lines are parallel
Given: ∠1 ≅ ∠5
Prove: $\ell \parallel m$
Given: ∠1 ≅ ∠5
Prove: $\ell \parallel m$
| Statements | Reasons |
|---|---|
| 1. ∠1 ≅ ∠5 | 1. Given |
| 2. ∠1 and ∠5 are corresponding angles | 2. Definition of corresponding angles |
| 3. $\ell \parallel m$ | 3. Converse of Corresponding Angles Postulate |
Example 2: Prove angle congruence
Given: $\ell \parallel m$, $m \parallel n$
Prove: $\ell \parallel n$
Given: $\ell \parallel m$, $m \parallel n$
Prove: $\ell \parallel n$
| Statements | Reasons |
|---|---|
| 1. $\ell \parallel m$ | 1. Given |
| 2. $m \parallel n$ | 2. Given |
| 3. $\ell \parallel n$ | 3. Transitive Property of Parallel Lines |
Example 3: Two-step proof
Given: ∠3 and ∠5 are supplementary
Prove: $\ell \parallel m$
Given: ∠3 and ∠5 are supplementary
Prove: $\ell \parallel m$
| Statements | Reasons |
|---|---|
| 1. ∠3 and ∠5 are supplementary | 1. Given |
| 2. ∠3 and ∠5 are consecutive interior angles | 2. Definition of consecutive interior angles |
| 3. $\ell \parallel m$ | 3. Converse of Consecutive Interior Angles Theorem |
Example 4: Multi-step proof
Given: $\ell \parallel m$, ∠1 ≅ ∠2
Prove: ∠2 ≅ ∠8
Given: $\ell \parallel m$, ∠1 ≅ ∠2
Prove: ∠2 ≅ ∠8
| Statements | Reasons |
|---|---|
| 1. $\ell \parallel m$ | 1. Given |
| 2. ∠1 ≅ ∠2 | 2. Given |
| 3. ∠1 ≅ ∠8 | 3. Alternate Exterior Angles Theorem |
| 4. ∠2 ≅ ∠8 | 4. Transitive Property of Congruence |
Line Relationships Summary
| Relationship | Definition | Coplanar? | Intersect? |
|---|---|---|---|
| Parallel Lines | Never intersect, same distance apart | YES (must be) | NO (never) |
| Intersecting Lines | Meet at exactly one point | YES (must be) | YES (at one point) |
| Perpendicular Lines | Intersect at 90° | YES (must be) | YES (at right angle) |
| Skew Lines | Not parallel, don't intersect | NO (different planes) | NO (but not parallel) |
Angle Pairs with Transversals
| Angle Pair | Location | If Lines Parallel | Pattern |
|---|---|---|---|
| Corresponding | Same position, same side of transversal | Congruent | "F" shape |
| Alternate Interior | Between lines, opposite sides | Congruent | "Z" shape |
| Alternate Exterior | Outside lines, opposite sides | Congruent | Outside "Z" |
| Consecutive Interior | Between lines, same side | Supplementary (180°) | "C" shape |
Theorems Quick Reference
| Theorem | Original (Given ∥) | Converse (Prove ∥) |
|---|---|---|
| Corresponding Angles | If $\ell \parallel m$, then corresp. ∠s ≅ | If corresp. ∠s ≅, then $\ell \parallel m$ |
| Alternate Interior | If $\ell \parallel m$, then alt. int. ∠s ≅ | If alt. int. ∠s ≅, then $\ell \parallel m$ |
| Alternate Exterior | If $\ell \parallel m$, then alt. ext. ∠s ≅ | If alt. ext. ∠s ≅, then $\ell \parallel m$ |
| Consecutive Interior | If $\ell \parallel m$, then consec. int. ∠s are supp. | If consec. int. ∠s are supp., then $\ell \parallel m$ |
Key Properties
| Property | Statement |
|---|---|
| Transitive Property | If $\ell \parallel m$ and $m \parallel n$, then $\ell \parallel n$ |
| Perpendicular Transversal | If line $\perp$ to one of two parallel lines, it's $\perp$ to the other |
| Parallel to Same Line | Two lines parallel to same line are parallel to each other |
| Unique Parallel | Through point not on line, exactly one line parallel to given line |
Success Tips for Parallel and Perpendicular Lines:
✓ Parallel lines NEVER intersect and must be coplanar
✓ Skew lines are NOT parallel and DON'T intersect (3D only)
✓ Transversal creates 8 angles with two lines
✓ Corresponding angles: "F" pattern, same position
✓ Alternate interior: "Z" pattern, between lines, opposite sides
✓ When lines are parallel: Corresponding, Alt. Interior, Alt. Exterior are CONGRUENT
✓ When lines are parallel: Consecutive Interior angles are SUPPLEMENTARY (180°)
✓ To PROVE lines parallel: Use CONVERSE theorems
✓ Congruent = equal measures; Supplementary = add to 180°
✓ Always identify angle pair relationship before solving!
✓ Parallel lines NEVER intersect and must be coplanar
✓ Skew lines are NOT parallel and DON'T intersect (3D only)
✓ Transversal creates 8 angles with two lines
✓ Corresponding angles: "F" pattern, same position
✓ Alternate interior: "Z" pattern, between lines, opposite sides
✓ When lines are parallel: Corresponding, Alt. Interior, Alt. Exterior are CONGRUENT
✓ When lines are parallel: Consecutive Interior angles are SUPPLEMENTARY (180°)
✓ To PROVE lines parallel: Use CONVERSE theorems
✓ Congruent = equal measures; Supplementary = add to 180°
✓ Always identify angle pair relationship before solving!