Parallel and Perpendicular Lines — Definitions and Proof

Lesson 2 of 2: Line Relationships and Proof Precision

In this lesson:

• Precise definitions of perpendicular and parallel lines
• Why "never meets" is informal — and what the formal definition says
• How precise definitions make geometric proof possible

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. State the precise definitions of perpendicular and parallel lines
2. Use definitions to identify whether a given pair of lines is perpendicular, parallel, or neither
3. Recognize why precision in mathematical definitions matters — and how it differs from informal language

Do You Really Know What Parallel Means?

You've seen parallel and perpendicular lines since middle school:

• Parallel: railroad tracks, ruled notebook lines, opposite sides of a rectangle
• Perpendicular: corner of a room, the letter "T", a cross

But "these lines look parallel" is informal reasoning — it can be wrong.

Today we make these ideas mathematically precise.

Perpendicular Lines — Precise Definition

• Two lines are perpendicular if they intersect and form right angles
• Right angle = 90° = one-quarter of a full rotation (360°)
• When two lines are perpendicular, all four angles formed are 90°

90° Means One Quarter Rotation

Why does perpendicular mean 90°?

• A full rotation = 360°
• A half rotation = 180° (a straight line)
• A quarter rotation = 90° = one right angle

Perpendicular lines divide the plane around their intersection into four equal angular regions — each exactly one-quarter of a full turn.

Quick Check: Is This Pair Perpendicular?

Two lines intersect at 75°. Are they perpendicular?

Apply the definition before advancing.

Answer: 75° Is Not a Right Angle

No — these lines are not perpendicular.

• Perpendicular requires all four angles at the intersection to equal 90°
• 75° ≠ 90°, so the first condition fails immediately
• The adjacent angle would be 180° − 75° = 105°, confirming the four angles are not all equal

Watch out: Perpendicular requires exactly 90° — lines that look nearly perpendicular are not perpendicular unless verified.

Parallel Lines — Precise Definition

• Two lines in a plane are parallel if they do not intersect
• They must lie in the same plane — this rules out skew lines in 3D
• Parallel lines maintain constant distance from each other

"Same Plane" Is Not Redundant

In three-dimensional space, two lines can fail to intersect without being parallel.

• Skew lines: not in the same plane, never intersect, but not parallel
• Example: a floor edge and an opposite ceiling edge — they don't intersect, yet they're not parallel

Parallel = same plane + no intersection — both conditions required

Classifying Line Pairs: Parallel, Perpendicular, or Neither

• Left pair: all four angles = 90° → perpendicular
• Middle pair: same plane, no intersection → parallel
• Right pair: intersect, angles ≠ 90° → neither

Quick Check: Can Perpendicular Lines Be Parallel?

Are perpendicular lines ever parallel? Explain using the definitions.

Think through both definitions before advancing.

Answer: Perpendicular and Parallel Are Mutually Exclusive

Never. Perpendicular lines cannot be parallel.

• Perpendicular requires the lines to intersect (and form right angles)
• Parallel requires the lines to not intersect
• A pair of lines cannot simultaneously intersect and not intersect

This is a logical consequence of the definitions — no diagram needed.

From Informal to Formal Reasoning

Informal reasoning (middle school):

"These lines look parallel — they seem to go in the same direction."

Formal reasoning (high school geometry):

"Lines and are parallel because they lie in the same plane and, extended in both directions, share no common point — by definition of parallel lines."

The difference: formal reasoning is checkable, logically airtight, and independent of the diagram.

Proof Setup: Two Lines Perpendicular to One Line

Claim: If two lines are both perpendicular to the same line, they are parallel to each other.

Given:

• Line line
• Line line

To show:

Before we prove this, ask: what does each definition give us?

Proof Reasoning — Contradiction Shows No Intersection

• → 90° at ; → 90° at
• If and met, they'd form a triangle with
• Two 90° angles already sum to 180° — leaving 0° for the third angle
• Contradiction → and cannot meet →

Citing Definitions in Proof

Notice where definitions were invoked:

• "m ⊥ l" → by definition of perpendicular, m meets l at 90°
• "m and n do not intersect" → by definition of parallel, m ∥ n

Precision matters: without the definitions, "the lines look parallel" is not a proof.

Mathematical proof requires naming which definition or theorem justifies each step.

Quick Check: Citing the Right Definition

In the proof above, what definition did we use to conclude that lines and are parallel?

Name the definition and what it says before advancing.

Answer: The Definition of Parallel Lines

We used the definition of parallel lines:

Two lines in the same plane that do not intersect are parallel.

• We showed and lie in the same plane (both intersect )
• We showed and do not intersect (the triangle angle-sum contradiction)
• Together: by definition

The Definition Tree: How It All Connects

Key Takeaways and Misconception Warnings

✓ Perpendicular: intersect at four 90° angles — 90° = one quarter rotation

✓ Parallel: same plane, never intersect — constant distance

✓ Precise definitions make proof possible — informal observation cannot

"Never meet" is informal — formal definition adds same plane

Perpendicular and parallel are mutually exclusive — one requires intersection, the other forbids it

What Comes Next in This Course

• HSG.CO.A.2 — Transformations: rely on distance and angle — the undefined terms
• HSG.CO.B — Congruence: theorems cite angle and segment definitions
• HSG.GPE.A.1 — Circle equation: applies the definition of circle directly
• HSG.C.A — Circle theorems: require the precise definition of circle

The five definitions from these two lessons appear in every proof this year.