In this lesson:
By the end of this lesson, you should be able to:
You've seen parallel and perpendicular lines since middle school:
But "these lines look parallel" is informal reasoning — it can be wrong.
Today we make these ideas mathematically precise.
Why does perpendicular mean 90°?
Perpendicular lines divide the plane around their intersection into four equal angular regions — each exactly one-quarter of a full turn.
Two lines intersect at 75°. Are they perpendicular?
Apply the definition before advancing.
No — these lines are not perpendicular.
Watch out: Perpendicular requires exactly 90° — lines that look nearly perpendicular are not perpendicular unless verified.
In three-dimensional space, two lines can fail to intersect without being parallel.
Parallel = same plane + no intersection — both conditions required
Are perpendicular lines ever parallel? Explain using the definitions.
Think through both definitions before advancing.
Never. Perpendicular lines cannot be parallel.
This is a logical consequence of the definitions — no diagram needed.
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.
Claim: If two lines are both perpendicular to the same line, they are parallel to each other.
Given:
To show:
Before we prove this, ask: what does each definition give us?
Notice where definitions were invoked:
Precision matters: without the definitions, "the lines look parallel" is not a proof.
Mathematical proof requires naming which definition or theorem justifies each step.
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.
We used the definition of parallel lines:
Two lines in the same plane that do not intersect are parallel.
✓ 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
The five definitions from these two lessons appear in every proof this year.
Click to begin the narrated lesson
Define geometric terms precisely