Precise Definitions: Perpendicular & Parallel Lines

Geometry demands precise definitions, because proofs can only use what a definition actually says. This lesson pins down three relationships between lines: perpendicular lines intersect to form a right angle, parallel lines are coplanar lines that never intersect, and skew lines never intersect and are not coplanar.

The word that trips students up is coplanar — lying in the same plane. Both parallel and skew lines never intersect; what separates them is whether a single flat plane can contain both. Get that distinction down and every classification question in this lesson becomes a two-part check.

Perpendicular lines

Perpendicular lines intersect to form a right angle. The symbol is $\ell \perp m$ , read “line $\ell$ is perpendicular to line $m$ .”

One right angle is all the definition requires, because the rest come free: the right angle's linear-pair partner must also measure $90^\circ$ , and vertical angles are congruent — so all four angles at the intersection are right angles.

Parallel lines

Parallel lines are coplanar lines that never intersect. The symbol is $\ell \parallel m$ . Both conditions matter: the lines must lie in the same plane, and they must have no point in common. In a figure, matching arrowheads drawn on two lines mark them as parallel.

The matching chevron marks in the figure are exactly those parallel marks — they tell you the two lines run in the same direction and never meet.

Skew lines

Skew lines never intersect and are not coplanar. They only exist in three dimensions — think of an edge along the top of a box and a non-touching edge along the bottom running in a different direction. No single plane contains both.

So the classification test has two questions. Do the lines intersect? If yes: perpendicular when they form a right angle, otherwise just intersecting. If no: parallel when they are coplanar, skew when they are not.

Worked examples

Example 1: classify coplanar lines that never meet

Two distinct lines lie in the same plane and never intersect. Classify them.

The lines do not intersect, so they are either parallel or skew

They are coplanar, which rules out skew

Coplanar and non-intersecting is the definition of parallel

Answer: The lines are parallel: $\ell \parallel m$ .

Example 2: classify lines meeting at a right angle

Two lines intersect, and one of the four angles they form measures $90^\circ$ . Classify the lines and find the other three angles.

Lines that intersect to form a right angle are perpendicular

\ell \perp m

The angle beside the right angle forms a linear pair with it

180^\circ - 90^\circ = 90^\circ

Vertical angles are congruent, so the remaining two angles are also

90^\circ

Answer: The lines are perpendicular, and all four angles measure $90^\circ$ .

Example 3: spot skew lines

An edge along the top of a rectangular box runs front-to-back. An edge along the bottom runs left-to-right and does not touch the first edge. Classify the two lines.

The lines never intersect — one is at the top of the box, the other at the bottom

Check coplanarity: one runs front-to-back, the other left-to-right, at different heights — no single plane contains both

Non-intersecting and not coplanar is the definition of skew

Answer: The lines are skew.

Try one yourself

Common questions

What does coplanar mean?

Lying in the same plane — a single flat surface can contain both lines. Parallel lines are coplanar; skew lines are not.

What is the difference between parallel and skew lines?

Both never intersect. Parallel lines lie in the same plane; skew lines do not. In a flat, two-dimensional figure every pair of lines is coplanar, so skew lines only show up in three-dimensional situations.

Do perpendicular lines form four right angles or just one?

The definition only requires one right angle, but linear pairs and vertical angles force the other three to be right angles as well. So in practice, all four are $90^\circ$ .

What do the symbols

\parallel

and

\perp

mean?

$\ell \parallel m$ means line $\ell$ is parallel to line $m$ . $\ell \perp m$ means the lines are perpendicular. In figures, matching arrowheads mark parallel lines and a small square marks a right angle.

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