Math Core

Look at a sheet of notebook paper. The blue lines never cross each other, and the red margin line crosses each of them at a square corner. Those two ideas have special names: parallel and perpendicular.

Lines that cross and lines that don't

Two lines in a flat picture can do only two things: cross, or never cross.

Intersecting lines cross at one point.
Perpendicular lines are intersecting lines that cross at a right angle. They make four square corners where they meet.
Parallel lines never cross, no matter how far you extend them. They stay the same distance apart everywhere, like the two rails of a train track.

Left to right: intersecting lines, perpendicular lines and parallel lines.

Definition

Parallel and perpendicular

Parallel lines are lines that never meet. Perpendicular lines are lines that meet at a right angle ( $90^\circ$ ).

We use symbols to save writing. $\parallel$ means "is parallel to" and $\perp$ means "is perpendicular to." For example, $\overleftrightarrow{AB} \perp \overleftrightarrow{CD}$ says line $AB$ is perpendicular to line $CD$ .

Segments and rays can be parallel or perpendicular too

The sides of a shape are segments. Two sides are parallel if the lines they lie on are parallel, and perpendicular if they meet at a right angle.

Worked example: Sides of a rectangle

Rectangle $ABCD$ is shown below. Name a pair of parallel sides and a pair of perpendicular sides.

The top and bottom never meet: $\overline{AB} \parallel \overline{DC}$ . The left and right sides are parallel too: $\overline{AD} \parallel \overline{BC}$ .
Sides $\overline{AB}$ and $\overline{BC}$ meet at a right angle at $B$ , so $\overline{AB} \perp \overline{BC}$ .

A rectangle has 2 pairs of parallel sides and 4 pairs of perpendicular sides, one pair at each corner.

Worked example: Letters

Which of these capital letters have perpendicular segments: L, V, H?

L: the two segments meet at a square corner. Perpendicular. ✓
V: the two segments meet at a point, but the corner is sharper than a square corner. Not perpendicular.
H: the crossbar meets each upright at square corners. Perpendicular. ✓ (The two uprights are also parallel to each other.)

So L and H have perpendicular segments.

Common mistake

Two lines that don't cross in the picture are not always parallel. A picture shows only part of each line. If the lines are getting closer together, they will cross somewhere if you extend them, so they are not parallel. Parallel lines stay the same distance apart the whole way.

Tip

Perpendicular lines don't have to go straight across and straight up. Tilted lines can be perpendicular too. To check, hold the corner of a sheet of paper at the spot where they cross. If the corner fits exactly, the lines are perpendicular.

Practice

Practice 1

What do we call two lines that never meet?

Practice 2

Two lines cross and make four right angles. What are they called?

Practice 3

What best describes these two lines?

Practice 4

Two perpendicular lines cross. How many right angles do they make?

Type a number.

Practice 5

How many pairs of parallel sides does a rectangle have?

Type a number.

Practice 6

Which capital letter has a pair of parallel segments but no perpendicular segments?

Practice 7

Look at shape $ABCD$ . Which side is parallel to $\overline{AB}$ ?

Practice 8

How many pairs of perpendicular sides does a square have?

Type a number.