Parallel, Perpendicular and Intersecting Lines | Grade 7 Math | LevelKite

You already know how to tell a line from a line segment and a ray. Hold on to one fact today: a line is straight and goes on without end in both directions. A drawing has edges, but a line has no ends. It keeps going past the drawing.

Now picture two different lines on the same flat surface. They might cross. They might cross at a perfect square corner. Or they might never meet, however far you follow them. Each case has its own name. In this lesson you'll learn those names and choose the right one from facts you can check, not from how a picture looks.

Key concept

Intersecting, perpendicular and parallel lines

A line is straight and has no end in either direction. Take two different lines on the same flat surface, called a plane. They share either exactly one point or no point at all.

Intersecting lines share exactly one point. They cross there.
Perpendicular lines are intersecting lines that cross at a right angle. A right angle is a square corner. Its size is $90^\circ$ , read as 90 degrees. Degrees are the unit for the size of an angle.
Parallel lines share no point, however far you follow them in either direction.

Every perpendicular pair is also an intersecting pair. A parallel pair is never an intersecting pair. Decide which name fits from facts you can check, never from how a drawing looks.

To get facts you can check, place the lines on a coordinate plane: a plane with two number lines, called axes. The axis that runs left to right is the $x$ -axis. The axis that runs up and down is the $y$ -axis. The two axes cross at a right angle.

Every point has two coordinates, written $(x, y)$ . The $x$ -coordinate gives the point's position left or right, measured from the $y$ -axis. The $y$ -coordinate gives its position up or down, measured from the $x$ -axis. Both are counted in grid units, where one unit is one step of the grid.

Two kinds of line are easy to describe with coordinates:

A horizontal line runs in the same direction as the $x$ -axis. Every point on it has the same $y$ -coordinate.
A vertical line runs in the same direction as the $y$ -axis. Every point on it has the same $x$ -coordinate.

The figure below has four lines on a grid that shows $x$ from $0$ to $8$ and $y$ from $0$ to $8$ . Each line continues past the edges of the grid without end.

Line A is horizontal. Every point on it has $y$ -coordinate $2$ .
Line B is horizontal. Every point on it has $y$ -coordinate $5$ .
Line C is vertical. Every point on it has $x$ -coordinate $3$ .
Line D is vertical. Every point on it has $x$ -coordinate $7$ .

These four facts are what you'll reason from, not the look of the drawing.

Line A and Line B are horizontal, with

y

-coordinates

2

and

5

. Line C and Line D are vertical, with

x

-coordinates

3

and

7

.

Worked example

Two horizontal lines

Line A is horizontal, with $y$ -coordinate $2$ at every point. Line B is horizontal, with $y$ -coordinate $5$ at every point. How are Line A and Line B related?

Choose any x-coordinate, on the grid or beyond it. There, Line A has a point with y-coordinate 2 and Line B has a point with y-coordinate 5, because a horizontal line keeps one y-coordinate at every point.
$(x, 2) \text{ and } (x, 5)$
Find the gap between those two points. Subtract the smaller y-coordinate from the larger one, so the gap comes out as a positive number of grid units.
$5 - 2 = 3$
The subtraction did not depend on which x-coordinate you chose, so the gap is 3 units at every x-coordinate. A shared point would need a gap of 0, and the gap is never 0. So the lines share no point.
$3 \neq 0$
Use the definition. Two different lines in the same plane that share no point are parallel.

Answer

Line A and Line B are parallel, because the gap between them is $3$ units at every $x$ -coordinate, so they never meet.

Worked example

A horizontal line and a vertical line

Line A is horizontal, with $y$ -coordinate $2$ at every point. Line C is vertical, with $x$ -coordinate $3$ at every point. How are Line A and Line C related?

Look for a point on both lines. It needs x-coordinate 3 to be on Line C and y-coordinate 2 to be on Line A.
$(3, 2)$
Check that no other point is shared. Every other point on Line C has x-coordinate 3 but a y-coordinate that is not 2, so it is not on Line A. The lines share exactly one point, so they are intersecting.
Find the angle from facts. Line A runs in the direction of the x-axis, and Line C runs in the direction of the y-axis. The axes cross at a right angle, so this horizontal line and this vertical line cross at a right angle too.
$\text{angle at } (3, 2) = 90^\circ$
Use the definition. Intersecting lines that cross at a right angle are perpendicular.

Answer

Line A and Line C are perpendicular: they share only the point $(3, 2)$ , and they cross there at a right angle.

For the next example: Line A has

y

-coordinate

2

at every point, and Line E is made of the points whose two coordinates are equal, such as

(1, 1)

and

(6, 6)

.

Worked example

A crossing with a stated angle

In the figure just before this example, Line A is the same horizontal line as before, with $y$ -coordinate $2$ at every point. Line E is neither horizontal nor vertical. It is made of every point whose $x$ -coordinate equals its $y$ -coordinate, such as $(1, 1)$ and $(6, 6)$ . Here is a stated fact: each angle where the two lines cross measures $45^\circ$ or $135^\circ$ . How are Line A and Line E related?

Look for a point on both lines. A point on Line A has y-coordinate 2. To be on Line E as well, its x-coordinate must equal its y-coordinate, so its x-coordinate must be 2.
$(2, 2)$
Only one point has x-coordinate 2 and y-coordinate 2, so the lines share exactly one point. They are intersecting.
Test for perpendicular. Line E is not vertical, so the reason from the last example does not apply. Use the stated angles, not the look of the crossing: perpendicular lines need a right angle there, and neither stated measure is 90 degrees.
$45^\circ \neq 90^\circ \quad \text{and} \quad 135^\circ \neq 90^\circ$

Answer

Line A and Line E are intersecting but not perpendicular: they share only the point $(2, 2)$ , and no angle where they cross is a right angle.

The grid shows

x

from

0

to

8

. There, Line G falls from

(0, 5)

to

(8, 4)

while Line F stays at

y

-coordinate

3

. They meet at

(16, 3)

, beyond the grid.

Here is the idea to keep. Two different lines in a plane are intersecting when they share exactly one point, perpendicular when they intersect at a right angle, and parallel when they share no point, however far they run.

Decide from facts, never from looks. Two horizontal lines, or two vertical lines, with a coordinate gap that is the same non-zero number everywhere are parallel. A horizontal line and a vertical line on a coordinate plane are perpendicular, and so are lines with a stated right angle where they meet. A shared point alone tells you only that the lines are intersecting.

Next, you'll measure and draw angles with a protractor, so you'll be able to measure the angle where two lines cross.