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Middle Years

9.03 Parallel lines and transversals

Lesson

Two different lines either intersect at a single point, or they have no intersection. If they have no intersection, we say they are parallel.

Parallel lines are marked with chevrons like this: $>$>, and if the number of chevrons on two lines match, they are parallel.

Two pairs of parallel lines.

If a pair of lines are not marked as parallel, how can we tell if they are or not? Is the intersection point off the edge of the diagram, or does it not exist at all? To tell these cases apart we introduce a third line, called a transversal, that intersects both lines.

The transversals (in blue) are drawn to test whether the lines they intersect are parallel.

The transversal forms angles at the intersection points, and there are three useful ways to identify them in pairs. The first kind are called cointerior, the two angles formed between the original lines, and on the same side of the transversal:

The cointerior angles for each pair of lines.

The second kind of pair are called alternate angles, lying between the original lines on opposite sides of the transversal:

The alternate angles for each pair of lines.

The final kind of pair are called corresponding angles, lying in the same relative position on each line:

The corresponding angles for each pair of lines.

We can tell whether lines are parallel using one of these criteria:

  • Cointerior angles formed on parallel lines are supplementary. If they are not supplementary, the lines are not parallel.
  • Alternate angles formed on parallel lines are equal. If they are not equal, the lines are not parallel.
  • Corresponding angles formed on parallel lines are equal. If they are not equal, the lines are not parallel.
Caution

This lesson refers to lines, but all of these ideas also apply to rays and segments. Rays and segments are parallel if the lines through their defining points are parallel.

You can explore the three kinds of angle pairs formed on parallel lines in the applets below:

If we know two lines are parallel, we can use these criteria to find the values of other angles.

Worked example

If $\angle APB=82^\circ$APB=82°, what is $\angle DQE$DQE?

 

Think: The line through $A$A and $C$C is marked as parallel to the line through $F$F and $D$D. The line through $B$B and $E$E is a transversal. This means the cointerior angles at $P$P and $Q$Q are supplementary, the alternate angles are equal, and the corresponding angles are equal.

Do: $\angle APB=\angle FQB$APB=FQB, since they are corresponding angles on parallel lines.

$\angle FQB=\angle DQE$FQB=DQE, since these are vertically opposite angles.

This means that $\angle APB=\angle DQE$APB=DQE, so $\angle DQE=82^\circ$DQE=82°.

Reflect: We could have found the angle many different ways, using some of the other criteria. Here is another path we could have taken:

$\angle APB=\angle CPE$APB=CPE, since these are vertically opposite angles.

$\angle CPE=\angle DQE$CPE=DQE, since they are corresponding angles on parallel lines.

This means that $\angle APB=\angle DQE$APB=DQE, so $\angle DQE=82^\circ$DQE=82°, as before.

Can you find any others?

Summary

  • Cointerior angles formed on parallel lines are supplementary. If they are not supplementary, the lines are not parallel.
  • Alternate angles formed on parallel lines are equal. If they are not equal, the lines are not parallel.
  • Corresponding angles formed on parallel lines are equal. If they are not equal, the lines are not parallel.

Practice questions

Question 1

Select the diagram showing a pair of parallel lines:

  1. A

    B

    C

    D
Question 2

Consider this diagram and answer the questions that follow:

  1. Which two of the following are true statements?

    The marked angles are equal.

    A

    The marked angles are not supplementary.

    B

    The marked angles are alternate.

    C

    The marked angles are supplementary.

    D

    The marked angles are cointerior.

    E

    The marked angles are corresponding.

    F
  2. Is there a pair of parallel lines in the diagram?

    Yes

    A

    No

    B
Question 3

Solve for the value of $x$x.

Enter each line of working as an equation.

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