7.01 Parallel lines and transversals

Two different lines will either intersect at a single point or they will never intersect. If lines never intersect, 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.

If a pair of lines are not marked with chevrons, how can we tell if they are parallel or not? Is the point of intersection off the edge of the diagram or do the lines not intersect at all? To tell these cases apart we introduce a third line, called a transversal, that intersects both of the original 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 pairs of angles that are created, which we will look at in the next sections.

Examples

Example 1

Select the diagram showing a pair of parallel lines:

A

B

Worked Solution

Create a strategy

Observe the lines and think about if they were drawn continously, if they might intersect.

Apply the idea

The lines in option B, if they were drawn continuosly, would appear that they might intersect near the bottom of the image.

The lines in option A, if they were drawn continuosly appear that they won't intersect, and have the same markings on them showing that the lines are parallel, like this: >. So the answer is option A.

The first type of angles, called consecutive interior angles, are formed between the original lines and on the same side of the transversal:

The second type of angles, called alternate interior angles, lie between the original lines on opposite sides of the transversal:

The alternate interior angles are congruent and are marked the same way on a pair of parallel lines.

Examples

Example 2

Select the diagram showing a pair of consecutive interior angles:

A

B

C

Worked Solution

Create a strategy

Choose the option that has two angles formed between the original lines and that are on the same side of the transversal.

Apply the idea

The angles in option B are between the original lines and are both on the left side of the transversal. So the answer is option B.

Example 3

Select the diagram showing a pair of alternate interior angles:

A

B

C

Worked Solution

Create a strategy

Choose the option that has two angles formed between the original lines and that are on opposite sides of the transversal.

Apply the idea

The angles in option C are between the original lines and are on opposite sides of the transversal. So the answer is option C.

The final type of angles, called corresponding angles, lie in the same relative position on each line:

The corresponding angles are marked the same way because they are equal.

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

• Consecutive interior angles formed on parallel lines are supplementary. If they are not supplementary, the lines are not parallel.

• Alternate interior 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.

Note: 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.

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

Examples

Example 4

If \angle APB=82^\circ , what is the measure of \angle DQE?

Worked Solution

Create a strategy

Find the corresponding angle of the given angle and use a property of congruence.

Apply the idea

\angle APB has a corresponding angle \angle FQB, so \angle APB \cong \angle FQB.

But \angle FQB and\angle DQE are vertical angles, so \angle FQB \cong \angle DQE.

By transitivity, if \angle APB \cong \angle FQB and \angle FQB \cong \angle DQE, then \angle APB \cong \angle DQE.

So \angle DQE=82\degree.

Reflect and check

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, since these are vertical angles.

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

This means that \angle APB = \angle DQE, so \angle DQE = 82\degree, as before.

Example 5

Consider the diagram below:

a

Which two of the following are true statements?

A

The marked angles are consecutive interior.

B

The marked angles are corresponding.

C

The marked angles are alternate interior.

D

The marked angles are equal.

E

The marked angles are not supplementary.

F

The marked angles are supplementary.

Worked Solution

Create a strategy

Recall the following types of angles on parallel lines:

• The corresponding angles are equal

• The alternate interior angles are equal

• The consecutive interior angles are supplementary

Apply the idea

\displaystyle 113+67

\displaystyle =

\displaystyle 180

Add the angles

Since the angles add to 180\degree they are supplementary. Since the angles lie on the same side of the transversal and between the parallel lines, they are consecutive interior angles.

The answers are options A and F.

b

Is there a pair of parallel lines in the diagram?

Worked Solution

Apply the idea

Yes, there is a pair of parallel lines in the diagram, since the consecutive interior angles were supplementary.

Example 6

Solve for the value of x.

Worked Solution

Create a strategy

Recall the following types of angles on parallel lines:

• The corresponding angles are equal

• The alternate interior angles are equal

• The consecutive interior angles are supplementary

Apply the idea

The angles are corresponding angles which are equal.

\displaystyle x

\displaystyle =

\displaystyle 125

Equate the corresponding angles