12.03 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.

Examples

Example 1

Select the diagram showing a pair of parallel lines:

A

B

C

D

Worked Solution

Create a strategy

Choose the option that has the same markings with chevrons that look like this: >.

Apply the idea

The lines in option A have the same markings on them showing that the lines are parallel. So the answer is option A.

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The first kind of angles are called cointerior, the two angles formed between the original lines, and on the same side of the transversal:

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

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

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:

• 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.

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 2

Consider the diagram below:

a

Which of the following are true statements?

A

The marked angles are equal.

B

The marked angles are not supplementary.

C

The marked angles are alternate.

D

The marked angles are supplementary.

E

The marked angles are cointerior.

F

The marked angles are corresponding.

Worked Solution

Create a strategy

Recall the following types of angles on parallel lines:

• The corresponding angles are equal

• The alternate angles are equal

• The cointerior angles are supplementary

Apply the idea

\displaystyle 124+46

\displaystyle =

\displaystyle 170

Add the angles

Since the angles add to 170\degree, they are not supplementary. But the angles lie on the same side of the transversal and between the parallel lines, so they are cointerior angles.

The answers are options B and E.

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b

Is there a pair of parallel lines in the diagram?

A

Yes

B

No

Worked Solution

Apply the idea

No, there is no pair of parallel lines in the diagram, since the cointerior angles were not supplementary.

So the correct answer is option A.

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Example 3

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 angles are equal

• The cointerior angles are supplementary

Apply the idea

The angles are cointerior angles which are supplementary.

\displaystyle x+128	\displaystyle =	\displaystyle 180	Equate the sum of the angles to 180
\displaystyle x	\displaystyle =	\displaystyle 180-128	Subtract 128 from both sides
\displaystyle x	\displaystyle =	\displaystyle 52	Evaluate

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