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12.03 Parallel lines and transversals

Lesson

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.

This image shows different pairs of lines that are parallel or not. Ask your teacher for more information.
Two pairs of parallel lines.

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.

Two pairs of parallel lines where each pair are intersected by a transversal.

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
This image shows a pair of lines. Ask your teacher for more information.
B
This image shows a pair of lines. Ask your teacher for more information.
C
This image shows a pair of lines. Ask your teacher for more information.
D
This image shows a pair of lines. Ask your teacher for more information.
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.

Idea summary

Parallel lines are lines that never intersect and are marked with chevrons like this: >.

A transversal is a line that intersects a set of parallel lines.

Cointerior and alternate angles

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

This image shows cointerior angles on parallel lines. Ask your teacher for more information.

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

This image shows alternate angles on parallel lines. Ask your teacher for more information.

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

Exploration

We can investigate these angle relationships more using the applet below.

  1. Click and drag any blue points to adjust the lines.

  2. Check the boxes to highlight different angle pairs.

  3. Do the relationships change as the points on the lines are moved?

Loading interactive...

Alternate angles are equal while the cointerior angles are supplementary or have a sum of 180\degree. These relationships do not change as the points on the lines are moved.

Idea summary

Cointerior angles formed on parallel lines are supplementary.

Alternate angles formed on parallel lines are equal.

Corresponding angles

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

This image shows corresponding angles on parallel lines. Ask your teacher for more information.

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

Exploration

We can investigate this angle relationship using the applet below.

  1. Click and drag any blue points to adjust the lines.

  2. Check the box to show all corresponding angles.

  3. Do the relationships change as the points on the lines are moved?

Loading interactive...

All pairs of corresponding angles on parallel lines are equal and this relationship does not change as points on the line are moved.

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.

This image shows the criteria to find the values of other angles using parallel lines. Ask your teacher for more information.
This image shows criteria that will not make a pair of lines parallel. Ask your teacher for more information.

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:

Parallel lines crossed by a transversal forming angles of 124 and 46 degrees. Ask your teacher for more information.
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 170Add 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.

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.

Example 3

Solve for the value of x.

Parallel lines with a transversal forming cointerior angles 128 degrees and x degrees.
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 180Equate the sum of the angles to 180
\displaystyle x\displaystyle =\displaystyle 180-128Subtract 128 from both sides
\displaystyle x\displaystyle =\displaystyle 52Evaluate
Idea summary

Corresponding angles, lie in the same relative position on each parallel line.

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.

Outcomes

MA4-18MG

identifies and uses angle relationships, including those related to transversals on sets of parallel lines

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