Topics
3. Geometry: Pythagorean Theorem, Congruence & Similarity Transformations
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United States of AmericaNJ
Grade 8

3.08 Alternate interior angles

Lesson
Practice
Lesson

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.

Two pairs of parallel lines.

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.

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

The consecutive interior angles for each pair of lines.

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

The alternate interior angles for each pair of lines.

The final type of angles, called corresponding angles, lie 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:

  • 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.

Exploration

We can investigate these angle relationships more using the applets 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?

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

Worked example

Question 1

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 interior 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 vertical 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 vertical 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

  • 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.

Practice questions

Question 1

Select the diagram showing a pair of parallel lines:

  1. Two lines with arrowheads at their endpoints indicating that the lines extend indefinitely in both directions. Both lines have an identical arrow marking, pointing in the same direction.

    A

    Two lines with arrowheads at their endpoints indicating that the lines extend indefinitely in both directions. The arrowheads pointing at the top right direction are closer to each other than the arrowheads pointing at the bottom left direction.

    B

    Two lines with arrowheads at their endpoints indicating that the lines extend indefinitely in both directions. The arrowheads pointing at the left direction are closer to each other than the arrowheads pointing at the right direction.

    C

    Two lines with arrowheads at their endpoints indicating that the lines extend indefinitely in both directions. The arrowheads pointing at the top left direction are closer to each other than the arrowheads pointing at the bottom right direction.

    D
Question 2

Consider this diagram and answer the questions that follow:

  1. Which of the following are true statements? Select the two correct options.

    The marked angles are consecutive interior.

    A

    The marked angles are corresponding.

    B

    The marked angles are alternate interior.

    C

    The marked angles are equal.

    D

    The marked angles are not supplementary.

    E

    The marked angles are supplementary.

    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 your answer as an equation.

Outcomes

8.G.A.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

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