Unit 1, Lesson 14
Alternate Interior Angles
Grade 8
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To Gather
Geometry toolkits
The purpose of this activity is for students to recall prior work with supplementary angles and to connect vertical angles and 180-degree rotations of intersecting lines. As students find the angle measures, listen to their conversations, specifically for the use of vocabulary such as “supplementary angles,” “vertical angles,” and “rotations.”
Provide access to geometry toolkits, including protractors and tracing paper. If needed, display the image from the problem and invite a student to state the name of the
Find the measure of angle
Find and label a second
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Display the image for all to see. Invite students to share their responses, adding onto the image as needed to help make clear student thinking.
If no students use supplementary angles or the property that a straight line is
MLR2: Collect and Display
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To Gather
Geometry toolkits
In this task, students explore the relationship between angles formed when two parallel lines are cut by a transversal line. Students investigate whether knowing the measure of one angle is sufficient to figure out all the angle measures in the diagram.
Monitor for students who use these different strategies to find the angle measures:
Select these students to share during the whole-class discussion.
A transversal (or transversal line) for a pair of parallel lines is a line that meets each of the parallel lines at exactly one point. Introduce this idea, and provide a picture such as this picture where line
Arrange students in groups of 2. Provide access to geometry toolkits. Give students 2–3 minutes of quiet think time, then 5–8 minutes of partner work time, followed by a whole-class discussion.
The goal of this discussion is for students to describe the relationships they notice between the angles formed when two parallel lines are cut by a transversal.
Display the image from the problem for all to see and invite groups to share what they noticed. Encourage students to use precise vocabulary, such as "supplementary angles" and "vertical angles," when describing how they figured out the different angle measurements (MP6). After students point out the matching angles at the two vertices, define the term alternate interior angles: Alternate interior angles are created when two parallel lines are crossed by another line called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.
Ask a few students to identify the pairs of alternate interior angles from the activity.
MLR8: Discussion Supports
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To Gather
Geometry toolkits
The goal of this task is for students to connect their work with rigid transformations with the property that alternate interior angles are congruent. In this activity, students use properties of 180-degree rotations to find corresponding angles in the figure. As students describe their rigid transformations to their partner and listen to their partner’s reasoning, they construct and critique the argument that these angles are congruent (MP3).
Listen for different strategies students use to show that the angles are congruent and select these students to share their strategies during the discussion. Approaches might include:
Arrange students in groups of 2. Give 2–3 minutes of quiet work time, followed by 2–3 minutes of partner discussion, then a whole-class discussion.
Provide access to geometry toolkits. Tell students that in this activity, we will try to figure out why we saw all the matching angles we did in the last activity.
Invite previously selected students to share their explanations. Ask students to describe and demonstrate the transformations they used to show that alternate interior angles are congruent. If any students connect this diagram to earlier work with 180-degree rotations or their justifications that vertical angles are congruent, invite them to share their observations.
Consider displaying this image for all to see as students share their thinking:
If any students use a translation to take
Optional
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None
Lines
Line A C contains point B. Line D F contains point E. Line J H contains points B and E. Angle A B J is labeled question mark. Angle A B E is labeled 63 degrees. Angle E B C is labeled question mark. Angle C B J is labeled question mark. Angle D E H is labeled question mark. Angle H E F is labeled 108 degrees. Angle F E B is labeled question mark. Angle B E D is labeled question mark.
Find the missing angle measures around point
What do you notice about the angles in this diagram?
Point
Can you find a rigid transformation that shows angle
The goal of this discussion is for students to articulate that for pairs of alternate interior angles to be formed, a transversal must cut two parallel lines. If the lines are not parallel, then we cannot use rigid transformations to show these pairs of angles are congruent.
Display both of these images for all to see:
Here are some questions for discussion:
The goal of this discussion is for students to articulate which angles are congruent to one another and give an example of a rigid transformation that explains why.
Display the image of two parallel lines cut by a transversal. Tell students that in cases like this, translations and rotations can be particularly useful in figuring out angle measurements since they move angles to new positions, but the angle measure does not change.
Invite students to identify pairs of alternate interior, vertical, and supplementary angles in the image.
Here are some questions for discussion:
When two lines intersect, vertical angles are congruent, and adjacent angles are supplementary, so their measures sum to 180. For example, in this figure angles 1 and 3 are congruent, angles 2 and 4 are congruent, angles 1 and 4 are supplementary, and angles 2 and 3 are supplementary.
Two intersecting lines. Angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees.
When two parallel lines are cut by another line, called a transversal, two pairs of alternate interior angles are created. (“Interior” means on the inside, or between, the two parallel lines.) For example, in this figure angles 3 and 5 are alternate interior angles and angles 4 and 6 are also alternate interior angles.
Two lines that do not intersect. A third line intersects with both lines. At the first intersection, angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees. At the second intersection, angle 5 is marked 70 degrees. Angle 6 is marked 110 degrees. Angle 7 is marked 70 degrees. Angle 8 is marked 110 degrees.
Alternate interior angles are equal because a
Using what we know about vertical angles, adjacent angles, and alternate interior angles, we can find the measures of any of the eight angles created by a transversal if we know just one of them. For example, starting with the fact that angle 1 is
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Lines
Line A C contains point B. Line D F contains point E. Line J H contains points B and E. Angle A B J is labeled question mark. Angle A B E is labeled 63 degrees. Angle E B C is labeled question mark. Angle C B J is labeled question mark. Angle D E H is labeled question mark. Angle H E F is labeled question mark. Angle F E B is labeled question mark. Angle B E D is labeled question mark.
With your partner, find the seven unknown angle measures in the diagram. Explain your reasoning.
What do you notice about the angles with vertex
Using what you noticed, find the measures of the four angles at point
Line A C contains point B. Line D F contains point E. Line H G contains points B and E. Angle A B H is labeled with a question mark. Angle A B E is labeled with a question mark. Angle E B C is labeled with a question mark. Angle C B H is labeled with a question mark. Angle G E F is labeled 34 degrees.
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Students may fill in congruent angle measurements based on the argument that they look the same size. Ask students how they can be certain that the angles don't differ in measure by 1 degree. Encourage them to explain how we can know for sure that the angles are exactly the same measure.
Lines
Find a rigid transformation showing that angles
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