Lesson 12

Alternate Interior Angles

12.1: Angle Pairs (10 minutes)

Warm-up

In this activity, students practice determining the measurement of unknown angles using their relationship to angles with known angle measure and what students already know about the measurement of right angles and straight angles. After they have worked on the activity and shared their solutions, they are introduced to the vocabulary terms complementary and supplementary. Complementary describes angles whose measures sum to 90 degrees and supplementary describes angles whose measures sum to 180 degrees.

Some students may wish to use protractors, either to double check work or to investigate the different angle measures. This is an appropriate use of technology (MP5), but ask these students what other methods they could use instead.

Launch

Provide access to geometry toolkits.

Student Facing

  1. Find the measure of angle \(JGH\). Explain or show your reasoning.
    Lines F H and J I intersect at point G. Angle J G F is labeled 30 degrees.
  2. Find and label a second \(30^\circ\) angle in the diagram. Find and label an angle congruent to angle \(JGH\).
  3. Angle \(BAC\) is a right angle. Find the measure of angle \(CAD\).
    Angle C, A, B is shown. segment D A, lies in the interior of angle C, A, B. Angle D, A, B, is labeled 64 degrees.

    ​​​​​

Student Response

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Anticipated Misconceptions

If needed, remind students that right angles have a measure of \(90^\circ\) and straight angles have a measure of \(180^\circ\).

Activity Synthesis

The goal of this discussion is to introduce students to the terms complementary and supplementary for describing relationships between pairs of angles. Display the images for all to see and select students to briefly share their response to each question, recording any calculations used.

Explain to students that the term complementary describes a pair of angles whose measures sum to 90 degrees, and the term supplementary describes a pair of angles whose measures sum to 180 degrees. This is true even for pairs of angles that are not adjacent. Conclude the discussion by inviting students to identify pairs of angles that are complementary (\(CAD\) and \(DAB\)), supplementary (\(FGJ\) and \(JGH\), \(JGH\) and \(HGI\), \(HGI\) and \(IGF\), or \(IGF\) and \(FGJ\)), or vertical (\(FGJ\) and \(HGI\) or \(JGH\) and \(IGF\)) from the activity.

12.2: Cutting Parallel Lines with a Transversal (15 minutes)

Activity

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 this situation. They also consider whether the relationships they found hold true for any two lines cut by a transversal. 

The last two questions in this activity are optional, to be completed if time allows. Make sure to leave enough time for the next activity, “Alternate Interior Angles are Congruent.”

As students work with their partners, they begin to fill in supplementary angles and vertical angles. To find the measures of corresponding and alternate interior, students may use tracing paper and some of the strategies found earlier in the unit. For example, they may use tracing paper to translate vertex \(B\) to vertex \(E\). They might try to translate \(B\) to \(E\) in the third picture and observe that the angles at those two vertices are not congruent. Similarly, to find measures of vertical angles students may use a \(180^\circ\) rotation like they did earlier in this unit when showing that vertical angles are congruent. Monitor for students who use these different strategies and select them to share during the discussion.

For students who finish early, ask them to think of different methods they could use to determine the angles: For example, all of the congruent angles can be shown to be congruent with transformations.

Launch

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 \(k\) is a transversal for parallel lines \(\ell\) and \(m\):

Parallel lines l and m with transversal k.

Arrange students in groups of 2. Provide access to geometry toolkits. Give students 1 minute of quiet think time to plan on how to find the angle measures in the picture then time to share their plan with their partner. Give partners time for the rest of the task, followed by a whole-class discussion. Instruct students to stop after the third question if you've decided to skip the last two questions.

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of these terms. Include the following term and maintain the display for reference throughout the unit: transversal.
Supports accessibility for: Conceptual processing; Language
Conversing, Representing: MLR2 Collect and Display. As students discuss with a partner, listen for and collect vocabulary, gestures, and diagrams students use to describe the relationships they notice between the angles formed when two parallel lines are cut by a transversal. Capture student language that reflects a variety of ways to determine determine angle congruence. Record students’ words, phrases, and diagrams onto a visual display and update it throughout the lesson. Remind students to borrow language from the display as needed. This will help students use mathematical language during small-group and whole-class discussions.
Design Principle(s): Support sense-making

Student Facing

Lines \(AC\) and \(DF\) are parallel. They are cut by transversal \(HJ\).

3 lines. 
  1. With your partner, find the seven unknown angle measures in the diagram. Explain your reasoning.

  2. What do you notice about the angles with vertex \(B\) and the angles with vertex \(E\)?
  3. Using what you noticed, find the measures of the four angles at point \(B\) in the second diagram. Lines \(AC\) and \(DF\) are parallel.

    3 lines in a plane. 
  4. The next diagram resembles the first one, but the lines form slightly different angles. Work with your partner to find the six unknown angles with vertices at points \(B\) and \(E\).

    Three lines in a plane. 
  5. What do you notice about the angles in this diagram as compared to the earlier diagram? How are the two diagrams different? How are they the same?

Student Response

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Student Facing

Are you ready for more?

Four lines on a grid.

Parallel lines \(\ell\) and \(m\) are cut by two transversals which intersect \(\ell\) in the same point. Two angles are marked in the figure. Find the measure \(x\) of the third angle.

Student Response

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Anticipated Misconceptions

In the first image, 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 a way to explain how we can know for sure that the angles are exactly the same measure.

Activity Synthesis

The purpose of this discussion is to make sure students noticed relationships between the angles formed when two parallel lines are cut by a transversal and to introduce the term alternate interior angles to students. Display the images from the Task Statement for all to see one at a time and invite groups to share their responses. Encourage students to use precise vocabulary, such as supplementary and vertical angles, when describing how they figured out the different angle measurements. After students point out the matching angles at the two vertices, define the term alternate interior angles and ask a few students to identify some pairs of angles from the activity.

Consider asking some of the following questions:

  • "What were some tools you used to find or confirm angle measures?" (Tracing paper, protractor, transformations)
  • "What were some angle relationships you used to find missing measures?" (Vertical angles, supplementary angles)
  • "What do you notice about the angles at vertex \(B\) compared to the angles at vertex \(E\)?" (They have the same angle measures for angles in the same position relative to the transversal.)
  • "Which angle relationships were true for all three pictures? Which were true for only one or two of the pictures?" (Congruent vertical and supplementary angles around a vertex were always true. Congruent angles in corresponding positions at the two vertices were only true in the first two pictures, which had parallel lines.)

12.3: Alternate Interior Angles Are Congruent (15 minutes)

Activity

The goal of this task is to experiment with rigid motions to help visualize why alternate interior angles (made by a transversal connecting two parallel lines) are congruent. This result will be used in a future lesson to establish that the sum of the angles in a triangle is 180 degrees. The second question is optional if time allows. This provides a deeper understanding of the relationship between the angles made by a pair of (not necessarily parallel) lines cut by a transversal.

Expect informal arguments as students are only beginning to develop a formal understanding of parallel lines and rigid motions. They have, however, studied the idea of 180 degree rotations in a previous lesson where they used this technique to show that a pair of vertical angles made by intersecting lines are congruent. Consider recalling this technique especially to students who get stuck and suggesting the use of tracing paper.

Given the diagram, the tracing paper, and what they have learned in this unit, students should be looking for ways to demonstrate that alternate interior angles are congruent using transformations. Make note of the different strategies (including different transformations) students use to show that the angles are congruent and invite them to share their strategies during the discussion. Approaches might include:

  • A 180 degree rotation about \(M\)
  • First translating \(P\) to \(Q\) and then applying a 180-degree rotation with center \(Q\)
Teacher Notes for IM 6–8 Math Accelerated.
Adjust this activity to 10 minutes.

Launch

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.

Action and Expression: Develop Expression and Communication. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “First, I _____ because...”, “I noticed _____ so I...”, “Why did you...?”, “I agree/disagree because….”
Supports accessibility for: Language; Organization

Student Facing

  1. Lines \(\ell\) and \(k\) are parallel and \(t\) is a transversal. Point \(M\) is the midpoint of segment \(PQ\).
    Line l contains points B and Q. Line k contains points P and A. Line t contains points P, M, and Q.

    Find a rigid transformation showing that angles \(MPA\) and \(MQB\) are congruent.
  2. In this picture, lines \(\ell\) and \(k\) are no longer parallel. \(M\) is still the midpoint of segment \(PQ\).
    Line l contains points B and Q. Line k contains points P and A. Line t contains points P and Q.

    Does your argument in the earlier problem apply in this situation? Explain.

Student Response

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Activity Synthesis

Select students to share their explanations. Pay close attention to which transformations students use in the first question and make sure to highlight different possibilities if they arise. Ask students to describe and demonstrate the transformations they used to show that alternate interior angles are congruent.

Highlight the fact that students are using many of the transformations from earlier sections of this unit. The argument here is especially close to the one used to show that vertical angles made by intersecting lines are congruent. In both cases a 180 degree rotation exchanges pairs of angles. For vertical angles, the center of rotation is the common point of intersecting lines. For alternate interior angles, the center of rotation is the midpoint of the transverse between the two parallel lines. But the structure of these arguments is identical. 

Speaking: MLR8 Discussion Supports. Use this routine to amplify students’ mathematical uses of language when describing and demonstrating transformations used for showing alternate interior angles are congruent. After a student shares their response, invite another student to repeat the reasoning using the following mathematical language: vertical angles, 180 degree rotation, center of rotation, intersecting lines, midpoint, parallel lines. Invite all students to chorally repeat the phrases that include these words in context.
Design Principle(s): Support sense-making, Optimize output (for explanation)

12.4: Info Gap: Angle Finding (20 minutes)

Activity

The purpose of this info gap activity is for students to see how they can use different pieces of information to solve for an unknown angle measure in a multi-step problem. During the whole-class discussion, students are introduced to writing and solving equations to represent the relationships between angles.

The info gap structure requires students to make sense of problems by determining what information is necessary. This may take several rounds of discussion (MP1). Since there are many different pieces of information that could be used to solve the problem but are not given on the data card, consider using a variation on the typical info gap structure: Instead of the student with the problem card asking for specific pieces of information, the student with the data card chooses a piece of information to share, and the student with the problem card explains how they use that piece of information. If enough information hasn’t been given, the student with the data card chooses another piece of information to share.

You will need the blackline master for this activity. Here is the text of the cards for reference and planning:

Info Gap cards.
Info Gap cards.

As students work, monitor for those who use different strategies or start with different pieces of information. Also, monitor for students who choose to show their reasoning by writing and solving equations.

Teacher Notes for IM 6–8 Math Accelerated.
This Info Gap activity is optional. If time allows, use this activity to give students additional practice calculating angle measurements using their understanding of complementary, supplementary, and vertical angles.

Launch

If desired, explain this variation from the typical info gap: instead of the student with the problem card asking for each piece of information, the student with the data card chooses a piece of information to share. The student with the problem card still needs to explain how they can use each piece of information. If more information is needed to solve the problem, the student with the data card chooses another piece of information to share. Also, students need to listen to their partner carefully because they may be asked to explain their partner’s reasoning to the class.

Arrange students in groups of 2. Distribute a problem card to one student and a data card to the other student in each group.

Action and Expression: Internalize Executive Functions. Begin with a small-group or whole-class demonstration and think aloud of a sample situation problem card and data card to remind students how to use the info gap structure. Keep the worked-out equations and angle drawing on display for students to reference as they work.
Supports accessibility for: Memory; Conceptual processing
Conversing: Use this modified version of MLR4 Information Gap to give students an opportunity to discuss the information necessary to solve for an unknown angle measure. Display the follow sentence frames to support student discussion: “Can you tell me . . . (specific piece of information)”, “Why do you need to know . . . (that piece of information)?", and “I can use this information to . . . .”
Design Principle(s): Cultivate conversation

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

  1. Silently read your card and think about what information you need to be able to answer the question.

  2. Ask your partner for the specific information that you need.

  3. Explain how you are using the information to solve the problem.

    Continue to ask questions until you have enough information to solve the problem.

  4. Share the problem card and solve the problem independently.

  5. Read the data card and discuss your reasoning.

If your teacher gives you the data card:

  1. Silently read your card.

  2. Ask your partner “What specific information do you need?” and wait for them to ask for information.

    If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

  3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

  4. Read the problem card and solve the problem independently.

  5. Share the data card and discuss your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

Student Response

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Anticipated Misconceptions

For the second set of cards, students may struggle to find the connection between the lower half of the figure and the upper half. Remind them that supplementary angles do not need to be next to one another, but they can be.

Activity Synthesis

The goal of this discussion is for students to see that writing and solving equations is an efficient strategy to show their reasoning about multi-step angle problems.

For each problem, select a student who had the information card to share their partner’s reasoning. Record their reasoning using equations and display for all to see. Listen carefully and make sure the equation matches how they explain their reasoning. Here are some sample equations for the first problem.

solving for \(e\) given \(d\)

\(\begin{align} e + d &= 180 \\ e + 124 &= 180 \\ e &= 180 - 124 \\ e &= 56 \end{align}\)

solving for \(a\) given \(c\)

\(\begin{align} a + c &= 90 \\ a + 56 &= 90 \\ a &= 90 - 56 \\ a &= 34 \end{align}\)

solving for \(e\) given \(c\)

\(\displaystyle c = 56\)

\(\displaystyle e = c \\ e = 56\)

solving for \(b\) given that \(a\) and \(c\) are complementary

\(\displaystyle a + c = 90\)

\(\begin{align} a + b + c &= 180 \\ b + (a + c) &= 180 \\ b + 90 &= 180 \\ b &= 180 - 90 \\ b &= 90 \end{align}\)

Display the equations that represent different students’ strategies side by side and have students contrast the different methods, for example, the difference between how two students worked the problem if one was given the measure of angle \(d\) first but the other was given the measure of angle \(c\) first.

Lesson Synthesis

Lesson Synthesis

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.

Two parallel lines cut by a transversal.

Select students to point out examples of alternate interior, vertical, and supplementary angles in the picture. They should also be able to articulate which angles are congruent to one another and give an example of a rigid transformation that explains why.

In particular, make sure students can articulate:

  • \(c = 60\) because it is the measure of an angle forming an alternate interior angle with the given 60 degree angle.
  • \(e = d = 120\) because they are also alternate interior angles, each supplementary to a 60 degree angle.
  • The rest of the angle measures can be found using vertical or supplementary angles.

12.5: Cool-down - All The Rest (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

If two angle measures add up to \(90^\circ\), then we say the angles are complementary. If two angle measures add up to \(180^\circ\), then we say the angles are supplementary. When two lines intersect, vertical angles are equal and adjacent angles are supplementary. For example, in this figure angles 1 and 3 are equal, angles 2 and 4 are equal, angles 1 and 4 are supplementary, and angles 2 and 3 are supplementary.

Two intersecting lines.

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.

Alternate interior angles are equal because a \(180^\circ\) rotation around the midpoint of the segment that joins their vertices takes each angle to the other. Imagine a point \(M\) halfway between the two intersections—can you see how rotating \(180^\circ\) about \(M\) takes angle 3 to angle 5?

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 \(70^\circ\) we use vertical angles to see that angle 3 is \(70^\circ\), then we use alternate interior angles to see that angle 5 is \(70^\circ\), then we use the fact that angle 5 is supplementary to angle 8 to see that angle 8 is \(110^\circ\) since \(180 -70 = 110\). It turns out that there are only two different measures. In this example, angles 1, 3, 5, and 7 measure \(70^\circ\), and angles 2, 4, 6, and 8 measure \(110^\circ\).