Lesson 20

Transformations, Transversals, and Proof

20.1: Math Talk: Angle Relationships (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for angle relationships in parallel lines cut by a transversal. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to prove these relationships using transformational arguments. In this activity, students have an opportunity to notice and make use of structure (MP7) as they identify congruent angle pairs formed by parallel lines cut by a transversal.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Lines \(\ell\) and \(m\) are parallel. Mentally evaluate the measure \(x\) in each figure.

Figure A

Parallel lines l and m cut by transversal line p, creating two congruent angles above l and m and to the right to the right of the transversal. Upper angle is 40 degrees and lower angle is x degrees.

Figure B

Parallel lines l and m cut by transversal line p, creating two alternate interior angles marked congruent. Upper left angle is 61 degrees and lower right angle is x degrees.

Figure C

Parallel lines l and m cut by transversal line p, creating two same-side inside angles marked congruent. Upper angle is 98 degrees and lower angle is x degrees.

Figure D

Parallel lines l and m cut by transversal line p, creating two congruent angles above l and m and to the right to the right of the transversal. Upper angle is 40 degrees and lower angle is x degrees.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. Encourage students to use precise language to express their ideas. If students do not recall phrases such as alternate interior angles, that language can wait until the synthesis of the subsequent activity.

To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ."  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

20.2: Make a Mark? Give a Reason. (15 minutes)

Activity

In this activity, students translate one line in a pair of intersecting lines to create parallel lines cut by a transversal. Using the definition and properties of translations, students conclude that pairs of corresponding angles are congruent. Students work with a partner and trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Arrange students in groups of 2. Ask students to take turns marking angles as congruent. The first partner identifies a pair of congruent angles and explains why they think the angles are congruent while the other listens and works to understand. Then they switch roles.

Consider providing sentence starters like: Angle _____ is congruent to angle _____ because _____.

Instruct students to create a diagram with two intersecting lines and labeled points using this image as a guide.

Here are intersecting lines \(AE\) and \(CD\):

Intersecting lines AE and CD.

Student Facing

Here are intersecting lines \(AE\) and \(CD\):

Intersecting lines AE and CD.
  1. Translate lines \(AE\) and \(CD\) by the directed line segment from \(B\) to \(C\). Label the images of \(A, B, C, D, E\) as \(A’, B’, C’, D’, E’\).
  2. What is true about lines \(AE\) and \(A’E’\)? Explain your reasoning.
  3. Take turns with your partner to identify congruent angles.
    1. For each pair of congruent angles that you find, explain to your partner how you know the angles are congruent.
    2. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Arrange students in groups of 2. Ask students to take turns marking angles as congruent: the first partner identifies a pair of congruent angles and explains why they think the angles are congruent, while the other listens and works to understand. Then they switch roles.

Consider providing sentence starters like: Angle _____ is congruent to angle _____ because _____.

Student Facing

Here are intersecting lines \(AE\) and \(CD\):

Intersecting lines AE and CD.
  1. Translate lines \(AE\) and \(CD\) by the directed line segment from \(B\) to \(C\). Label the images of \(A, B, C, D, E\) as \(A’, B’, C’, D’, E’\).
  2. What is true about lines \(AE\) and \(A’E’\)? Explain your reasoning.
  3. Take turns with your partner to identify congruent angles.
    1. For each pair of congruent angles that you find, explain to your partner how you know the angles are congruent.
    2. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may have difficulty drawing a reasonably accurate image of the figure. Remind them of the tools in their geometry toolkits, such as tracing paper and a straightedge.

Activity Synthesis

The purpose of discussion is to refine student explanations with more formal language.

Ask students to share their responses. As students share, record what they say by writing a congruence statement (\(\angle ABC \cong \angle EBD\)) and marking the figure. Insist that whenever the figure is marked with a congruence, students need to write a congruence statement and give a reason that references a definition or properties of translations. If students get stuck when justifying congruence statements, ask them to look for properties of translations in their reference charts for help.

If not mentioned by students, introduce the vocabulary of alternate interior angles and corresponding angles.

Writing, Speaking: MLR 1 Stronger and Clearer Each Time. Use this with successive pair shares to give students a structured opportunity to revise and refine their response to “For each pair of congruent angles that you find, explain to your partner how you know the angles are congruent.” Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help teams strengthen their ideas and clarify their language (for example, "Can you explain how…?" "You should expand on...,"). Students can borrow ideas and language from each partner to strengthen the final product. Design Principle(s): Optimize output (for generalization)
Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, write the congruence statement \(\angle ABC \cong \angle EBD\) and mark the angles in the figure, \(\angle ABC\) and \(\angle EBD\), the same color.
Supports accessibility for: Visual-spatial processing

20.3: An Alternate Explanation (10 minutes)

Activity

In this activity, students rotate one line in a pair of intersecting lines by 180 degrees to create parallel lines cut by a transversal. Using the definition and properties of rotations, students conclude that pairs of corresponding angles are congruent. Students work with a partner and trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Tell students they will be looking at a similar set-up as in the previous activity, but they will be doing a 180 degree rotation instead. Emphasize that one important property of 180 degree rotations is that they take lines either to themselves if the center of rotation is on the line or to parallel lines if the center of rotation is off the line. Students can verify this experimentally by using tracing paper to rotate line \(CD\) by 180 degrees around various points on line \(AE\), including \(B\), then translating along line \(AE\) until the line returns to where it began.

Ask students to add this as an assertion in their reference charts as you add it to the class reference chart:

Rotation by 180 degrees takes lines to parallel lines or to themselves. (Assertion)

Rigid motion.

Instruct students to create intersecting lines with some labeled points using this image as a guide.

Here are intersecting lines \(AE\) and \(CD\):

Intersecting lines AE and CD.

Student Facing

  1. Rotate line \(AE\) by 180 degrees around point \(C\). Label the images of \(A, B, C, D, E\) as \(A’, B’, C’, D’, E’\).
  2. What is true about lines \(AB\) and \(A’B’\)? Explain your reasoning.
  3. Take turns with your partner to identify congruent angles.
    1. For each pair of congruent angles that you find, explain to your partner how you know the angles are congruent.
    2. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Tell students they will be looking at a similar set-up as in the previous activity, but they will be doing a 180 degree rotation instead. Emphasize that one important property of 180 degree rotations is that they take lines either to themselves if the center of rotation is on the line or to parallel lines if the center of rotation is off the line. Students can verify this experimentally by using tracing paper to rotate line \(CD\) by 180 degrees around various points on line \(AE\), including \(B\), then translating along line \(AE\) until the line returns to where it began.

Ask students to add this as an assertion in their reference charts as you add it to the class reference chart:

Rotation by 180 degrees takes lines to parallel lines or to themselves. (Assertion)

Rigid motion.

Student Facing

Here are intersecting lines \(AE\) and \(CD\):

Intersecting lines AE and CD.
  1. Rotate line \(AE\) by 180 degrees around point \(C\). Label the images of \(A, B, C, D, E\) as \(A’, B’, C’, D’, E’\).
  2. What is true about lines \(AB\) and \(A’B’\)? Explain your reasoning.
  3. Take turns with your partner to identify congruent angles.
    1. For each pair of congruent angles that you find, explain to your partner how you know the angles are congruent.
    2. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. Prove that 180 degree rotations take lines that do not pass through the center of rotation to parallel lines.
  2. What is the image of a line that is rotated 180 degrees around a point on the line?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If student struggle to visually estimate the result of the 180 degree rotation invite them to trace line \(BC\) onto tracing paper and ask how they will know when they have rotated 180 degrees. Then they can trace the entire diagram and repeat the process.

Activity Synthesis

The purpose of discussion is to refine student explanations that alternate interior angles are congruent with more formal language.

Ask for students to share their responses. As students share, record what they say using congruence symbols and marking the figure. Insist that whenever the figure is marked with a congruence, students need to write a congruence statement and give a reason that references the definition and properties of translations.

Ask students how this activity is different from the previous activity. (The previous activity used translation but this one used rotation.)

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, write the congruence statement \(\angle ABC \cong \angle A’B’C\) and mark the angles in the figure, \(\angle ABC\) and \(\angle A’B’C\), the same color.
Supports accessibility for: Visual-spatial processing

Lesson Synthesis

Lesson Synthesis

During the lesson, students used transformations to create parallel lines and then observed which angles are congruent. But what if they started with parallel lines? Display an image of two parallel lines with corresponding angles emphasized:

Parallel lines cut by transversal.

Tell students that lines \(AI\) and \(GJ\) are parallel, and line \(FE\) is a transversal that intersects them. Here are some questions for discussion:

  • “What transformation would take angle \(EBI \) to angle \(BCJ\)?” (Translation.)
  • “How do we know that a translation along the directed line segment from \(B\) to \(C\) takes line \(AI\) to line \(GJ\)?” (Translations take lines to parallel lines and \(B\) is taken to \(C\), so line \(AI\) is taken to a parallel line that goes through \(C\). There is only one such line, which is line \(GJ\).)

Students will complete the rest of the proof in their cool-down. They will also have the opportunity to prove the converses in practice problems. Converse statements are studied in a subsequent lesson. For now, point out that the two statements are related but have different given information.

Distribute new copies of the blackline master Blank Reference Chart. Inform students they will continue to need the first page along with this one so they should keep the pages together.

Ask students to add these theorems to their reference charts as you add them to the class reference chart:

Alternate Interior Angle Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent. 

Conversely, if two lines are cut by a transversal and alternate interior angles are congruent, then the lines have to be parallel. 

(Theorem)

2 lines cut by transversal line. Alternate interior angles marked with arcs and ticks.

Corresponding Angle Theorem: If two parallel lines are cut by a transversal, then corresponding angles are congruent. 

Conversely, if two lines are cut by a transversal and corresponding angles are congruent, then the lines have to be parallel. 

(Theorem)

2 lines cut by a transversal line. The upper right angles are marked congruent with arcs and tick marks.

20.4: Cool-down - Transformations on Parallel Lines (10 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

There are often several different ways to explain why statements are true. Comparing the different ways can lead to new insights or more flexible understanding. Consider the angles formed when 2 parallel lines \(\ell\) and \(m\) are cut by a transversal:

Parallel lines l and m cut by transversal EF at points B and C and having midpoint M. Point A is to the left of the transversal on l and point G is to the right of the transversal on m.

Suppose we want to explain why angle \(ABE\) is congruent to angle \(GCF\). Label the midpoint of \(BC\) as \(M\). Rotating 180 degrees around \(M\) takes angle \(ABE\) to angle \(GCF\). Why? Well, \(B\) and \(C\) are equidistant from \(M\), so the rotation takes \(B\) to \(C\). Also, it takes the transversal to itself, so it takes the ray \(BE\) to the ray \(CF\). Finally, the rotation takes line \(\ell\) onto line \(m\) because 180 degree rotations take lines onto parallel lines and \(m\) is the only line parallel to \(\ell\) that also goes through \(C\).

A different explanation can prove the same fact using a translation and the idea that vertical angles are congruent. Try thinking of that explanation yourself.