Unit 1, Lesson 9
Transformations, Transversals, and Proof
Integrated Math 2
9.1
Warm-up
Instructional Routines
NoneMaterials
Activity Narrative
The purpose of this Warm-up is to remind students of angle relationships in parallel lines cut by a transversal. 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
NoneActivity
NoneLines and are parallel. Find the value of in each figure.
Figure A
Figure B
Figure C
Figure D
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 precise phrases, such as “alternate interior angles,” that language can wait until the Activity Synthesis of the subsequent activity.
Math Community
After the Warm-up, display the revised Math Community Chart created from student responses in Exercise 3. Tell students that today they are going to monitor for two things:
- “Doing Math” actions from the chart that they see or hear happening.
- “Doing Math” actions that they see or hear that they think should be added to the chart.
Provide sticky notes for students to record what they see and hear during the lesson.
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 this image:
Tell students that lines and are parallel, and line is a transversal that intersects them. Ask students to name a pair of corresponding angles. (Possible pairs: and , and , and , and .)
Students will have the opportunity to prove the following theorems in the Cool-down and the converses in the practice problems and section checkpoint. 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.
Add the following theorems to the class reference chart, and ask students to add it to their reference charts:
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)
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)
Math Community
Invite 2–3 students to share what “Doing Math” actions they noticed. Record and display their responses for all to see, such as by adding check marks to any already listed items or adding new items near the chart for the class to consider adding. Next, give students 1–2 minutes with a partner to discuss any changes or revisions they think the chart needs. Tell students they can suggest revisions during the Cool-down.
Student Lesson Summary
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 and are cut by a transversal:
Suppose we want to explain why angle is congruent to angle . Label the midpoint of as . Rotating 180 degrees around takes angle to angle . Why? Well, and are equidistant from , so the rotation takes to . Also, it takes the transversal to itself, so it takes the ray to the ray . Finally, the rotation takes line onto line because 180-degree rotations take lines onto parallel lines, and is the only line parallel to that also goes through .
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.