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Unit 1, Lesson 20

Transformations, Transversals, and Proof

Geometry

20.1

Warm-up

5 mins

Angle Relationships

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

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

None

Activity

None

Student Task Statement

Lines and are parallel. Find the value of in each figure.

<p>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.</p> — Figure A

<p>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.</p> — Figure B

<p>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.</p> — Figure C

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.

20.2

Activity

15 mins

Make a Mark? Give a Reason

Instructional Routines

Draw It

MLR1: Stronger and Clearer Each Time

20.3

Activity

10 mins

An Alternate Explanation

Instructional Routines

Draw It

Take Turns

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:

<p>Parallel lines cut by transversal.</p>

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)

<p>2 lines cut by transversal line. Alternate interior angles marked with arcs and ticks.</p>

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)

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

to access 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:

<p>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.</p>

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.

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Lesson 20

Transformations, Transversals, and Proof

Warm-up

Angle Relationships

Standards Alignment

8.G.A.5

HSG-CO.A.1

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

Make a Mark? Give a Reason

Instructional Routines

Activity

An Alternate Explanation

Instructional Routines

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Student Response

Building on Student Thinking

Are You Ready for More?

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Materials

Activity Narrative

Launch

Activity Synthesis

Standards Alignment

8.G.A.5

HSG-CO.A.4

HSG-CO.A.2

HSG-CO.C.9

HSG-CO.C.10

Standards Alignment

8.G.A.5

HSG-CO.A.4

HSG-CO.A.2

HSG-CO.C.9

HSG-CO.C.10

Student Response

Building on Student Thinking

Are You Ready for More?

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?