Illustrative Math

GEOMETRY

Lesson 20
Transformations, Transversals, and Proof


G.LA.1, G.TRF.1, G.TRF.2M, G.GF.8, G.SC.6, G.LA.5, G.LA.8

Unit 1
Constructions and Rigid Transformations

Learning

Goal

Geometry

Let’s prove statements about parallel lines.

Unit 1 ● Lesson 20

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics.

Warm-up: Math Talk

Lines l and m are parallel. Evaluate the measure x in each figure.

Angle Relationships

Unit 1 ● Lesson 20 ● Activity 1

Fill in the Blanks

Multiple Choice

How did you know the value of x?

• The missing angle measure is the same as the given angle measure because the angle is an image after a translation of the given angle

• The missing angle measure is the same as the given angle measure because the angle is an image after a rotation of the given angle

• The missing angle measure is the same as the given angle measure because the angle is an image after a reflection of the given angle

Fill in the Blanks

Multiple Choice

How did you know the value of x?

• The missing angle measure is the same as the given angle measure because the angle is an image after a translation of the given angle

• The missing angle measure is the same as the given angle measure because the angle is an image after a reflection of the given angle

• The missing angle measure is the same as the given angle measure because the angle is an image after a rotation of the given angle

Fill in the Blanks

Multiple Choice

How did you know the value of x?

• The missing angle measure is the same as the measure of the angle that is supplementary to the given angle measure

• The missing angle measure is the same as the measure of the angle that is complementary to the given angle measure

• The missing angle measure is the same as the measure of the angle that is congruent to the given angle measure

• The missing angle measure is the same as the measure of the angle that is adjacent to the given angle measure

Fill in the Blanks

Multiple Choice

How did you know the value of x?

• The missing angle measure is the same as the given angle measure because the angle is an image after a translation of the given angle

• The missing angle measure is the same as the given angle measure because the angle is an image after a rotation of the given angle

• The missing angle measure is the same as the given angle measure because the angle is an image after a reflection of the given angle

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics.

Open the Lesson using GeoGebra.

Here are intersecting lines AE and CD:
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’.

Make a Mark? Give a Reason

Unit 1 ● Lesson 20 ● Activity 2

Multiple Choice

What is true about lines AE and A'E'?

• They are parallel.

• They are congruent.

• They are perpendicular.

• They are intersecting.

Multiple Choice

Why do we know that lines AE and A'E' are parallel?

• Lines AE and A'E' are parallel because translations take lines to parallel lines.

• Lines AE and A'E' are parallel because reflections take lines to parallel lines.

Multiple Select

Which pairs of angles are congruent?

• vertical angles ABC and EBD

• vertical angles ABD and EBC

• adjacent angles ABC and ABD

• adjacent angles EBC and EBD

Multiple Select

Which pairs of angles are congruent?

• translation takes EBD to E'CB, so they are congruent

• translation takes EBC to E'CC', so they are congruent

• translation takes ABD to A'CB, so they are congruent

• translation takes ABC to A'CC', so they are congruent

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. 

Corresponding Angle Theorem

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics.

Rotate line AE by 180 degrees around point C. Label the images of A, B, C, D, E as A’, B’, C’, D’, E’.

An Alternate Explanation

Unit 1 ● Lesson 20 ● Activity 3

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics.

Rotate line AE by 180 degrees around point C. Label the images of A, B, C, D, E as A’, B’, C’, D’, E’.

An Alternate Explanation

Unit 1 ● Lesson 20 ● Activity 3

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics.

Rotate line AE by 180 degrees around point C. Label the images of A, B, C, D, E as A’, B’, C’, D’, E’.

An Alternate Explanation

Unit 1 ● Lesson 20 ● Activity 3

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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. 

Alternate Interior Angle Theorem

Multiple Choice

What transformation would take angle EBI to angle BCJ?

• Translation

• Rotation

• Reflection

Multiple Select

How do we know that a translation along the directed line segment from B to C takes line AI to line GJ?

• Translation takes lines to parallel lines

• Rotation of 180 degrees takes lines to parallel lines

• Reflection takes lines to perpendicular lines

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics.

In this question, lines AI and GJ are parallel and intersected by the transversal line FE.

Cool Down: Transformations on Parallel Lines

Unit 1 ● Lesson 20 ● Activity 4

Open Ended

Angles EBI and BCJ are corresponding angles. Use a transformation that takes angle EBI to angle BCJ to prove that corresponding angles are congruent.

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics.

In this question, lines AI and GJ are parallel and intersected by the transversal line FE.

Cool Down: Transformations on Parallel Lines

Unit 1 ● Lesson 20 ● Activity 4

Open Ended

Angles ABC and BCJ are alternate interior angles. Use a transformation that takes angle ABC to angle BCJ to prove that alternate interior angles are congruent. Label any
other points on the figure that will help to define a transformation.

Learning
Targets

Geometry

Unit 1 ● Lesson 20

● I can prove alternate interior angles are congruent

● I can prove corresponding angles are congruent

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