Transformations and Angle Relationships

Lesson 18

Math

Unit 3

8th Grade

Lesson 18 of 22

Objective


Define and identify alternate interior and alternate exterior angles in parallel line diagrams. Find missing angles in parallel line diagrams.

Common Core Standards


Core Standards

  • 8.G.A.2 — Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
  • 8.G.A.5 — Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Foundational Standards

  • 7.G.B.5

Criteria for Success


  1. Argue that alternate interior angles are congruent using a rotation.
  2. Argue that alternate exterior angles are congruent using a translation and reflection.
  3. Given an angle measure in a parallel line diagram, find other angle measurements.

Tips for Teachers


Lesson 18 builds on Lesson 17 by introducing two new angle relationships in parallel line diagrams: alternate interior and alternate exterior angles. Students use their knowledge of transformations to prove why these angles are congruent and to deepen their understanding of the relationships of angles in parallel line diagrams. 

Lesson Materials

  • 180° Protractor (1 per student)
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Anchor Problems


Problem 1

Suppose $$l$$ and $$m$$ are parallel lines, with point $$Q$$ on line $$l$$ and point $$P$$ on line $$m$$, as pictured below. Also labeled in the picture is point $$M$$, the midpoint of $$\overline{PQ}$$, and two angles $$a$$ and $$b$$.

 

How can you use a rotation to prove that angles $$a$$ and $$b$$ are congruent?

Guiding Questions

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References

Illustrative Mathematics Congruence of Alternate Interior Angles via Rotations

Congruence of Alternate Interior Angles via Rotations, accessed on Oct. 13, 2017, 4:15 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Problem 2

In the picture below, $$m$$ and $$n$$ are parallel lines and angles $$a$$ and $$b$$ are shown. Show that angle $$a$$ is congruent to angle $$b$$ using rigid motions.

Guiding Questions

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References

Illustrative Mathematics Rigid Motions and Congruent Angles

Rigid Motions and Congruent Angles, accessed on Oct. 13, 2017, 4:16 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Problem 3

In the diagrams below, lines $$b$$ and $$c$$ are parallel; lines $$f$$ and $$d$$ are not parallel.

a.   Identify all of the angles congruent to $${{\angle 1}}$$ in the first diagram. Explain why each angle is congruent to $${{\angle 1}}$$.

b.   Identify all of the angles congruent to $${{\angle 1}}0$$ in the second diagram. Explain why each angle is congruent to $${{\angle 1}}0$$.

Guiding Questions

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Problem Set

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Target Task


In the figure below, $$L1$$ and $$L2$$ are parallel and $$m$$ is a transversal that intersects them both.

a.   What is the relationship between the measures of ∠4 and ∠6? Explain how you know.

b.   What is the relationship between the measures of ∠1 and ∠7? Explain how you know.

Student Response

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Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Examples where students are given one angle measure in a parallel line diagram and asked to determine all of the other angle measures with a brief explanation of how they determined the measurement
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Lesson 17

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Lesson 19

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Congruence and Rigid Transformations

Topic B: Similarity and Dilations

Topic C: Angle Relationships

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