Constructions, Proof, and Rigid Motion

Lesson 12

Math

Unit 1

10th Grade

Lesson 12 of 19

Objective


Prove angle relationships in parallel line diagrams.

Common Core Standards


Core Standards

  • G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
  • G.CO.C.9 — Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Foundational Standards

  • 8.G.A.5

Criteria for Success


  1. Identify relationships in parallel line diagrams beginning with corresponding angles, vertical angles, and supplementary angles. 
  2. Use notation to mark up a parallel line diagram as an initial step to developing a proof. 
  3. Use established relationships in parallel line diagrams to prove theorems about alternate interior, alternate exterior, same side interior, and same side exterior angles in parallel line diagrams. 
  4. Apply the transitive property in the development of proofs. 
  5. Determine where converse theorems apply, and establish proofs.

Tips for Teachers


  • This lesson formalizes understanding from 8th grade geometry in parallel line diagrams. Students should have a running record of the parallel line diagram angle relationships and be fluid in using these throughout the year.
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Anchor Problems


Problem 1

Find all the angle measures that equal 127° and all the angle measures that are supplementary to 127°.

Guiding Questions

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Problem 2

In the figure, $${\overline{AB} \parallel \overline{CD}}$$ and $${\overline{EF} \parallel \overline{GH}}$$. Prove that $${\angle AFE \cong \angle DKH}$$.

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 1 > Topic C > Lesson 18Problem Set #4

Geometry > Module 1 > Topic C > Lesson 18 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Target Task


$${\overline{AD} \parallel \overline{BC}}$$ and $${\angle EJB}$$ is supplementary to $${\angle JBK}$$. Prove that $${\overline {AD} \parallel \overline{JE}}$$.

References

EngageNY Mathematics Geometry > Module 1 > Topic C > Lesson 18Problem Set #7

Geometry > Module 1 > Topic C > Lesson 18 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

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.

  •  Include a variety of proofs that students need to complete, find missing statements/reasons, find alternative statements/reasons, etc.
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Lesson 11

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Lesson 13

Lesson Map

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

Topic A: Constructions of Basic Geometric Figures

Topic B: Justification and Proof of Angle Measure

Topic C: Translations of Points, Line Segments, and Angles, and Parallel Line Relationships

Topic D: Reflections and Rotations of Points, Line Segments, and Angles

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