Curriculum  /  Math  /  10th Grade  /  Unit 1: Constructions, Proof, and Rigid Motion  /  Lesson 13

Constructions, Proof, and Rigid Motion

Lesson 13

Math

Unit 1

10th Grade

Lesson 13 of 19

Objective

Construct auxiliary parallel lines and use these in the development of proofs and identification of missing measures.

Common Core Standards

Core Standards

  • 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.
  • G.CO.D.12 — Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Foundational Standards

  • 8.G.A.5

Criteria for Success

  1. Identify when there is not sufficient angle relationships to establish a proof. 
  2. Construct parallel auxiliary lines and assess how the relationships that result can help to develop a proof. 
  3. Use established angle relationships and auxiliary lines to develop proofs. 

Tips for Teachers

  • This lesson is an extension of G-CO.9 and G-CO.12, but provides opportunities for students to become more fluid in finding relationships within a diagram, and adding necessary information to see additional relationships. 
  • Students need to practice their skills of annotating the diagram, notice/wonder about relationships, and organizing statements and reasons to develop an argument. 
  • A good extension to this lesson is Dan Meyer’s Three Act Task “Pool Bounce.” Description of the activity can be found here with the actual lesson and associated documents found here.
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Anchor Problems

25-30 minutes

Problem 1

The lines shown are parallel. Find the measure of angle 1.

Guiding Questions

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Problem 2

Given the two pairs of parallel lines shown below, find the measure of all the variables.

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 1 > Topic B > Lesson 7 — Exit Ticket

Geometry > Module 1 > Topic B > Lesson 7 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

5-10 minutes

Problem 1

In the figure below, $${\overline {AB} \parallel \overline{CD}}$$.

Prove that a°=b°.

References

EngageNY Mathematics Geometry > Module 1 > Topic B > Lesson 10 — Exit Ticket

Geometry > Module 1 > Topic B > Lesson 10 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..

Problem 2

Prove $${m\angle p = m\angle r}$$.

References

EngageNY Mathematics Geometry > Module 1 > Topic B > Lesson 10 — Exit Ticket

Geometry > Module 1 > Topic B > Lesson 10 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 problem such as: 
    • Give students two lines with a transversal and several degree measures and ask, “Are these lines parallel? How do you know? Justify your answer. Include both lines that are parallel and those that are not.” 
    • Error analysis problems where students are expected to identify incorrect reasons/statements, jumps in understanding of the development of the proof. 
    • Review drawing a figure by giving a written explanation of a parallel line diagram and having students draw the appropriate figure.

Next

Perform reflections on a coordinate plane across axes and other defined lines. Identify characteristics and an algebraic rule for the reflection.

Lesson 14
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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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