Lesson 13 | Constructions, Proof, and Rigid Motion | 10th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Congruence

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.

Congruence

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

The foundational standards covered in this lesson

8.G.A.5

Geometry

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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

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

Tips for Teachers

Suggestions for teachers to help them teach this lesson

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

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

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

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

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

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.

Illustrative Mathematics Find the Missing Angle
Illustrative Mathematics Street Intersections
EngageNY Mathematics Geometry > Module 1 > Topic B > Lesson 7 — Exercises 2-10 (Ensure that students are using each term correctly and, in essence, creating a two-column proof.)
New Visions for Public Schools Unit 2 Big Idea 1 — Angle Diagrams and Angle Theorems Resource
EngageNY Mathematics Geometry > Module 1 > Topic B > Lesson 10 — Problem Set
Divisible By 3 Transversals, Tape, and Stickies — The link has handouts. Add an additional activity by naming the angles based on the description or “reason.”
Beauty, Rigor, Surprise Elementary Euclidean Geometry — Chapter 4 Parallels, 4.4 WS The Playfair Postulate and its Consequences Pt. 2

Lesson 12

Lesson 14

Lesson Map

Topic A: Constructions of Basic Geometric Figures

Describe the precise definition and notation for foundational geometric figures.

G.CO.A.1

Construct an equilateral triangle with only a straight-edge and a compass. Copy a line segment.

G.CO.A.1 G.CO.D.12 G.CO.D.13

Construct, bisect, and copy an angle.

G.CO.A.1 G.CO.D.12

Construct a perpendicular bisector.

G.CO.A.1 G.CO.D.12

Construct the altitudes and perpendicular bisectors of sides of triangles.

G.CO.C.10 G.CO.D.12

Topic B: Justification and Proof of Angle Measure

Use principles of proof to justify each step in solving an equation.

A.REI.A.1

Use angle relationships around a point to find missing measures. Prove angle relationships around a point using geometric statements and reasons.

G.CO.C.9

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

Describe and identify rigid motions.

G.CO.A.1 G.CO.A.2 G.CO.B.6

Describe rigid motions. Use algebraic rules to translate points and line segments and describe translations on the coordinate plane.

G.CO.A.2 G.CO.B.6

Translate points and line segments not on the coordinate plane using constructions. Describe properties of translations with respect to line segments and angles.

G.CO.A.2 G.CO.A.4 G.CO.A.5

Construct parallel lines. Prove the relationship between corresponding angles. Use this relationship to find missing measures directly and algebraically.

G.CO.A.1 G.CO.A.4 G.CO.C.9 G.CO.D.12

Prove angle relationships in parallel line diagrams.

G.CO.A.1 G.CO.C.9

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

G.CO.C.9 G.CO.D.12

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

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

G.CO.A.2 G.CO.A.4 G.CO.A.5 G.CO.B.6

Use construction and patty paper to reflect a figure not on the coordinate plane. Describe the properties of a reflection.

G.CO.A.2 G.CO.A.4 G.CO.A.5 G.CO.B.6

Perform rotations on a coordinate plane. Identify characteristics and algebraic rules for the rotation.

G.CO.A.2 G.CO.A.4 G.CO.A.5 G.CO.B.6

Use construction and patty paper to rotate a figure not on the coordinate plane. Describe the properties of a rotation.

G.CO.A.2 G.CO.A.5 G.CO.B.6

Describe a sequence of rigid motions that will map a point, line segment, or angle onto another figure.

G.CO.A.5 G.CO.B.6

Describe rigid motions, or sequences of rigid motions that have the same effect on a figure.

G.CO.A.5 G.CO.B.6

Constructions, Proof, and Rigid Motion

Objective

Common Core Standards

Core Standards

Congruence

Congruence

Foundational Standards

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

References

Target Task

Problem 1

References

Problem 2

References

Additional Practice

Lesson Map

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