Curriculum  /  Math  /  10th Grade  /  Unit 1: Constructions, Proof, and Rigid Motion  /  Lesson 7

Constructions, Proof, and Rigid Motion

Lesson 7

Math

Unit 1

10th Grade

Lesson 7 of 19

Objective

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

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.

Foundational Standards

  • 7.G.B.5

Criteria for Success

  1. Describe the definition and properties of adjacent angles. 
    1. Adjacent angles are two angles with a common ray. 
    2. Adjacent angles are supplementary if their angle measures sum to 180°. 
    3. Adjacent angles are complementary if their angle measures sum to 90°.
  2. Use the accepted notion that there are 360 degrees around the center of a circle and that vertical angles are equal in measure to solve problems, using direct solutions for angle measures and using algebraic expressions to find angle measures.
  3. Use the definition of a straight angle, an angle that is formed when the vertex and the point on each ray of the angle are collinear, to solve problems, involving direct solutions for angle measures and using algebraic expressions to find angle measures.
  4. Use theorems to establish a logical proof of statements and reasons that are accepted as true to show a new relationship.

Tips for Teachers

  • Ask Dr. Math “Why is a Circle 360 Degrees?” may be helpful in answering a common question associated with this topic.  
  • In this lesson, you will continue to build students’ ability to complete proofs. Proofs are a major way that mathematicians have developed “facts” or can come to conclusions about geometric figures.
  • Until this point in the unit, students have done a fair amount of exploration, and development of conjectures about relationships. This lesson marks the first in which students will need to develop a working bank of theorems and properties to use in proofs. Determine the method that is best for you and students to reference and ensure that students have consistent notes to use as reference in the development of proofs. 
  • Ensure throughout the lesson that students use correct symbols when indicating rays, line segments, lines, etc.
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Anchor Problems

Problem 1

Find the value of $$x$$, $$y$$ and $$z$$. All values are in degrees. Drawings not to scale. 

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 1 > Topic B > Lesson 6

Geometry > Module 1 > Topic B > Lesson 6 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

Find the value of $$x$$ and the missing angle measures.

Guiding Questions

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Target Task

  • Use the diagram below to determine x and y. $${\overleftrightarrow{AB}}$$ and $${\overleftrightarrow{CD}}$$ are straight lines.
  • Name a pair of vertical angles.
  • Find the measure of $${\angle BOF}$$. Justify your calculation.

References

EngageNY Mathematics Geometry > Module 1 > Topic B > Lesson 7 — Opening Exercise (drawing recreated in Geogebra)

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..

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.

  • Make sure to include problem types like: 
    • Draw a pair of adjacent supplementary angles with a vertex of C. (How many different diagrams can you get from this? What do you need to do to be more specific?)
    • Error analysis problems. Give incorrect statements or reasons, or mixed up statements and reasons and have students correct them. 
    • “Select all that apply” problems. Give an angle diagram with one angle given and one angle to find. Have students select the reasons WHY that angle measure is what it is. 
    • “Always, sometimes, never” problems. Vertical angles are always equal.
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Lesson 6

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Lesson 8

Lesson Map

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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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