Lesson plan

Lesson 19: Evidence, Angles, and Proof

teaches Alabama State Standards Geo-31.a.
teaches Alabama State Standards Geo-31.
teaches Arizona State Standards G.G-CO.C.9
teaches Common Core State Standards MP7 http://corestandards.org/Math/Practice/MP7
teaches Common Core State Standards MP5 http://corestandards.org/Math/Practice/MP5
teaches Common Core State Standards HSG-CO.C.9 http://corestandards.org/Math/Content/HSG/CO/C/9
teaches Common Core State Standards MP1 http://corestandards.org/Math/Practice/MP1
teaches Common Core State Standards MP3 http://corestandards.org/Math/Practice/MP3
teaches Colorado State Standards HS.G-CO.C.9.
teaches Georgia State Standards MGSE9-12.G.CO.9.
teaches Kansas State Standards G.CO.7.
teaches Minnesota State Standards
teaches Minnesota State Standards
teaches Minnesota State Standards
teaches Minnesota State Standards
teaches Minnesota State Standards
teaches Ohio State Standards G.CO.9.
teaches Pennsylvania State Standards CC.2.3.HS.A.3.

Lesson 19: Evidence, Angles, and Proof

In previous grades, students used facts about supplementary, complementary, vertical, and adjacent angles to solve problems. In previous lessons, students made conjectures, developed definitions of the basic rigid motions, and explained why they think certain claims are true or false. Over the next several lessons, students will learn ways to express their reasoning more formally. In this lesson, students create conjectures about angle relationships and prove them using what they know about rigid transformations. Students begin to label and mark figures to indicate congruence which helps them communicate more precisely. Students are asked to make viable arguments and critique the reasoning of others when they write convincing explanations for why vertical angles are congruent (MP3).

The proofs in these materials are all written in narrative form. The narrative format matches the discussion students might have to convince their partner, and it also matches the way mathematicians write proofs. While students may use other formats to support their organization, it is important that students can see the flow of reasoning that exists in a well-written proof.  A two-column proof can be thought of like an outline for an essay. Outlines help organize thoughts but an outline is less persuasive than a well-written essay. Students should learn to write a well-written justification in the form of a narrative proof. This is an opportunity for them to make sense of problems and persevere in solving them (MP1).

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

Lesson overview

  • 19.1 Warm-up: Math Talk: Supplementary Angles (5 minutes)
  • 19.2 Activity: That Can’t Be Right, Can It? (15 minutes)
  • 19.3 Activity: Convince Me (15 minutes)
    • Includes "Are you Ready for More?" extension problem 
  • Lesson Synthesis
  • 19.4 Cool-down: Plead Your Case (5 minutes) 

Learning goals:

  • Label diagrams and explain conjectures (orally and in writing).
  • Prove (in writing) that vertical angles are congruent.

Learning goals (student facing):

  • Let’s make convincing explanations.

Learning targets (student facing):

  • I can label and make conjectures from diagrams.
  • I can prove vertical angles are congruent.


  • This lesson builds on the standards:CCSS.7.G.B.5MS.7.G.5MO.7.GM.B.5CCSS.HSG-CO.A.1MS.G-CO.1MO.G.CO.A.1
  • This lesson builds towards the standard:CCSS.HSG-CO.C.9MS.G-CO.9MO.G.CO.C.8






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