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.
- Label diagrams and explain conjectures (orally and in writing).
- Prove (in writing) that vertical angles are congruent.
- Let’s make convincing explanations.
- I can label and make conjectures from diagrams.
- I can prove vertical angles are congruent.