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# Triangle Congruence and CPCTC Practice

Lesson 6 of 10

## Objective: Students will be able to apply and prove properties of special triangles and quadrilaterals.

As we begin today's lesson I want students to recall our brief discussion about **congruent triangles having congruent corresponding parts**. So, I give students a pair of triangles which they can conclude are congruent by AAS. For this proof, I will introduce students to a new proof format for our class, the **two-column proof**.

To construct the proof, we organize our statements and reasons in the example proof and then, shortly after, students work on the practice proof. After a few minutes, I will ask a student volunteer to present his/her proof to get feedback from the class. Feedback and discussion is especially important at this point in the unit because students have to use the triangle congruence shortcuts along with the precise definition of geometry terms like **midpoint**, which adds another layer to students’ proof writing (**MP6**).

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This is a **station work activity**. In their work, students will be asked to use precise definitions of special triangles and quadrilaterals to prove triangles are congruent. I differentiate this activity by letting students choose four of the five proofs to complete and whether they would like to work individually or with the people around them. There are often multiple ways to write the proofs, which fosters opportunities for students to agree or disagree with each other while realizing what information they must include to write a high-quality proof.

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#### Debrief

*10 min*

When we debrief today's classwork, I will project my own answer key. Throughout our closure discussion, I ask students questions like:

- Is there another way to prove the triangles congruent?
- What evidence did you need to prove the triangles congruent?

As the discussion unfolds, I expect students will volunteer different ways to prove the triangles congruent, which they can present to the class or have me record on their behalf.

Additionally, I hope students will ask questions about SSA as this is still a relatively new idea for our class; if students do not ask about SSA, I will eventually pose the question, "What about SSA? Is the SSA condition sufficient to prove triangles are congruent?"

#### Resources

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- UNIT 1: Creating Classroom Culture to Develop the Math Practices
- UNIT 2: Introducing Geometry
- UNIT 3: Transformations
- UNIT 4: Discovering and Proving Angle Relationships
- UNIT 5: Constructions
- UNIT 6: Midterm Exam Review
- UNIT 7: Discovering and Proving Triangle Properties
- UNIT 8: Discovering and Proving Polygon Properties
- UNIT 9: Discovering and Proving Circles Properties
- UNIT 10: Geometric Measurement and Dimension
- UNIT 11: The Pythagorean Theorem
- UNIT 12: Triangle Similarity and Trigonometric Ratios
- UNIT 13: Final Exam Review

- LESSON 1: The Language and Properties of Proof
- LESSON 2: Triangle Sum Theorem and Special Triangles
- LESSON 3: Triangle Inequality and Side-Angle Relationships
- LESSON 4: Discovering Triangle Congruence Shortcuts
- LESSON 5: Proofs with Triangle Congruence Shortcuts
- LESSON 6: Triangle Congruence and CPCTC Practice
- LESSON 7: Proving Isosceles Triangle Conjectures
- LESSON 8: Group Assessment: Triangle Congruence and Proof
- LESSON 9: Triangle Properties, Triangle Congruence, and Proof Review
- LESSON 10: Discovering and Proving Triangle Properties Unit Assessment