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* *Reflection: Connection to Prior Knowledge
Triangle Sum Theorem and Special Triangles - Section 4: Prove the Triangle Sum Theorem

I thought proving the triangle sum theorem went really well this year. It’s not like there were fireworks, but rather, flickering light bulbs that seemed to stay “on” for much longer.

There was one few key phrase I used during this time that helped to reinforce the notion of proof, which I have used throughout the year: “convince a skeptic.” This phrase helped my students appreciate that despite the fact they investigated the angles of four different triangles (acute, obtuse, right, isosceles), tearing off the angles of each triangle, then gluing them so they are adjacent, this Visual Demonstration of a straight angle did not constitute a proof.

Since this was the first “hefty” proof of the year, it was important to me that students consider multiple pathways to prove the angles of a triangle are supplementary. I want my students to see multiple pathways because it helps them to make connections between the mathematics that drives the reasoning behind proof. Also, I think that students feel encouraged by the notion that if they don’t see one particular path, another one is waiting to be discovered.

The conversation in this lesson tends to follow a particular construction, create an auxiliary parallel line. Moreover, if students had used alternate interior angles in that construction, they tended to favor using alternate interior angles in their proof. **For this reason, if students favored alternate interior angles, I listened for and encouraged proofs that might have used corresponding and vertical angles, or consecutive (same side) interior angles**. This might sound inefficent, but being intentional about encouraging other ways to write the proof enables more of my students to achieve that “Aha!” light bulb moment. In addition, it encourages deeper engagement in mathematical practices. Giving students more time to wrestle with ideas and persevere, allows insight into the structure of a problem and prepares students to share and defend their ideas in conversation with others.

# Triangle Sum Theorem and Special Triangles

Lesson 2 of 10

## Objective: Students will be able to discover and prove the triangle sum theorem.

In this Warm-Up, students write and solve algebraic equations to review the idea that the opposite sides of a parallelogram are congruent. While I circulate the room, I look for a student who will volunteer to present their work on the first problem. At this point, I want a student who can give a model explanation for how they used the given information from the problem (the quadrilateral is a parallelogram) to draw certain conclusions (the opposite sides are congruent, so I can write and solve equations).

When they construct a proof, I want my students to begin by using the given information to draw their own diagram. As I circulate the room, I want to find a student who can logically show why the vertical angles are congruent by using the ideas we discussed in our work with the Language and Properties of Proof during yesterday's last lesson.

#### Resources

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During the Triangle Sum Investigation, students work in groups of four to "discover" the interior angle sum of a triangle. The structure of this investigation requires each student to take on a different “case” to explore, compare results, and then draw conclusions. In this investigation, the group collectively explores the angles of **acute**, **right**, **obtuse**, and **isosceles **triangles to then conjecture about the interior angle sum of any triangle.

I include a check-in point for this investigation, which provides me with an opportunity to assess student understanding. Groups call me over to check in, which essentially involves one student explaining what the group has conjectured. I like to shuffle all of my students’ papers behind my back to “randomly” choose the student who I will question—since students are familiar with this structure, they prepare each other for this moment, which holds them accountable to the group’s learning. After the check-in, I ask groups to move on to the next part of the task, which gives them an opportunity to apply and practice what they have just discovered.

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#### Debrief Notes

*15 min*

As in my other lessons, I like to formally debrief the discoveries we have made by having students take notes in their note-takers. We discuss students’ conjecture about the interior angle sum of any triangle, and extend this understanding to consider other conclusions we can draw about the angles of triangles. I introduce the idea of an **auxiliary line** to students, which they need when proving the Triangle Sum Theorem.

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In the Triangle Sum Theorem Proof, I ask students to construct a parallel line to the base of the triangle. I choose to do this because students, through construction, have to consider the angle relationships that will yield parallel lines, which gives them a way into the proof. (For example, students might use alternate interior angles to construct their auxiliary line, which then gives them a possible path to pursue.)

Because the proof of the triangle sum theorem requires students to go out of their comfort zone (work with auxiliary lines, name extra angles that they might use), I have found it really important to have at least two different students present their work (**MP1**).

- Many students use alternate interior angles in this proof and make connections to the Triangle Sum Investigation, where students arranged the angles of a triangle on a straight angle.
- An interesting proof that students often arrive at involves using consecutive interior angles to show that the interior angles of a triangle are supplementary.

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In this Exit Ticket, I want to assess students' understanding of the **Triangle Sum Theorem**. I want them to "explain to a 4th grader" because I want to get a good sense of how clearly they can articulate their understanding of the triangle sum theorem in a way that would make sense to anyone.

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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