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* *Reflection: Developing a Conceptual Understanding
Angles of Triangles - Section 1: Do Now

When students first learn about the sum of the measures of the angles in a triangle, they are told that the sum is 180 degrees. Through this activity, the students were able to actually understand why the sum is 180 degrees. I didn't initially tell them why they were gluing the angles next to each other, I had the students figure out why on their own. They were also able to see why the sum of the measures of the remote interior angles is equal to the measure of the exterior angle. Even though students can use numbers to work out these theorems, actually performing a hands on investigation helped them to develop a conceptual understanding of the theorems.

*Developing a Conceptual Understanding: Angles of a Triangle Reflection*

# Angles of Triangles

Lesson 3 of 11

## Objective: SWBAT prove theorems about the measures of angles of a triangles.

#### Do Now

*7 min*

As the students enter the room, they are given a paper triangle. I cut the triangles out before the lesson using a thin paper like Origami paper and a paper cutter to make the sides more exact. Students will use their triangles to investigate the sum of the angles by tearing off two angles and reassembling them next to the third angle. I show the students what to do or I project the Angles of Triangles Do Now Instructions.

I circulate while the students work to ensure they put the angles together correctly. Students often put a torn side next to a straight edge. To help prevent this, instruct the students not to tear their angles neatly when they tear them.

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#### Mini-Lesson

*10 min*

At the beginning of the Mini-Lesson, I ask the students, “What can the activity from the Do Now tell us about triangles?” Students can see that when the angles were laid next to each other, they formed a straight line. Using what students know about the measure of the line, they can see that the sum of the measures of the interior angles in a triangle is 180^{o}.

We then look at another theorem that can be demonstrated by the activity. The sum of the measures of the remote interior angles in a triangle is equal to the measure of the exterior angle. I show the students the presentation, which illustrates the two theorems (G.CO.10).

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#### Activity

*23 min*

In the Angles of a Triangle Activity students write proofs involving theorems about the angles of a triangle. The first two problems show proofs of the two theorems from the Mini-Lesson. The third problem can be used as a challenge problem. It is a proof showing the sum of the measures of the exterior angles of a triangle.

Students work in groups to complete their proofs. Some students may choose to work on the proofs on their own. Students who find writing proofs challenging are given a list of statements in the order needed to prove the statement. Their task is to match a list of reasons to the statements. Before I give the students the statements and reasons, I cut them out so the reasons are not in the correct order.

Each table has between three and five students. Group roles: cutting out the reasons, pasting the statements on the chart paper, drawing the diagrams, and timekeeper. Students then work together to decide which statements go with which reasons.

Students who complete their proofs before the time ends, can begin to work on the third proof.

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#### Summary

*5 min*

At the end of today's proof activity, students hang their chart paper on the wall so that all of the proofs are visible. I plan to call on two students to present their proofs. One student from each group stands by their chart and checks their group’s work. As a class, we will discuss the proofs and the theorems that can be proven from the proofs (**MP3**). To bring the lesson to closure, I plan to ask my students to write a summary of the proofs on their sheets.

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- UNIT 1: Preparing for the Geometry Course
- UNIT 2: Geometric Constructions
- UNIT 3: Transformational Geometry
- UNIT 4: Rigid Motions
- UNIT 5: Fall Interim Assessment: Geometry Intro, Constructions and Rigid Motion
- UNIT 6: Introduction to Geometric Proofs
- UNIT 7: Proofs about Triangles
- UNIT 8: Common Core Geometry Midcourse Assessment
- UNIT 9: Proofs about Parallelograms
- UNIT 10: Similarity in Triangles
- UNIT 11: Geometric Trigonometry
- UNIT 12: The Third Dimension
- UNIT 13: Geometric Modeling
- UNIT 14: Final Assessment

- LESSON 1: Triangle Inequality Theorem Investigation
- LESSON 2: Side-Side-Side
- LESSON 3: Angles of Triangles
- LESSON 4: Relationships between Angles and Sides of a Triangle
- LESSON 5: Applying Triangle Angle Theorems
- LESSON 6: Side-Angle-Side
- LESSON 7: Angle-Side-Angle
- LESSON 8: CPCTC
- LESSON 9: Additional Properties of Triangles
- LESSON 10: Properties of Isosceles Triangles
- LESSON 11: Properties of Isosceles Triangles Alternative Lesson with Dynamic Geometry Software