##
* *Reflection: Perseverance
Angles of a Triangle - Section 1: Discussion of the Homework

It was interesting to see the student reactions to the last two problems in this problem set. Some students were indignant that I had expected them to solve problems that had not yet been taught to them. A greater number of students, however, had wrestled with these problems, tried different approaches, and were so psyched when they were able to figure the problems out on their own (**MP1**).

This gave me a lot of information about my students - not just about their understanding of the material, but about them as problem solvers and mathematicians.

As we progressed through this unit, I also noticed that those students who had worked to devise their own method for solving these problems were completely comfortable when faced with similar problems. Because they really understood the solutions they had created, they "owned" that material!

*Perseverance: Problems They've Never Seen Before*

# Angles of a Triangle

Lesson 3 of 5

## Objective: SWBAT prove that the sum of the measures of the angles of a triangle is 180 degrees, and SWBAT solve problems involving the interior and exterior angles of triangles.

## Big Idea: Students learn about the different types of triangles, the relationships between the interior and exterior angles of a triangle, and apply their knowledge to numerical problems and constructions.

*90 minutes*

#### Discussion of the Homework

*15 min*

The students' first task as they enter the room is to get out their homework, entitled Parallel Line Problems. I run through the answers to the problems, and we discuss any problems for which the students have questions.

I anticipate that we will need to spend time on the last two problems, both of which require the students to add an auxiliary line (**MP7**). There are several different approaches to these problems, and I ask students to present and explain their techniques for solving these (**MP3**).

#### Resources

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#### Types of Triangles

*10 min*

The students learned to identify the different types of triangles in previous grades. However we take the time to review the terminology in order to ensure precision with regard to vocabulary (**MP6**), discussing the classifications of triangles by sides and by angles.

On my Smart Board I have prepared several pages of diagrams of triangles, similar to these. As a class, we run through these diagrams quickly, classifying each triangle as specifically as we can.

#### Resources

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I explain that next we will explore the angles of a triangle. We talk about the concepts of interior and exterior angles of triangles, and then I charge the class with exploring these angles using dynamic software (**MP5**).

**Teacher's Note**: In my classes I use GeoGebra. A brief explanation of GeoGebra can be found by looking at GeoGebra Quickstart.

The first time we used GeoGebra, we worked with a grid. Today we will use the geometry view, without a grid, as the coordinate plane will not add anything to this discussion. I hand out the instructions to the students, have them all create points A, B, and C on their screen, and then they take it from there!

I am expecting that the students will arrive at hypotheses for the interior and exterior angle sums of a triangle, and the exterior angle theorem (an exterior angle is equal to the sum of the two remote interior angles).

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#### Proving Our Theorems

*15 min*

Now we move onto the task of proving our hypotheses. We do the proofs for the sum of the interior angles and the exterior angle theorem as two-column proofs. However I use the proof for the sum of the exterior angles as the students' first exposure to a paragraph proof.

There is also a very nice GeoGebra applet that can be found here that demonstrates the exterior angle theorem using a rotation and a translation of angles. (It hints very nicely at our next unit on transformations!)

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#### Constructing Triangles

*20 min*

Next we work on constructing triangles (**MP5**). These constructions are new to the students. However, as we are doing the constructions, I ask for suggestions from them: *where should we start? what does it make sense to do next? what does swinging this arc accomplish? *(**MP2**)

First I familiarize the students to copying a line segment. Then I ask the students to:

- Construct an equilateral triangle
- Construct an isosceles right triangle, given the length of the legs.

We use the Triangle Constructions handout. I give the students time to work on the isosceles right triangle without help from me, as I suspect many will be able to figure this construction out on their own.

#### Resources

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I hand out the homework assignment and the class and I look it over together. We discuss the geometric concepts that are being targeted in the homework, and then we review the more challenging algebraic skills that are included (factoring and solving a system of equations).

#### Resources

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