## Triangle Sums

I. Section Objectives

- Identify interior and exterior angles in a triangle.
- Understand and apply the Triangle Sum Theorem.
- Utilize the complementary relationship of acute angles in a right triangle.
- Identify the relationship of the exterior angles in a triangle.

II. Cross- curricular-Hang Gliders

- Have students examine the triangles in the image from Wikipedia.
- This is Figure 04.01.01.
- www.en.wikipedia.org/wiki/Hang_gliding
- Ask students what they notice about the number of triangles that are in the hang glider.
- Then have the students identify all of the different angles of the triangles, also include the interior and exterior angles.
- There are angles created by the strings too.
- Complete this as a whole class discussion.

III. Technology Integration

- Have students complete a google search on triangles in nature.
- There they will find hundreds of different images of how triangles are found in nature.
- Ask the students to look for triangles and assign them the task of finding a real example of triangles in nature.
- Have students bring in these examples the next day and show them to the class.

IV. Notes on Assessment

- This class is a discussion class.
- You want the students to see the connection between the different angles of the triangles both interior and exterior.
- Although it is not directly mentioned, you can draw students back to the Triangle Sum Theorem and explain how the measurement of the angles still equal .
- Also make note of any congruent triangles and why they are important to the hang glider being able to fly.
- If the top sails weren’t congruent, what would happen then?

## Congruent Figures

I. Section Objectives

- Define congruence in triangles.
- Create accurate congruence statements.
- Understand that if two angles of a triangle are congruent to two angles of another triangle, the remaining angles will also be congruent.
- Explore properties of triangle congruence.

II. Cross- curricular-Bridge Construction

- Begin by showing students some truss bridge designs.
- For this activity, the students are going to use popsicle sticks or toothpicks to build a truss bridge.
- In younger grades, there are several workbooks on how to do this.
- Given that this is a high school course, have the students design and then build the bridge themselves.
- They need to draw a design first and get it approved.
- Then they can move on to the construction piece of the project.
- When finished, have students explain the importance of congruent triangles in building a solid bridge.

III. Technology Integration

- Students are going to work on a bridge exploration in this activity.
- Have the students google “triangles in bridges”
- Then the students need to look at the different types of bridges.
- Have the students explore two different types bridge designs.
- Ask the students to write congruence statements explaining the congruence of the triangles in the different bridge designs.
- When finished, allow students time to share their work.

IV. Notes on Assessment

- Grade student work in two parts.
- First, grade the design. Is it accurate? Is it neat? Is it labeled? Are the triangles congruent?
- Then grade the construction. Is it complete? Did the students demonstrate with congruent triangles? Is it accurate with the design?
- Provide students with feedback on their work.

## Triangle Congruence Using SSS

I. Section Objectives

- Use the distance formula to analyze triangles on a coordinate grid.
- Understand and apply the SSS postulate of triangle congruence.

II. Cross- curricular-Quiltmaking

- Students are going to be creating their own quilt squares.
- This will extend into the next two lessons.
- In this first lesson, the students are going to design a square that uses triangles that can be proven congruent using the SSS postulate.
- Students should use certain colors in this square and design a key card to explain the color code and that the triangles can be proven congruent using the SSS postulate.
- For example, this red and blue quilt square is made up of triangles that can be proven congruent using the SSS Postulate.
- Then the student would include measurements.
- Students can create this as a poster on poster board or using cloth and sewing by hand.
- If sewing, students could have a small quilt by the time they have finished this chapter.
- If working on poster board, they will have a poster when finished.

III. Technology Integration

- Have students search on quilt making.
- There they can find directions on making quilt squares as well as different stitches to use.
- Students can also see examples of different quilts and quilt squares on different websites.

IV. Notes on Assessment

- Assess the student’s quilt square.
- Does it represent the SSS postulate?
- Is it clearly explained on the note card?
- Provide students with feedback as needed.

## Triangle Congruence Using ASA and AAS

I. Section Objectives

- Understand and apply the ASA Congruence Postulate.
- Understand and apply the AAS Congruence Postulate.
- Understand and practice two- column proofs.
- Understand and practice flow proofs.

II. Cross- curricular-Quilt making

- In this lesson, students are going to add on to their quilts.
- They need to design two different squares.
- One is going to use triangles that can be proven congruent using the ASA Congruence Postulate.
- One is going to use triangles that can be proven congruent using the AAS Congruence Postulate.
- Students need to create a color card with these two squares as well.
- They need to include measurements and explain how the triangles are proven congruent using the different postulates.
- This can be done as an addition to the poster that was started in the last lesson.
- It can also be done as an addition to a sewn quilt.

III. Technology Integration

- Have students complete some research on the history of quilt making.
- Request that each student write a short report on its history and relevance in American society.
- Students can research their material at home or school and write the report as part of a final grade on the quilt.

IV. Notes on Assessment

- Check each quilt square.
- Is the postulate represented?
- Is the note card clearly written?
- Is the report on quilt making clearly written?
- Provide feedback as needed.

## Proof Using SAS and HL

I. Section Objectives

- Understand and apply the SAS Congruence Postulate.
- Identify the distinct characteristics and properties of right triangles.
- Understand and apply the HL Congruence Theorem.
- Understand that SSA does not necessarily prove triangles are congruent.

II. Cross- curricular- Quilt making

- Students are working on adding on to their quilts.
- In this lesson, they are going to be creating quilt squares using right triangles.
- After the student has created his/her square, ask them to create a color card to demonstrate congruence using the SAS Congruence Postulate.
- These quilt squares should contain right triangles.
- Add these quilt squares to the poster.
- Students can also sew these to the quilt.

III. Technology Integration

- Ask students to look at patterns using right triangles.
- Then have them find quilt patterns using right triangles.
- This investigation can impact their design work on the quilt squares.
- Have students look at some examples of Amish Quilts.
- How do they differ from other quilt designs?
- Ask the students to identify some of their favorite patterns and explain why they were selected.

IV. Notes on Assessment

- Examine student work.
- Is the student caught up on the work on the quilt?
- Is each square in today’s lesson using right triangles?
- Does the student understand the SAS Congruence Postulate?
- Is this clearly demonstrated on the color card?
- Provide students with feedback/coaching as needed.

## Using Congruent Triangles

I. Section Objectives

- Apply various triangles congruence postulates and theorems.
- Know the ways in which you can prove parts of a triangle congruent.
- Find distances using congruent triangles.
- Use construction techniques to create congruent triangles.

II. Cross- curricular-Quilt making

- Today have the students use what they have already been working on with regard to their quilts to explain the different congruence postulates.
- This can be a discussion piece that takes place in small groups.
- As the students discuss each of the triangles and how to prove congruence, the students will expand their understanding of the information.
- Next, allow time for students to “catch up” on unfinished work with regard to the quilts.
- If students are sewing, they will probably need an extra day to sew their quilt squares.

III. Technology Integration

- Have students go to the following website to explore the concepts behind proving triangles are congruent.
- www.onlinemathlearning.com/congruent-triangles.html
- This website not only has information for students to learn with, but also has short videos for students to watch.
- This is created as a support for students to expand what they have already learned.

IV. Notes on Assessment

- Listen to student explanations during the presentations.
- Listen for accuracy in student explanations.
- If the students are missing important information stop them and provide correction/feedback.
- If the students are not clear in their explanations, help them to clarify their explanation on how to determine congruence.
- You can also use this class as a way for students to complete their quilt squares.
- Help the students to make a backing for the quilt if it is made of cloth.
- If it is in poster form, then display the student quilt posters in the class.

## Isosceles and Equilateral Triangles

I. Section Objectives

- Prove and use the Base Angles Theorem.
- Prove that an equilateral triangle must also be equiangular.
- Use the converse of the Base Angles Theorem.
- Prove that an equiangular triangle must also be equilateral.

II. Cross- curricular- Geodesic Domes

- For this activity, students are going to examine the equilateral triangles in a geodesic dome.
- Use this website to see this image. This is Figure 04.07.01.
- www.en.wikipedia.org/wiki/File:Epcot07.jpg
- Ask students to use to image to justify the Base Angles Theorem.
- Ask students to use the image to prove that an equilateral triangle is also equiangular.
- Show how an equiangular triangle is also equilateral.
- Allow time for the students to share their work in small groups.
- Have students work on designing their own geodesic dome.
- They can draw it out on graph paper.
- Once they decide on the size of the equilateral triangle, the rest comes together quite easily by repeating the pattern.
- The technology integration can help with this.

III. Technology Integration

- In designing their geodesic domes, the students may want some support from technology.
- Students can research geodesic domes and look at some designs for them.
- Here is another option. Use this website.
- www.fetchaphrase.com/dome/index.html
- This website shows you how to build a geodesic dome out of cardboard.
- Students can use this to construct small geodesic domes.

IV. Notes on Assessment

- Collect student explanations of the different concepts and theorems from the text.
- Be sure that the students have an understanding of how to prove each one of them using the image of the geodesic dome.
- If anything is unclear, provide students with correction and feedback.

## Congruence Transformations

I. Section Objectives

- Identify and verify congruence transformations.
- Identify coordinate notation for translations.
- Identify coordinate notation for reflections over the axes.
- Identify coordinate notation for rotations about the origin.

II. Cross- curricular-Art Images

- Have the students select a singular image.
- They can choose any image that they would like to choose as long as it is singular and simple.
- Then have the students make several different copies of the image.
- If it is in book, they can use a copy machine.
- The students are going to create a piece of art using the image and what they have learned about transformations.
- Each of the transformations needs to be represented.
- Students are going to include a reflection, a rotation, a slide and a dilation of their image.
- Students should do this in a creative way.
- Students are welcome to use more copies of the image as long as at least one of the above listed transformations is in the art piece.

III. Technology Integration

- Have students use the following website to explore all of the different types of transformations.
- www.mathsnet.net/transform/index.html
- On this website, students can explore, understand and work with transformations in an interactive way.
- This is a great way to integrate technology into the lesson.
- You can have students work on this individually or in pairs.

IV. Notes on Assessment

- Examine each student’s piece of art.
- Does it contain each of the required transformations?
- Is there more that the student could have done creatively?
- Provide students with feedback/criticism.
- Display work in the classroom.

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