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# 3.4: Congruent Triangles

Created by: CK-12

## 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 $180^\circ$.
• 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.

## Date Created:

Feb 22, 2012

Feb 23, 2012
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