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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.

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CK.MAT.ENG.TE.1.Geometry.3.4

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