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# 3.5: Relationships Within Triangles

Created by: CK-12

## Midsegments of a Triangle

I. Section Objectives

• Identify the midsegment of a triangle.
• Apply the Midsegment Theorem to solve problems involving side lengths and midsegments of triangles.
• Use the Midsegment Theorem to solve problems involving variable side lengths and midsegments of triangles.

II. Cross- curricular-Mapping

• Use the following image of the Bermuda Triangle in this activity.
• This is Figure 05.01.01
• www.en.wikipedia.org/wiki/File:Bermuda_Triangle.png
• Each student will need a copy of the image to work with.
• Use a scale and a ruler to determine the distance between each of the vertices of the triangle.
• Then determine the midsegment of the triangle.
• Draw the midsegment into the image of the triangle.
• After drawing in the midsegment, write a proof that proves that this is the correct midsegment of the triangle.
• Students can work on this in pairs so that they have peer support when writing the proof.
• Students may want to name each of the vertices to help with writing the proof.
• When finished, allow time for the students to share their work.

III. Technology Integration

• Use Wikipedia or another website to research facts about the Bermuda Triangle.
• What are some of the mysteries surrounding this area?
• When was it discovered to be a “triangle” in shape?
• Be prepared to share your findings with the others in the class.

IV. Notes on Assessment

• Assess student understanding by examining the proof.
• Is it clear?
• Is the given information clear?
• Is it clear which information needs to be proven and which does not?
• Provide students with feedback on their work.
• Share strong example with the others in the class so that all can improve their proof writing.

## Perpendicular Bisectors in Triangles

I. Section Objectives

• Construct the perpendicular bisector of a line segment.
• Apply the Perpendicular Bisector Theorem to identify the point of concurrency of the perpendicular bisectors of the sides (the circumcenter).
• Use the Perpendicular Bisector Theorem to solve problems involving the circumcenter of triangles.

II. Cross- curricular-Origami

• There are several different origami designs that you can do that require the use of an equilateral triangle.
• First, use this website to help the students move from a circle to an equilateral triangle.
• www.cyffredin.co.uk/The equilateral triangle.htm
• This will help the students to have an equilateral triangle in design.
• Then you can move on to folding in perpendicular bisectors of the triangle.
• This will help you to identify and mark the circumcenter.
• After the exploration is complete, you can ask the students what they have learned about the perpendicular bisectors of a triangle and the circumcenter of the triangle.
• Brainstorm a list of conclusions on the board.

III. Technology Integration

• Complete a websearch on origami.
• There are several different sites and patterns that students can explore.
• Ask them to select patterns that begin with an equilateral triangle.
• Use this pattern and the equilateral triangle to fold a dolphin or another animal of choice.
• Allow students time to share their work.

IV. Notes on Assessment

• Assessment is completed through observation.
• You can walk around and see students working with the equilateral triangles and the perpendicular bisectors as they fold their designs.

## Angle Bisectors in Triangles

I. Section Objectives

• Construct the bisector of an angle.
• Apply the Angle Bisector Theorem to identify the point of concurrency of the perpendicular bisectors of the sides (the incenter).
• Use the Angle Bisector Theorem to solve problems involving the incenter of triangles.

II. Cross- curricular-Art

• This activity builds on the origami that the students completed in the last lesson.
• This time, students aren’t going to be working with equilateral triangles but with three different sized triangles.
• Ask the students to cut out triangles that are three different sizes.
• Then with each triangle, students are to fold the paper to show the three bisecting lines of each of the angles of the triangle.
• In the end, the students will have the point of concurrency.
• From there, they can inscribe the circle into the triangle.
• Students need to complete this with all three triangles.
• Allow time for the students to share their work in small groups when finished.

III. Technology Integration

• Students can use the following website to explore bisecting angles.
• www.geom.uiuc.edu/~demo5337/Group2/incenter.html
• When finished, ask the students to share what they discovered about angle bisectors and inscribing circles.
• Write the conclusions on the board.

IV. Notes on Assessment

• Walk around and observe the students as they work on the paper folding.
• Assist students who are having difficulty.
• Students should see this as a hands- on way to work through the point of concurrency and inscribing circles.

## Medians in Triangles

I. Section Objectives

• Construct the medians of a triangle.
• Apply the Concurrency of Medians Theorem to identify the point of concurrency of the medians of the triangle (the centroid).
• Use the Concurrency of Medians Theorem to solve problems involving the centroid of triangles.

II. Cross- curricular-Art

• Use the concept of Napolean’s Theorem to create a new design/stained glass window effect.
• Review Napolean’s Theorem and how it works.
• Then have students begin to work on a design.
• Students can explore trying different sizes of triangles.
• They can also see if it makes sense to integrate different shapes.
• Since the goal is a stained glass window of sorts, students are going to create a frame and then place different colored tissue paper inside the frame.
• Students can complete this and hang them in the window and the sunlight will come through the design.

III. Technology Integration

• One of the ways to integrate technology into this lesson is to have the students look at some of the other designs of Napolean’s Theorem.
• They can begin by googling Napolean’s Theorem.
• There will be several different websites that will come up where students can read about Napoleans Theorem and see different patterns and designs.
• When finished, students will have expanded their thinking on this theorem.

IV. Notes on Assessment

• Walk around and assist students as they work.
• When finished, assess student designs.
• Does it work according to Napolean’s Theorem?
• If not, what would make a difference?
• Is the shape incorrect or does there need to be more shapes?
• How detailed did the student get?
• Is the stained glass window colorful and creative?
• Provide students with feedback on their work.

## Altitudes in Triangles

I. Section Objectives

• Construct the altitude of a triangle.
• Apply the Concurrency of Altitudes Theorem to identify the point of concurrency of the altitudes of the triangle (the orthocenter).
• Use the Concurrency of Altitudes Theorem to solve problems involving the orthocenter of triangles.

II. Cross- curricular-Sculpture

• Investigate the concept of altitude by using the following image. This is Figure 05.05.01
• This is an image of the Mihashira Torii sculpture.
• www.en.wikipedia.org/wiki/File:Yamato_mihasira006.jpg
• Use this image and ask the students to share how they think the concept of altitude impacts this sculpture.
• Brainstorm ideas and write them on the board.
• Then set students to work on designing and building their own sculpture.
• You will need dowels, small hand saws, sand paper and wood glue or fast drying glue.
• You can use dowels with a small diameter so that they will be easy to cut.
• The students need to work with a triangle as the basic shape of the sculpture and demonstrate the altitude of the triangle in their sculpture.
• Allow time for the students to work and then present their sculptures when finished.

III. Technology Integration

• Investigate the concept of altitude using the computer.
• Ask the students to search all of the different ways that altitude impacts our way of life.
• Have them keep a list of the websites that they visit.
• They also need to make notes on at least ten different ways that altitude impacts how we live.
• Allow time for students to share their research when finished.

IV. Notes on Assessment

• Assessment is completed through observation.
• Walk around and see how the students are doing on their sculptures.
• Help out when needed.

## Inequalities in Triangles

I. Section Objectives

• Determine relationships among the angles and sides of a triangle.
• Apply the Triangle Inequality Theorem to solve problems.

II. Cross- curricular-Sculpture

• In this activity, you are going to design a sculpture using triangles.
• You want to show that your triangles represent an inequality.
• To do this, you will need to design your sculpture before building it.
• This design should have measurements and demonstrate an inequality.
• When finished with the design, students can use clay to build their triangles.
• Have tools available to work with.
• When finished, have students write a short explanation of their sculpture, what they designed, how it was created and how it demonstrates the concept of an inequality.

III. Technology Integration

• Have the students research triangular sculptures.
• There are so many different sites to select and sculptures to see.
• Students need to select one triangle sculpture that they appreciate.
• Then there is a writing piece to this assignment.
• Students need to write about why they selected the piece and to use principles already learned to describe it.
• What kind of triangle(s) are in the sculpture?
• Is it in three- dimensions or two?
• Are there congruent triangles involved?
• How can you determine congruence?

IV. Notes on Assessment

• Assess student designs and sculptures.
• Does the design and the sculpture match up?
• Does the sculpture represent an inequality?
• Is student writing clear?
• Does the student have an understanding of the concept of triangle inequalities?
• Provide students with feedback/notes.

## Inequalities in Two Triangles

I. Section Objectives

• Determine relationships among the angles and sides of two triangles.
• Apply the SAS and SSS Triangle Inequality Theorems to solve problems.

II. Cross- curricular- Architecture

• Use the concept of gable windows during this lesson.
• Students can use the two Theorems to determine the relationship between triangles in gable windows.
• Use the following image. This is Figure 05.07.01.
• www.loghomebuilders.org/files/images/log-home-bham-gable-windows.preview.jpg
• Have students work in small groups.
• In each group, the students need to come up with a way to prove the relationship between the two triangles in the image of gable windows.
• They are going to be using the SAS and the SSS Triangle Inequality Theorem to do this.
• When finished, have the students present their work to the class.

III. Technology Integration

• Expand this websearch into many different types of triangular windows.
• Some windows will show congruent triangles, but others won’t.
• Have the students select a window pair that is congruent and then prove congruency.
• Have the students select a window pair that demonstrates an inequality and then prove this using the theorems.
• Have the students repeat the exercise that they did in small groups with a new window design.

IV. Notes on Assessment

• Assessment is completed through observation.
• Walk around as students work.
• Ask questions and probe into student thinking.

## Indirect Proofs

I. Section Objectives

• Reason indirectly to develop proofs of statement.

II. Cross- curricular-Sports

• You have the job of being a sports announcer at a basketball game.
• To do this, you will be reporting on the actions of the game.
• However, you can only report your findings using if-then statements.
• You are going to prepare a short broadcast and then present it with a peer to the class.
• This is meant to be a fun short assignment to help students to see how to use if then statements in real life.
• Students will have fun with this.
• Give them time to work and props are fine to use as well.
• When finished, allow time for each pair to present their skit.

III. Technology Integration

• What is proof?
• Who uses proof?
• What kinds of careers or projects require people to prove something?
• Complete a web investigation on the topic of proof.
• Make a list of the websites that you visit.
• Keep a record of the data you discover.
• Write a one page paper to share/explain your findings.

IV. Notes on Assessment

• Assessment is complete through observation.
• Did the students use if- then statements?
• Were they prepared?
• Were they focused?
• Offer feedback to students as needed.

## Date Created:

Feb 22, 2012

Feb 23, 2012
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