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4.5: 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. Multiple Intelligences

  • The best way to differentiate this lesson is to be sure to draw out each of the diagrams in this text on the board/overhead.
  • Begin by reviewing the Parallel Postulate.
  • Review with a few examples.
  • Introduce students to the Midsegment Theorem through a diagram first.
  • Draw the diagram on the board. Ask students to brainstorm some of the conclusions that they can draw about the diagram.
  • List these conclusions on the board.
  • Then take the conclusions that have been generated and use them to write the Midsegment Theorem.
  • Include the two statements that are proven in the lesson.
  • 1. Parallel to the third side
  • 2. Half as long as the third side
  • Show students how to prove these two statements.
  • Use the examples in the text to do this, but be sure that the students are following through on the examples.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal

III. Special Needs/Modifications

  • Review symbols for parallel and congruent.
  • The notes on how numbers 1 and 2 are proven are written out in paragraph form. Rewrite these notes in step form so that the students can follow it easier. This will assist students who have any problem with visually tracking information.
  • Review solving multi- step equations.
  • Show students how to take the diagram and write an equation from the information.
  • Then review solving the equation.
  • Show how to apply solving to equation to Example 2 on page 268.

IV. Alternative Assessment

  • In this lesson, alternative assessment is done through observation.

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. Multiple Intelligences

  • Begin by reviewing notes on the perpendicular bisector of a line segment.
  • Show how the bisector divides the line segment into two congruent segments.
  • Show how it intersects the line at a right angle.
  • Ask students to draw two different line segments, measure them and draw in the perpendicular bisector.
  • Either walk around and check student work or do a peer check. It is important to establish understanding before moving on to the next section.
  • Go over the Perpendicular Bisector Theorem and its converse.
  • Have students use a compass and colored pencils in the next activity.
  • 1. Students draw a triangle of their own design.
  • 2. Students draw in the perpendicular bisectors of each line segment.
  • 3. Students use a compass to draw in a circle that encompasses the triangle.
  • 4. Students label the circumcenter of the diagram.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic

III. Special Needs/Modifications

  • Review the definitions of perpendicular and bisector.
  • Write the Theorems on the board.
  • Be sure that students copy those notes into their notebooks.
  • Define circumcenter.
  • Demonstrate how to draw in the segment bisectors and label the circumcenter.
  • Review using a compass.

IV. Alternative Assessment

  • Collect and examine student drawings/diagrams to assess student understanding.
  • Be sure to allow time for student questions.
  • If students are having a difficult time with the in class assignments, allow them the option of working with a peer.
  • Be sure that peer work is on task through observation and walking around.

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. Multiple Intelligences

  • Write the intention of this lesson on the board/overhead.
  • The intention is to inscribe circles in triangles.
  • Go through the material in the lesson.
  • After teaching the material in the lesson, give the students an opportunity to work with large triangles and inscribe circles in these triangles.
  • This can be a lot of fun.
  • Encourage students to use colored pencils and to make their diagram as colorful as they wish.
  • Also use large chart paper.
  • Allow students the option of working in a small group or by themselves.
  • Then ask them to draw a diagram that shows the Concurrency of Angle Bisector Theorem.
  • Be sure that they label each part of the diagram.

III. Special Needs/Modifications

  • Review how to bisect and angle.
  • Review that the bisector of an angle is the ray that divides the angle into two congruent angles.
  • Remind students that with the Concurrency of Angle Bisectors Theorem, that we are going to show the point of intersection.
  • Write these steps on the board.
  • 1. Draw in angle bisectors.
  • 2. Draw in perpendicular bisectors of each line segment.
  • 3. Show the point of intersection
  • 4. Use a compass to inscribe the circle inside the triangle.

IV. Alternative Assessment

  • Allow time for the students to present their work to the class or in small groups.
  • Walk around and listen to the students discuss and explain their work.
  • Use this as a way to assess student understanding.

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. Multiple Intelligences

  • Go through the initial material in this lesson first.
  • Review the median of a triangle and how to find it.
  • Show a diagram on the board that introduces the students to the Concurrency of Medians Theorem.
  • Use the diagram to show each median being drawn in then show the point of intersection, the centroid.
  • Introduce the vocabulary word as the material is covered.
  • You may want to allow time for the students to try this with a triangle of their own creation.
  • This will give them a good understanding of the concepts.
  • Then complete the second part of the lesson.
  • Pg. 300 -Complete the activity in Example 2 together as a class.
  • Ask the students to write down the answers to the two questions.
  • Then open up the discussion to a brainstorming session.
  • Write student responses on the board.

III. Special Needs/Modifications

  • Write all definitions on the board/overhead.
  • Define median of a triangle.
  • Define concurrent
  • Define centroid and show the connection between concurrent and centroid.
  • Review the midpoint formula.
  • Review the distance formula.
  • Review using the Geometer’s Sketchpad.
  • Define Napolean’s Theorem.

IV. Alternative Assessment

  • Assessment is completed through observation and discussion.

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. Multiple Intelligences

  • Begin this lesson with a real life example about altitude. You could use a plane and review the meaning of the word altitude with the students. This will help them to draw associations as they work with the concept.
  • Define altitude.
  • Use these steps to find the altitude of a triangle.
  • 1. Identify the vertex you are using.
  • 2. Find the side of the triangle opposite the vertex or where this side should be located.
  • 3. Draw a straight line from the vertex to that opposite side, draw the side in if it does not exist in the original triangle.
  • Show the two examples with the acute triangle and the obtuse triangle. Have students draw these two examples in their notebooks.
  • Define the Concurrency of Triangles Altitude Theorem
  • Define orthocenter
  • Pg. 307 Example 1- do each step of the example with the students.
  • “What do you observe?” Request that the students write their observations in their notebooks. Then allow some time for sharing and write these notes on the board.
  • 8- What do you observe about the four points? Repeat the brainstorming activity to expand student thinking.
  • Have students draw conjectures about whether or not the points would be collinear for all other kinds of triangles.
  • Expand this as an extra credit assignment or homework assignment for students to draw different triangles and demonstrate whether or not the four points are collinear. Request that they include a writing piece in their assignment and describe their findings in words.
  • Intelligences- linguistic, logical- mathematical, bodily- kinesthetic, visual- spatial, intrapersonal, interpersonal

III. Special Needs/Modifications

  • Write all vocabulary words on the board and request that students copy these notes into their notebooks.
  • Allow a lot of time for questions.
  • Be sure that the students have copied down the steps for locating the altitude of a triangle in their notes.
  • Students are most likely to be concerned when the altitude is drawn outside of the triangle. Use some extra time to show students why this is the case.

IV. Alternative Assessment

  • Assess student learning through observation and through feedback during brainstorming sessions.
  • Be alert to students who are not writing during the independent writing times.
  • Be sure that all students are following the lesson. Review concepts or use peer tutoring if students are having challenges.

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. Multiple Intelligences

  • Teach the information in this lesson, and then expand it using a coordinate grid.
  • The activity has the students draw triangles with different side lengths to prove the Triangle Inequality Theorem.
  • Students use graph paper, colored pencils and rulers for this activity.
  • Give students the dimensions for five different triangles on the board/overhead. For example, can we have a triangle with the lengths 6, 7, 12.
  • Students need to draw out the triangle on the coordinate grid to demonstrate whether it can be a triangle or not.
  • Then they also need to write an inequality demonstrating whether or not these side lengths work for a triangle.
  • When finished, allow time for the students to prove their findings.
  • Include the different theorems into their explanations when they occur. Point these out to the students.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • (Sides) Theorem- walk through the proof step by step. Provide students with a brief explanation of each step.
  • Review briefly each previously learned term as it is introduced. Example- ruler postulate, angle addition, substitution
  • (Angle) Theorem- on board
  • Review what is meant by indirect reasoning- by assumption or conjecture
  • Define corollary

IV. Alternative Assessment

  • Collect student work on with the triangles on the coordinate grid.
  • Use these diagrams and the inequality statements to assess student understanding.
  • You can use this as a classwork grade.
  • It is also useful to assess the students that are in need of assistance.

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.
  • Multiple Intelligences
  • Expand this lesson by creating a model for students to use throughout the lesson.
  • Before beginning, have students use three long strips of paper. Put fasteners to connect the strips of paper into a long strip.
  • The fasteners will be moveable so that the students can manipulate the pieces into different shaped triangles. This is how they can demonstrate proving each of the theorems in the lesson.
  • Have each student work with a partner since we will be working with two triangles.
  • After going through the first example on SAS Inequality Theorem, have students work in pairs to test it out themselves.
  • Offer time for feedback.
  • Then go through the other examples in the lesson, after each example, have students work to test out the theorems with their model.
  • Offer time for feedback.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Review inequalities in one triangle first.
  • Review congruent triangles.
  • Write all of the theorems on the board/overhead.
  • Request that students copy these notes into their notebooks.
  • Allow time for questions.

IV. Alternative Assessment

  • Assess student understanding through observation of the partner work.
  • Interact with students as they work through proving each theorem.
  • Listen to student feedback and correct any unclear information.
  • Expand this lesson into a writing assignment by having students write their observations and conclusions about the theorems in narrative form.
  • If time allows, have students share their conclusions with the whole class or in small groups.

Indirect Proofs

I. Section Objectives

  • Reason indirectly to develop proofs of statement.

II. Multiple Intelligences

  • This is a short lesson but scaffold it into three sections. This will work for both multiple intelligences and for special needs students.
  • Begin by defining an indirect proof.
  • Define conjecture and what is meant by a conjecture.
  • 1. Begin by writing if- then statements using real life examples.
  • For example- “If Mary plays soccer then she is an athlete.”
  • Request that the students write three if- then statements in their notebooks.
  • Allow time for the students to share their work.
  • 2. Algebraic Examples- use the one in the text to begin with.
  • Then have students write three more algebraic examples.
  • Exchange papers with a partner.
  • Each partner must prove the if- then statement as true or false.
  • Allow time for students to share their work.
  • This helps students to make the connection between if- then statements and whether the statement is true or false.
  • 3. Geometric Examples- use the example in the text.
  • Then divide students into small groups.
  • Request that they prove the following using the same diagram from the text.
  • \angle 2 =  \angle 3
  • After students are finished writing the proof, allow time for sharing.
  • Take the best parts of each written proof to compose a proof on the board.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Write all theorems on the board.
  • Use the above activity to scaffold this lesson for the students.

IV. Alternative Assessment

  • Prior to teaching the lesson, compose a list of essential elements for the proof that the students are going to write.
  • When composing the group proof on the board, be sure that the final example has each of these elements in it.
  • Request that students copy this proof into their notebooks.

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CK.MAT.ENG.TE.1.Geometry.4.5

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