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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. Multiple Intelligences

  • Begin this lesson by talking about interior and exterior angles of a triangle. Request that students draw an example in their notebooks of a triangle with interior and exterior angles.
  • Request that they label the interior angles red and the exterior angles in blue.
  • Move on to the Triangle Sum Theorem.
  • Remind students that this is information that they already know, but that it has now been written as a theorem.
  • One way to make this part of the lesson interactive is to draw a diagram of a triangle on the board and to play “fill in the blank” with the students. You can even have them work on teams and see which team can complete the addition the quickest.
  • Another fun way to do this is to use only mental math. This is great for students to practice their thinking skills. The big thing to watch for is students calling out- remind them to raise a hand when done or to stand up.
  • If you wish to keep this more traditional, then have students draw the triangles at their seats but use a protractor to measure angles and solve for missing angles.

III. Special Needs/Modifications

  • Review the following terms.
  • Interior angles
  • Triangle
  • Exterior angles
  • Write new theorems on the board.
  • Request that students copy this information down in their notebooks.
  • Allow time for questions to check on student understanding.

IV. Alternative Assessment

  • The best way to assess this lesson is through observation.
  • If you are playing the game, observe which students are actively participating and which ones aren’t.
  • If completing seat work, walk around and check in with students as they work. Answer questions and offer assistance as needed.

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. Multiple Intelligences

  • A way to differentiate this lesson is to work with the determining the congruence of triangles in a hands- on way.
  • Begin by teaching the material in the lesson, then move on to the activity.
  • Students begin by drawing two congruent triangles. They should use letters to label the vertices of each triangle.
  • Then they need to use a protractor to be certain that the two triangles are congruent.
  • Once they have determined congruency, the student adds in the tic marks to show congruent sides.
  • Finally, they exchange papers with a peer. Once they have exchanged papers, the students need to write three statements that demonstrate and explain that the two triangles are congruent.
  • Check student work as part of a class discussion.
  • Intelligences- linguistic, logical- mathematical, bodily- kinesthetic, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Review what it the word congruent means.
  • Review the properties of a triangle- meaning corresponding sides and corresponding angles being congruent.
  • Explain the use of tic marks to show congruency. Be sure that students understand that they may see tic marks on other figures as well.
  • Write the congruence properties on the board. Draw the similarities between these properties and the properties of equality.
  • Allow time for any questions.

IV. Alternative Assessment

  • Complete an assessment of student understanding by reviewing student diagrams and congruence statements during the class discussion.

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. Multiple Intelligences

  • After reviewing the first example, ask students to participate in this activity.
  • Student one draws a triangle on the coordinate grid and the passes their paper to the right. All students are doing this simultaneously.
  • The next student takes the triangle passed to them and uses the distance formula to figure out the lengths of each side of the triangle. Then he/she passes the paper to the right.
  • The next student takes the measurements and draws a triangle congruent to the first triangle somewhere on the coordinate grid. Then he/she passes the paper to the right.
  • The final student checks the work of all of the others.
  • Discuss work when all have finished.
  • This is great practice for the students and keeps them engaged because of the paper passing.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal

III. Special Needs/Modifications

  • Review the definition for a postulate.
  • Review distance formula and how to use it. Provide students with two examples.
  • Practice drawing triangles on the coordinate grid.
  • Review ordered pairs and how to use the ordered pairs with the distance formula.
  • Write the SSS Triangle Congruence Postulate on the board/overhead. Request that students copy these notes down in their notebooks.

IV. Alternative Assessment

  • Students can be assessed during the class discussion.
  • Also, walk around during the paper passing exercise. You will be able to observe students as they work.
  • Allow time for questions.
  • Make a note of any students who are having difficulty during the lesson.

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. Multiple Intelligences

  • When you introduce the ASA Congruence Postulate, review that a postulate is assumed true.
  • Then go through the directions in the text, but have the students follow along with you and do the steps themselves in their seats.
  • Once students have a good grasp on the ASA Congruence Postulate, then move on to the AAS Congruence Theorem. Be sure that the students understand that a theorem can be proved.
  • Demonstrate the example.
  • Ask the students what they can notice about the ASA Congruence Postulate and the AAS Congruence Theorem. Write their ideas on the board.
  • You want the students to realize that they can be used equally. If the students aren’t making this connection on their own, use an example from the text to guide them in discovering it. Having them discover it on their own is much more valuable than telling them the information.
  • Move on to the two- column proofs. Go through the material. Request that the students participate in completing the proof.
  • Flow proofs- go through the material.
  • If time allows, have students write their own flow proofs and share them in small groups. You could also assign this as a homework assignment.

III. Special Needs/Modifications

  • Write the new terms and vocabulary on the board/overhead.
  • ASA Congruence Postulate
  • AAS Congruence Theorem
  • Review the difference between a postulate and a theorem.
  • Review the Third Angle Theorem
  • Review two- column proofs
  • Allow time for questions.

IV. Alternative Assessment

  • The best way to assess student learning in this lesson is through question and answer sessions.
  • Be sure that you allow time for the students to participate in the lesson. Do not assume that they understand the material. Verify that they do through their responses.

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. Multiple Intelligences

  • Write these two points on the board/overhead. Write that students are going to know all of the theorems and postulates that can prove congruence, and that they are going to understand all of the combinations of sides and angles that do not prove congruence.
  • Use the uncooked spaghetti throughout this lesson with the protractors.
  • As each exercise is described in the text, walk the students through using the uncooked spaghetti and the protractors to test out each theorem.
  • Then use the uncooked spaghetti to show how the AAA does not prove congruence but similarity.
  • Demonstrate this by creating two different size triangles that have the same angle measurements. Then the students will see that although the angle measurements are the same, the triangles are not congruent.
  • Teach the Pythagorean Theorem. Connect this theorem with the HL Congruence Theorem.
  • Make this lesson as interactive as possible by using the protractors and the uncooked spaghetti to model each part of the lesson.
  • Intelligences- linguistic, logical- mathematical, bodily- kinesthetic, visual- spatial, interpersonal.

III. Special Needs/Modifications

  • Write all of the theorems on the board/overhead as they are taught. Request that the students copy these notes down in their notebooks.
  • Review the different types of triangles: acute, obtuse, equilateral, right.
  • Review the parts of a triangle- sides and hypotenuse
  • Review Pythagorean Theorem
  • Show students how to connect the Pythagorean Theorem to the HL Congruence Theorem.

IV. Alternate Assessment

  • Observe students as they work through the lesson with the uncooked spaghetti.
  • Allow time for student thinking and feedback.

sing 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. Multiple Intelligences

  • Divide the students into seven groups. Each group is assigned one of the theorems from the review.
  • Then each group must create a diagram to show how each illustrates or does not illustrate congruence.
  • Allow time for the students to work.
  • When they have finished, allow time for each group to present their work to the class.
  • When completing constructions, be sure that students have both a compass and a straightedge.
  • Review the steps in the text on drawing a perpendicular bisector of the segment.
  • Expand Example 4- Have the students practice drawing segments and practice drawing perpendicular bisectors of each segment.
  • Students could work in pairs on this lesson.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal

III. Special Needs/Modifications

  • Complete the Congruence Theorem Review. Create a chart with all of this information and be sure that students copy it into their notebooks.
  • Write out the notes for proving parts congruent.
  • Be sure students understand that they need to use the distance formula, and the reflexive property of congruence, but that using a protractor does not necessarily mean that the triangles are congruent but similar.

IV. Alternative Assessment

  • Create a rubric to grade students on their group presentations.
  • Share the rubric with them prior to assigning the group work.
  • Then use the rubric to give students a quiz or class work grade.

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. Multiple Intelligences

  • In this lesson, the students are going to work with the Base Angles Theorem.
  • They are going to need to prove the Base Angles Theorem with both isosceles and equilateral triangles.
  • Prior to teaching the lesson, ask students to recall information about isosceles and equilateral triangles. Ask them to make a list of the characteristics of each in their notebooks.
  • When finished, use a class discussion to generate a list of characteristics for both isosceles and equilateral triangles on the board.
  • Then present the information in the text.
  • As you teach about the Base Angles Theorem, point out which characteristics apply when working with this theorem.
  • Do this for both the isosceles triangle and the equilateral triangle.
  • Be sure that the students take notes on both triangle examples.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal.

III. Special Needs/Modifications

  • Review isosceles triangles and their parts on the board.
  • Present the material in words and in a diagram.
  • Review converse statements. Be sure that students understand converse statements.
  • Review equilateral triangles and their parts on the board.
  • Present the material in words and in a diagram.
  • Write out any points or conclusions that you make while discussing this lesson with the students.
  • Write this information out on the board and request that the students copy these notes down in their notebooks.

IV. Alternative Assessment

  • Alternative Assessment can be completed through observation and listening during the brainstorming session and during the discussion.

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. Multiple Intelligences

  • To differentiate this lesson, begin by teaching all of the content in the lesson. The content will be necessary to do this activity.
  • Begin the activity by giving students the coordinates of one triangle on the coordinate grid.
  • Have students use colored pencils and a ruler during this lesson.
  • Request that the students graph this triangle on the coordinate grid in one color.
  • Then have students take this triangle and draw each of the following by using this triangle as a starting point.
  • Students are going to have multiple pages of diagrams when they are finished.
  • One page consists of a translation or slide.
  • One page consists of a reflection or flip.
  • One page consists of a rotation or turn.
  • One page consists of a dilation.
  • Stress the point that a dilation is the only image where the two images are not congruent. The two images are similar.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Write all new vocabulary words on the board.
  • Request that students copy these words in their notebooks.
  • Review the meaning of clockwise and counterclockwise.
  • Review the difference between things that are congruent and things that are similar.

IV. Alternative Assessment

  • Collect the packet of diagrams/drawings that the students have created during this lesson.
  • This packet will allow you to assess student understanding of the information presented in this lesson.
  • If letter grades are used, than a classwork or quiz grade can be given for this packet.

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