Student/Teacher Actions (what students and teachers should be doing to facilitate learning)
Define similar triangles, and compare the definition to that of congruent triangles. Review congruence shortcuts with students, and discuss why AAA is not a congruence shortcut. Tell students that they will be exploring similarity shortcuts.
Have students work in pairs to complete Activity Sheet 1. Alternatively, applets on the similarity theorems can be found at Web sites for National Council of Teachers of Mathematics, Math Open Reference, or the National Library of Virtual Manipulatives, by searching for similar triangles applets. AA can be explored using congruence theorems. Each student should record his/her own findings. Have students discuss findings with their partners. Discuss findings as a whole group.
Have students work in pairs to complete Activity Sheet 2. Each student should record his/her own findings. Have students discuss findings with their partners. Discuss findings as a whole group.
Have students work in pairs to complete Activity Sheet 3. Each student should record his/her own findings. Have students discuss findings with their partners. Discuss findings as a whole group.
Why don’t we call AA Similarity AAA Similarity?
What are three methods for proving triangles are similar? Explain each, and include diagrams.
Is it true that all isosceles triangles with a 100-degree angle are similar? Explain.
Is it true that all isosceles triangles with a 40-degree angle are similar? Explain.
Are all isosceles triangles similar? Explain why or why not, or give a counterexample.
Complete a journal entry summarizing the activities.
Describe a real-world example we have not seen in class that uses ratio and proportion.
Describe a real-world example we have not seen in class that uses similar triangles.
Explain how congruence and similarity are related.
Have students complete Activity 3 for a different triangle.
Have groups of students create problems by drawing two triangles and labeling them with information that can be used to determine whether the triangles are similar. Each group should create one problem with a separate solution for each method (AA similarity, SSS similarity, and SAS similarity). Have groups swap sets of problems with other groups.
Extensions and Connections (for all students)
Have students investigate patterns of congruence and similarity in the “real world.”
Have students explore scale drawings.
Have students explore indirect measurement, the use of a hypsometer, or surveying methods.
Have students explore similar right triangles formed by an altitude drawn to the hypotenuse of a right triangle.
Does SSSS similarity work for quadrilaterals? Have students draw two quadrilaterals whose corresponding sides have ratio 1:2 but are not similar.
Invite a surveyor or carpenter to come to the class to demonstrate how he or she uses indirect measurement when surveying land or measuring unreachable spaces.
Assign students to work in groups of three-to-four to plan and construct models out of paper sticks using the midpoint of a triangle. This could illustrate why using this practice is aesthetically pleasing.
Working in groups, assign students a real-world situation such as planning a triangular-shaped garden to be broken into equal-sized portions.
Strategies for Differentiation
Provide dynamic geometry software packages for students to use.
Have students use a spreadsheet to tabulate and graph class records.
Use colors so that corresponding sides are the same color. Mark the vertices of corresponding angles with dots of the same color. (Mark sides and opposite angles with the same color.)
Have students color-code each segment of the triangle to visually discriminate differences.
Provide students with paper with a triangle on it, and have them use string to outline each side of the triangle. Have them fold the string in half to arrive at the midpoint of the segment. Once students have found the three midpoints, have them connect the three strings to get the inner triangle.
Provide a large piece of felt (2 ft by 3 ft by 4-1/2 ft). Using additional felt the same size but different colors, cut one of the triangles on the midsegments so there are four similar triangles. Have students use these triangles to demonstrate their understanding of the key points of the lesson, including which triangles are congruent. The felt could be cut ahead of time.
Divide the vocabulary into two groups. One group is the definitions: undefined terms, defined terms, postulates, and theorems. The other group contains geometric shapes: coordinates, midpoints, line segments, similar triangles, and congruent angles. Ask students to divide the subgroups into a vocabulary notebook. For the first list, have students write or dictate an original short story using those terms. For the second list, have students draw or describe true items that appear in their environment and that illustrate these shapes.
Have students cut a sheet of notebook or computer paper into a triangle, folding each tip across its midpoint so that the imprint of four triangles is present on the paper. Have students use the imprinted triangles to illustrate or otherwise identify the key points of the lesson.
Develop a graphic organizer to support the activity sheet.
Activity Sheet 1: Which Triangles Are Similar?
Answer the questions, using the diagram and angle measures shown below.
Is there enough information to determine whether the triangles are congruent? Explain.
What can you say about some of the triangles, using the angle measures? The lengths of the sides are given below.
Are any of the triangles congruent to ? If so, write congruence statements (such as ) for the congruent triangles. (Remember the order of the letters matters!)
Are any of the triangles similar but not congruent to ? If so, write similarity statements (such as ) for the similar triangles.
According to the table of angle measures, the sum of the angle measures foris not 180°. Why do you think this is?
Activity Sheet 2: Similar Triangles Shortcuts
Complete the following tasks, and answer the questions, using a dynamic geometry software package.
Part 1: Is AA a Similarity Shortcut?
In this activity you will explore the following question: If two angles of one triangle are congruent to two angles of another triangle, are the triangles similar?
Open a new sketch.
Draw a triangle, and label it .
Construct a second triangle with and . What can you say about and ? What postulate or theorem allowed you to conclude this?
Measure all the angles of the two triangles to verify that corresponding angles are congruent.
Look at the definition of similar triangles from #1. You know the angles are congruent. What else do you need to know to determine if the triangles are similar?
Measure all the lengths of the sides of the two triangles. Use the software to compute the ratios of lengths of corresponding sides. Are the sides proportional? What can you say about the triangles? This is Trial 1. Complete Trial 1 for the table below.
Are Sides Proportional?
Are Triangles Similar?
Gently move the vertices of , and notice what happens to the angle measures and ratios of corresponding sides. This is Trial 2. Record your findings in the table above.
Repeat step 7 for Trial 3.
Do you think the sides will always be proportional if the angles of two triangles are congruent? Write a conjecture (prediction) about the sides of two triangles with congruent angles.
Write a conjecture about two triangles with congruent angles.
Save your sketch as directed by your teacher.
For the next sketch, you will construct a dilation of a triangle. To do this, you will need to choose a scale factor or dilation factor and a point as the center of the dilation. This will take the lengths of the sides of the original triangle, multiply them by the dilation factor, and create a triangle with those new side lengths.
Part 2: Is SSS a Similarity Shortcut?
In this activity you will explore the following question: If the sides of two triangles are proportional, are the triangles similar?
Dilate by the scale factor you selected, about the point D.
If you cannot see the dilation, or it overlaps , move the point D, or change your scale factor, until you can see the dilation, and it does not overlap .
Measure the lengths of the sides of and . Compute the ratios of the lengths of corresponding sides. Confirm (check) that the sides are proportional.
Gently move the points of the triangles to adjust the lengths of the sides. Confirm that the sides are still proportional.
Measure the angles of the two triangles. What do you notice?
Do you have enough information to determine that and are similar? Explain.
Write a conjecture about two triangles with proportional sides.
Part 3: SAS Similarity
In this activity you will explore the following question: If two sides of two triangles are proportional, and the included angles are congruent, are the triangles similar? Use paper, a ruler and compass for this activity.
Use a ruler, pencil, and paper to draw an angle and label it .
Use a ruler and compass to construct an angle congruent to . Label it .
Mark a point on one of the rays of that is 9 cm from A. Label this point B. Mark a point on the other ray that is 12 cm from A. Label this point C.
Mark a point on one of the rays of that is 6 cm from D. Label this point E. Mark a point on the other ray that is 8 cm from D. Label this point F.
Compute and . Are these two pairs of sides proportional?
Draw and . Measure the other angles of the triangles. Are the triangles similar?
Write a conjecture about two triangles with two pairs of proportional sides and congruent included angles.
Activity Sheet 3: Triangles from Midpoints
Complete the following tasks and questions, using a dynamic geometry software package or paper, straightedge, and compass:
Construct triangle , and label the vertices A, B, and C.
Construct the midpoint of each segment. Label the midpoint of segment , F. Label the midpoint of , D. Label the midpoint of , E.
Draw triangle .
Measure each of the six angles of the triangles and , and record the angle measures here.
What do you notice? Can you show that triangles and are similar? If not, what triangles can you show are similar? Explain the process for planning your proof.
What postulates or theorems would you use to show that your triangles are similar?
Your drawing includes four small triangles. Find the length of each segment, and investigate the relationship among these triangles. (You may need to create the segments first.) Record your findings here.
List all congruent angles and segments. Record your findings here.
What conclusions can you draw about these four small triangles?
How can you prove your conclusion? What postulates or theorems would you use?
Assuming this doesn’t work for all triangles, find a counterexample.
Discuss where it might be helpful to have patterns such as this.