• Your observation during pair and group activities must isolate accuracy of measurement of side lengths and angle measure, identification of corresponding sides and angles, and appropriate use of proportions. Note that misuse of the correct proportion and accurate computation will result in incorrect results. Remind students when evaluating their own results to check the reasonableness of the answer. For example, when using a ratio of corresponding sides of similar polygons, make sure that each proportion is in accordance with the relative sizes of the polygons. [IS.6 - All Students]
• Pair activities are conducive to partners evaluating each other’s results. Direct the evaluator to identify the specific outcome being evaluated, the reasonableness of the result, and the accuracy of the computation. Identify the error as specifically as possible, such as incorrect side or angle correspondence or incorrect use of the similarity ratio, such as using  instead of .

Using the overhead projector and a single volunteer, [IS.3 - Struggling Learners] place the triangle cutout directly on the overhead and have the student trace it. [IS.4 - Struggling Learners] Then, take the triangle and raise it closer to the overhead projector, casting a larger shadow on the overhead. Have the student trace the outline of the shadow onto the overhead. Ask the student to measure both triangles and record the measurements on the overhead projector while you engage the class in a discussion about whether the triangles are congruent or not, and what is the same about the two triangles. The following questions can be used to introduce the concept of similarity:

• “Has the overall shape of the triangle changed?” (no)
• “Have the measures of the angles changed?” (no)
• “Have the measures of the lengths changed?” (yes)

Direct students to draw the triangles, [IS.5 - All Students] both before and after the shapes have changed. Label the triangles First Triangle and Second Triangle and draw them to represent the shapes from the projector as closely as possible. They should use a straightedge, but it is not necessary to measure the sides and angles. The important feature is to record the changes in size of the two triangles while preserving the similarity of their shapes.

Once the measurements of the triangle are recorded, the student can return to his/her seat.

Ask students to make observations about the recorded angle measures (which should be the same) and the recorded side measures. Guide students toward comparing the relative lengths of the size. Use guiding questions like, “Are the sides of the larger triangle two times as large? Three times as large?”

Ask students to determine how many times larger the larger triangle is, compared to the smaller. Record this number on the overhead and label it scale factor. Tell students that these are the concepts they’ll be exploring today.

Provide the following definitions:

Similar Triangles: Two triangles are similar if the three angles of the first triangle are congruent to the three angles of the second triangle and the corresponding sides are all in the same proportion.

Scale Factor: The ratio that relates the lengths of the sides of two similar triangles. Since it is a ratio, it can be written as a fraction , 1:2, or 1 to 2.

Activity 1: Initial Exploration of AA

Divide the class into pairs. Have each pair decide on two angle measurements (they should sum to less than 150 degrees). To make measuring and drawing the triangles as simple as possible, suggest that one angle be at least 10 degrees. Each student should construct a triangle using these two angle measures. The sides can be any length. Ask students:

• “Are the triangles you and your partner constructed congruent?”

(most responses: no)

• “Do they look similar?” (yes)
• “How can you verify that they are similar?”

Tell students to note that errors of measurement, thickness of lines, and straightness of lines may result in angle sums for triangles that are not precisely equal to 180 degrees. These errors occur naturally as a result of drawing and are not necessarily due to work that is incorrect. Have students verify that their triangles are similar by measuring the third angle and also the side lengths. Have them determine the scale factor and also write a similarity statement (i.e., ΔABC is similar to ΔDEF; remind them of their knowledge about writing congruence statements for triangles). Ask students:

• “How much information is needed to know that triangles are similar?” (AA)
• “Is it necessary to measure all the sides and all the angles?” (no)

Introduce the Angle-Angle (AA) Similarity Postulate to the class:

“If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar. Why is this true?” (Because the third angle must also be congruent.)

Activity 2: Real-life Similarity

With setup, obtaining permissions, and moving location of the group, this activity may take a full class period.  Assemble the class into groups of three or four students. Have the groups work outside or in a large room (gymnasium, etc.) where there are shadows of the sun present. If direct sunlight is not available or practical, use instructional spaces where direct lighting is available in sufficient intensity to cast measureable shadows. Make sure that students have paper and pencil. Have one student hold a meter stick perpendicular to the ground, and have another student measure the shadow. A third student should record the lengths and construct a triangle on the paper to represent the real-world situation. This activity also works well with a flashlight.

Ask students:

“What determines the length of the shadow?” (the angle of the sun)

“What causes a shadow to exist?” (the absence of light)

Once they realize that the length of a shadow is determined by the angle of the light, note that the angle of the light does not change significantly if we perform measurements relatively quickly (and the angle of light does not change at all if the activity is being performed indoors).

Now have a student in each group stand straight up and have another student in each group measure the shadow. A third student should construct a triangle to represent this situation.

Ask students what is the same about the two triangles (the one representing the meter stick and the one representing the student). Remind them that the angle of the light has not changed significantly. Guide students toward realizing that the two triangles are similar by AA. Ask students, “Why is the postulate called AA and not AAA?” (AA is the minimum requirement for similarity.)

Have students determine the scale factor (comparing the small triangle to the large triangle). Have each group determine the height of the group member who cast a shadow using scale factor and similar triangles, and then have them measure the actual height of the student to see how accurate they were. Each group should repeat this activity for each other member of the group.

Ask students if they could have, just as easily, measured their heights at the beginning of the activity. Follow up by asking them if it is easier to measure the height of a flagpole, for example, or if it is easier to measure the length of the flagpole’s shadow. Ask for examples of other objects that people may want to determine the heights of, but may be unable to measure directly.

Remind students that the two triangles are similar by the AA postulate, and the two angles that are congruent in each triangle are the right angle at the ground and the angle the sunlight makes with the top of the object (or the ground). By proving the two triangles similar, we can use scale factor to determine unknown lengths.

Activity 3: Creating Similar-Triangle Problems

Have students develop a similar-triangle problem for a classmate to solve. Their problems should include a triangle where the height and shadow of a single object is known (they can use a triangle from the previous activity) and the length of a shadow of a given object is known. Have students trade and solve problems. In the process of solving the problems, students should find and clearly indicate the scale factor for each problem.

Distribute the Similarity worksheet (M-G-4-3_Similarity Worksheet and KEY.doc) as a summarizing activity and review the proportionality relationships between corresponding sides and angles.

Extension:

• The Side Splitting Theorem states that when a line parallel to one side of a triangle intersects the other two sides of the triangle, it divides the two sides into proportional segments.

In ΔGHJ, line KL is parallel to the base GJ. What is the relationship between ΔKLH and ΔGHJ, and how is that relationship proved?

(Answer: Because  is parallel to , the corresponding angles of the transversals HG and HJ are congruent. The two triangles are similar by the AA postulate. Prove that ΔUVW is not congruent to ΔXYZ?)