Lesson 9

The purpose of this Math Talk is to elicit strategies and understandings students have for using the triangle congruence theorems (in particular, Angle-Side-Angle) as part of proving that triangles are similar. These understandings will be helpful later in this lesson when students will need to be able to use the Angle-Side-Angle Triangle Congruence Theorem to prove the Angle-Angle Triangle Similarity Theorem by showing that a strategic dilation of one of the triangles will create the conditions for the Angle-Side-Angle Triangle Congruence Theorem, and that is enough to prove similarity.

In this activity, students have an opportunity to notice and make use of structure (MP7), because they are asked to make a connection between a theorem they have used many times before, and a new situation in which they can apply it if they figure out how to exploit the structure of the situation.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

While the Angle-Angle Triangle Similarity Theorem follows logically from the Angle-Side-Angle Triangle Congruence Theorem (dilate one triangle by a scale factor determined by the ratio of the sides between the known corresponding angle pairs, and the dilated triangle will be congruent to the target by the Angle-Side-Angle Triangle Congruence Theorem, so the original triangle must be similar to the target), it can still be unintuitive. Students experienced the Angle-Angle Triangle Similarity Theorem briefly in a previous unit when they looked at examples of triangles constructed so their angles were the same but side lengths were undetermined, and in middle school, students confirmed the Angle-Angle Triangle Similarity Theorem informally. However, their proofs until this point have tended to rely on knowing something about the proportionality of side lengths. Students might be unsure that the same scale factor will work for all three pairs of corresponding sides when all they know is the measure of two angles of the triangle. It is a surprising result when you put it that way. This activity starts by building students’ intuition about the Angle-Angle Triangle Similarity Theorem, to motivate the proof of the theorem.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Give students a few minutes of work time to draw the triangles. If multiple students are struggling, pause for a brief whole-class discussion. Invite a student to demonstrate their technique for drawing a triangle with one 45 degree angle and one 30 degree angle. Remind students they can choose whatever side lengths they want if that information isn’t specified.

If students struggle with the proof of their conjecture, invite them to go back to their examples. Here are some questions that can help scaffold the reasoning.

• Is there a dilation you could use that would make those triangles congruent? How does that help you prove they are similar? (Yes, dilate the smaller triangle with a scale factor of 3. Then I can show there’s a dilation and sequence of rigid motions that takes one triangle onto the other, so they’re similar.)
• How many sides did you need to have to figure out a dilation for the triangles with sides given? Why? Could you measure fewer sides? (I only needed one side to figure out I could scale the triangle by a scale factor of 3. I could measure just one pair of sides, and scale up by the right scale factor to make those sides match.)

The goal of this discussion is for students to be convinced that the Angle-Angle Triangle Similarity Theorem is a valid way to prove two triangles are similar. 

Invite students to summarize the main points of the proof:

• Choose the pair of corresponding sides between the given angles to generate a scale factor.
• Dilate one triangle using that scale factor.
• The triangles must be congruent now based on the Angle-Side-Angle Triangle Congruence Theorem.
• The triangles can be taken onto each other by dilation and rigid motions. Therefore, they are similar.

Ask students to add this theorem to their reference charts as you add it to the class reference chart:

Angle-Angle Triangle Similarity Theorem: In two triangles, if two pairs of corresponding angles are congruent, then the triangles must be similar. (Theorem)

\(\angle A \cong \angle C, \angle DEA \cong \angle BEC,\) so \(\triangle DEA \sim \triangle BEC\)

In later lessons, students will encounter situations in which they need to figure out if they have enough information to be sure two triangles are similar, based on angle measure alone. This activity introduces that concept as students grapple with whether the given information is enough to figure out that at least two pairs of corresponding angles are congruent.

Students will need to use their knowledge of the Triangle Angle Sum Theorem to figure out the third angle in each triangle.

If students are struggling unproductively, ask them to share what’s hard about this problem. (I’m stuck because I don’t have two pairs of corresponding angles.) Then ask them if it’s possible to find another corresponding angle using the information provided. (Yes, using the Triangle Angle Sum Theorem to figure out the third angle and check if the third pair of angles might be congruent.)

The key idea is that the Triangle Angle Sum Theorem allows us to use the Angle-Angle Triangle Similarity Theorem even when we haven’t been given two pairs of congruent corresponding angles. Invite students to explain how they could tell the triangles were similar.