Lesson 9

Conditions for Triangle Similarity

9.1: Math Talk: Angle-Side-Angle As A Helpful Tool (10 minutes)

Warm-up

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.

Launch

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.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

How could you justify each statement?

Triangle $$P'Q'R'$$ is congruent to triangle $$STU$$

Triangle $$PQR$$ is similar to triangle $$STU$$

Triangle $$G'H'I'$$ is congruent to triangle $$MNO$$.

Triangle $$GHI$$ is similar to triangle $$MNO$$.

Activity Synthesis

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?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ."  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

9.2: How Many Pieces? (15 minutes)

Activity

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).

Launch

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.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the proof that triangles with two pairs of congruent corresponding angles are similar. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What is the definition of similarity?”, “What is the center and scale factor of the dilation?”, and “How do you know that triangle $$A’B’C’$$ is congruent to $$PQR$$?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why triangles with two pairs of congruent corresponding angles are similar.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Provide Access for Physical Action. Provide access to tools and assistive technologies such as dynamic geometry software, whiteboards or large scale grid paper.
Supports accessibility for: Visual-spatial processing; Conceptual processing; Organization

Student Facing

For each problem, draw 2 triangles that have the listed properties. Try to make them as different as possible.

1. One angle is 45 degrees.
2. One angle is 45 degrees and another angle is 30 degrees.
3. One angle is 45 degrees and another angle is 30 degrees. The lengths of a pair of corresponding sides are 2 cm and 6 cm.
4. Compare your triangles with your neighbors’ triangles. Which ones seem to be similar no matter what?

Anticipated Misconceptions

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.)

Activity Synthesis

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)

9.3: Any Two Angles? (10 minutes)

Activity

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.

Launch

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to increase awareness of the language used to talk about the features of similar triangles. Before revealing the questions in this activity, display the image of both triangles. Ask students to write down possible mathematical questions that could be asked about the image. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the features of both triangles. For example, “Are the triangles similar?”, “What are the values of $$x$$ and $$y$$?”, and “What are the measures of the missing angles?”
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

Here are 2 triangles. One triangle has a 60 degree angle and a 40 degree angle. The other triangle has a 40 degree angle and an 80 degree angle.

1. Explain how you know the triangles are similar.
2. How long are the sides labeled $$x$$ and $$y$$?

Anticipated Misconceptions

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.)

Activity Synthesis

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.

Lesson Synthesis

Lesson Synthesis

The main ideas to draw out of this lesson are:

• the Angle-Angle Triangle Similarity Theorem: if two pairs of corresponding angles in triangles are congruent, then the triangles are similar
• Knowing the measures of any two angles from one triangle, and any two angles of the other triangle, is enough information to determine if the Angle-Angle Triangle Similarity Theorem can be used. The given angles do not have to be corresponding.

Invite students to make up their own examples of pairs of triangles that are similar and pairs of triangles that are not similar. If students are familiar with the game, they could set up their examples as two truths and a lie. Ask students to trade with a partner and justify which triangles are similar.

9.4: Cool-down - Any Four Angles? (5 minutes)

Cool-Down

In the diagram, angle $$A$$ is congruent to angle $$D$$, and angle $$B$$ is congruent to angle $$E$$. If a sequence of rigid motions and dilations moves the first figure so that it fits exactly over the second, then we have shown that the Angle-Angle Triangle Similarity Theorem is true.
Dilate triangle $$ABC$$ by the ratio $$\frac{DE}{AB}$$, so that $$A’B’$$ is congruent to $$DE$$. Now triangle $$A’B’C’$$ is congruent to triangle $$DEF$$ by the Angle-Side-Angle Triangle Congruence Theorem, which means there is a sequence of rotations, reflections, and translations that takes $$A’B’C’$$ onto $$DEF$$.
Therefore, a dilation followed by a sequence of rotations, reflections, and translations will take triangle $$ABC$$ onto triangle $$DEF$$, which is the definition of similarity. We have shown that a dilation and a sequence of rigid motions takes triangle $$ABC$$ to triangle $$DEF$$, so the triangles are similar.