# Lesson 7

Reasoning about Similarity with Transformations

## 7.1: Notice and Wonder: Nested Triangles (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit conjectures about similar triangles and dilation, which will be useful when students formalize and prove these conjectures in later activities in this lesson. While students may notice and wonder many things about these images, proportional lengths, congruent angles, and dilation are the important discussion points. This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that the smaller triangle is a dilation of the larger triangle.

### Launch

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

### Student Facing

What do you notice? What do you wonder?

### Student Response

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### Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

Rewrite a few of the students’ contributions as conjectures. Select things students wondered about proportional lengths, congruent angles, and dilation, if available.

## 7.2: Stretched or Distorted? Triangles (10 minutes)

### Activity

In previous lessons, students practiced coming up with sequences of rigid motions and dilations to take one figure onto another given similar figures. In this activity, students work to find a sequence of rigid motions and dilations that will take one triangle onto another for any pair of triangles with all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion.

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

### Launch

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.

Emphasize to students that their sketches are just a tool to help them think about the transformations needed. Since the sketches don’t need to be super-precise, their transformations might not line everything up perfectly.

*Provide Access for Physical Action.*Provide access to tools and assistive technologies such as dynamic geometry software or large-scale grid paper.

*Supports accessibility for: Visual-spatial processing; Conceptual processing; Organization*

### Student Facing

- Sketch 2 triangles with all pairs of corresponding angles congruent, and with all pairs of corresponding side lengths in the same proportion.
- Label your triangles \(ABC\) and \(DEF\) so that angle \(A\) is congruent to angle \(D\), angle \(B\) is congruent to angle \(E\), and angle \(C\) is congruent to angle \(F\). Label each side with its length.
- Do the 2 triangles you drew fit the definition of similar? Explain your reasoning.
- Switch sketches with your partner. Find a sequence of rigid motions and dilations that will take one of their triangles onto the other. Will the same sequence work for your triangles?

### Student Response

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### Student Facing

#### Are you ready for more?

How many sequences are there that take one similar triangle to the other? Explain or show your reasoning.

### Student Response

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### Anticipated Misconceptions

If students are stuck on sketching a pair of triangles that fit all the characteristics, suggest that they start by drawing any triangle. Then they can measure it and use that information to create their second triangle.

### Activity Synthesis

Invite a few students to demonstrate their transformations. Students will be generalizing by playing Invisible Triangles in a subsequent activity, so it is not necessary to generalize the sequence at this point.

*Speaking: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. As students share their sequence of transformations that take one triangle to the other, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped clarify the original explanation, such as, “Dilate triangle \(ABC\) using center _____ by the scale factor _____.” This provides more students with an opportunity to produce language as they interpret the reasoning of others.

*Design Principle(s): Support sense-making*

## 7.3: Invisible Triangles: Similarity (20 minutes)

### Activity

Students play a new version of the Invisible Triangles game to generalize, and then they work as a class to prove that they really have come up with a sequence of rigid motions and dilations that they can justify will take any pair of triangles with all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion onto each other.

Monitor for students who dilate first then use rigid motions as well as students who use rigid motions first and then dilate.

### Launch

Arrange students in groups of 2. Distribute one transformer card and one set of three playing cards to each group. If feasible, have students use folders or books so that students can’t see each other’s desktops.

*Conversing: MLR2 Collect and Display.*As students work on this activity, listen for and collect the language students use to describe the sequence of transformations that takes one triangle to the other. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a phrase such as, “Move the triangle so \(A\) matches with \(D\),” can be improved by restating it as, “Translate triangle \(ABC\) using the directed line segment from \(A\) to \(D\).” This will help students use the mathematical language necessary to precisely describe the sequence of transformations that takes triangle \(ABC\) to \(DEF\).

*Design Principle(s): Optimize output (for explanation); Maximize meta-awareness*

*Action and Expression: Provide Access for Physical Action.*Provide access to tools and assistive technologies such as dynamic geometry software.

*Supports accessibility for: Visual-spatial processing; Conceptual processing; Organization*

### Student Facing

Player 1: You are the transformer. Take the transformer card.

Player 2: Select a triangle card. Do not show it to anyone. Study the diagram to figure out which sides and which angles correspond. Tell Player 1 what you have figured out.

Player 1: Take notes about what they tell you so that you know which parts of their triangles correspond. Think of a sequence of rigid motions and dilations you could tell your partner to get them to take one of their triangles onto the other. Be specific in your language. The notes on your card can help with this.

Player 2: Listen to the instructions from the transformer. Use tracing paper to follow their instructions. Draw the image after each step. Let them know when they have lined up 1, 2, or all 3 pairs of vertices on your triangles.

### Student Response

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### Activity Synthesis

Invite students who dilated first to share their sequence:

- Ask students to justify why they knew that the dilation of one triangle would create at least one pair of congruent corresponding sides. (Since the lengths of the corresponding sides are in the same proportion, dilating by the right scale factor would make them congruent. Then we know a sequence of rigid motions exists to take one triangle to the other triangle.)

Invite students who lined up one angle first and then dilated to share their sequence:

- Ask students to justify why they knew it had to be possible to get corresponding angles to coincide using rigid motions. (The angles are congruent, so there must be a sequence of rigid motions to take one to the other.)
- Ask students to justify how they knew a dilation would get all pairs of corresponding vertices to coincide if they used rigid motions first and dilated second. (After we lined up a corresponding angle, since all the pairs of corresponding side lengths are in the same proportion, we knew the triangles were dilations of each other and dilating by the right scale factor would get them to line up exactly.)

Ask students why the proof is shorter if they dilate first. (Once we dilate to get at least one pair of corresponding sides to be congruent, we can see if we have the criteria to use one of the triangle congruence theorems.) If no student dilated first, outline the proof. This technique will come in very handy for proving the Angle-Angle Triangle Similarity Theorem.

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

If two triangles have all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion, then the two triangles are similar. (Theorem)

## Lesson Synthesis

### Lesson Synthesis

A primary goal of this lesson is for students to prove that triangles with all pairs of corresponding side lengths in the same proportion and all pairs of corresponding angles congruent must be similar, because there is a sequence of rigid motions and dilations that will always work to take one triangle onto the other.

Ask students to write an explanation of how we proved that triangles with all pairs of corresponding side lengths in the same proportion and all pairs of corresponding angles congruent must be similar using rigid transformations and dilations.

## 7.4: Cool-down - Not quite similar (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second. By the properties of dilations and rigid motions, similar figures have corresponding angles congruent and pairs of corresponding side lengths in the same proportion.

In the case of triangles, the converse of this statement is true as well. If a pair of triangles has all pairs of corresponding side lengths in the same proportion, and all pairs of corresponding angles congruent, then the triangles must be similar. Imagine any pair of triangles with all pairs of corresponding side lengths in the same proportion, and all pairs of corresponding angles congruent. The same sequence of rigid motions and dilations will work to show that the triangles are similar.

For example, triangle \(EFI\) was dilated using \(E\) as the center by the scale factor given by \(\frac{BC}{EF}\). Because we wisely chose the scale factor this way, we know that side \(BC\) is congruent to side \(E’F’\). We already know that all pairs of corresponding angles are congruent, which means we have enough information to use the Angle-Side-Angle Triangle Congruence Theorem to prove that triangle \(E’F’I’\) is congruent to triangle \(ABC\).

That means that triangle \(ABC\) can be lined up exactly with a dilation of triangle \(EFI\), which is the definition of similarity. It doesn’t matter what the triangles look like or where we start. We can always define a dilation that made one pair of corresponding sides congruent, and then use the Angle-Side-Angle Triangle Congruence Theorem to finish proving that there is a sequence of dilations and rigid motions that takes one triangle onto the other.