# Lesson 10

Other Conditions for Triangle Similarity

## 10.1: Math Talk: Triangle Congruence (5 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for triangle congruence shortcuts. These understandings help students recall what they learned in a previous unit and will be helpful later in this lesson when students will need to be able to use these shortcuts in proofs of similarity. In this activity, students have an opportunity to notice and make use of structure (MP7) as they choose which criteria apply to a certain case. Since the images below have multiple ways they could be proven to be congruent, comparing ideas brings out structure.

### 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

Evaluate mentally. Is there enough information to determine if the pairs of triangles are congruent? If so, what theorem(s) would you use? If not, what additional piece of information could you use?

### 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 _____’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 _____’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)

## 10.2: Side-Angle-Side Triangle Similarity? (15 minutes)

### Optional activity

In a previous lesson, students studied the connection between the Angle-Side-Angle Triangle Congruence Theorem and the Angle-Angle Triangle Similarity Theorem. This activity invites students to conjecture about the Side-Angle-Side Triangle Similarity Theorem. A key concept is that students need to define what must be true about the sides for “Side-Angle-Side” to stand for something useful for proving that triangles are similar.

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

### Launch

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion about the proof of the Side-Angle-Side Triangle Similarity Theorem. After students prove there is a sequence of transformations that takes triangle $$ABC$$ to $$DEF$$, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their proofs. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the various methods of proving the Side-Angle-Side Triangle Similarity Theorem.
Design Principle(s): Cultivate conversation
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “The sides in similar triangles are . . .”
Supports accessibility for: Language; Social-emotional skills

### Student Facing

Andre remembers lots of ways to prove triangles congruent. He asks Clare, “Can we use Angle-Side-Angle to prove triangles are similar?”

Clare: “Sure, but we don’t need the Side part because Angle-Angle is enough to prove triangles are similar.”

Andre: “Hmm, what about Side-Angle-Side or Side-Side-Side? What if we don’t know 2 angles?”

Clare: “Oh! I don’t know. Let’s draw a picture and see if we can prove it.”

Andre: “Uh-oh. If ‘side’ means corresponding sides with the same length, then we’ll only get congruent triangles.”

1. What could ‘side’ stand for to prove triangles similar?
2. Draw a diagram that would help you prove the Side-Angle-Side Triangle Similarity Theorem.
3. Write a proof.

### Anticipated Misconceptions

Make sure that students are only labeling the information they know in their diagram, so only a single pair of corresponding angles should be labeled as congruent.

### Activity Synthesis

The main idea to draw out of this activity is that knowing that the Side-Angle-Side Triangle Congruence Theorem is true makes it much easier to prove the Side-Angle-Side Triangle Similarity Theorem.

Some students might use the dilation-first argument, and other students might define a specific sequence of rigid motions and a dilation without mentioning the Side-Angle-Side Triangle Congruence Theorem. Compare the two methods and discuss how the Side-Angle-Side Triangle Congruence Theorem gives us an opportunity to shorten our proof by making use of structure.

## 10.3: Side-Side-Side Triangle Similarity (15 minutes)

### Optional activity

In a previous lesson, students studied the connection between the Angle-Side-Angle Triangle Congruence Theorem and the Angle-Angle Triangle Similarity Theorem. In a previous activity, students proved the Side-Angle-Side Triangle Similarity Theorem. This activity gives students an opportunity to apply what they learned about that proof to prove the Side-Side-Side Triangle Similarity Theorem.

### Launch

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the proof of the Side-Side-Side Triangle Similarity Theorem. 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 $$DEF$$?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why any pair of triangles with three pairs of corresponding proportional sides must be similar.
Design Principle(s): Optimize output (for justification); Cultivate conversation

### Student Facing

Prove that these 2 triangles must be similar.

### Student Facing

#### Are you ready for more?

Prove or disprove the Side-Side-Angle Triangle Similarity Theorem.

### Activity Synthesis

Some students might use the dilation-first argument outlined in the student response, and other students might define a specific sequence of rigid motions and a dilation without mentioning the Side-Side-Side Triangle Similarity Theorem. This provides an opportunity to compare the two methods and discuss how the Side-Side-Side Triangle Similarity Theorem gives an opportunity to shorten the proof by making use of structure.

## Lesson Synthesis

### Lesson Synthesis

Ask students, “Which of these pairs of triangles can we now prove are similar using one of our shortcuts?” (The pair of triangles in A are similar by the Angle-Angle Triangle Similarity Theorem. The pair of triangles in B are similar by the Side-Angle-Side Triangle Similarity Theorem. There’s not enough information to decide if the triangles in C or D are similar. The triangles in E are congruent by the Side-Side-Side Triangle Congruence Theorem and all congruent triangles are similar.)

## Student Lesson Summary

### Student Facing

Besides the Angle-Angle Triangle Similarity Theorem, what other criteria are sufficient to prove triangles similar?

When 2 sides of one triangle are proportional to 2 corresponding sides of a second triangle using the same scale factor $$k$$, and the pair of angles between these sides are congruent, the triangles are similar by the Side-Angle-Side Triangle Similarity Theorem.

For example, angles $$EDF$$ and $$BDC$$ are vertical angles and so they are congruent, and there are 2 pairs of corresponding sides with the same scale factor.

Dilate triangle $$DEF$$ using center $$D$$ and scale factor $$k$$. Since $$\frac{BD}{ED}=\frac{CD}{FD} = k$$, $$BD$$ is now congruent to $$E'D$$, and $$CD$$ is congruent to $$F'D$$. The dilation did not change the size of the angles. Therefore, triangle $$E'DF'$$ is congruent to triangle $$BDC$$ by the Side-Angle-Side Triangle Congruence Theorem. This means there is a sequence of rigid motions that takes triangle $$E'DF'$$ to triangle $$BDC$$. That means triangle $$BDC$$ is similar to triangle $$EDF$$ because there is a dilation and a sequence of rigid motions that takes one to the other. There wasn’t anything special about these 2 triangles, therefore, any pair of triangles with 2 pairs of sides whose lengths are in the same proportion and with the angle between them congruent must be similar.
We can also show that if all 3 pairs of corresponding sides are proportional and use the same scale factor $$k$$, this is sufficient to prove the triangles are similar. We call this the Side-Side-Side Triangle Similarity Theorem.