# Lesson 3

Congruent Triangles, Part 1

## 3.1: True or . . . Sometimes True?: Triangles (5 minutes)

### Warm-up

This warm-up both reviews previous work on corresponding parts of congruent figures and previews the subsequent activity in which students will use rigid transformations to take one figure onto another.

Students start by drawing conclusions based on a congruence statement, and then are given a specific figure to reason about. This gives the opportunity to talk about how, often in geometry, the static picture is just one example of many different possible images that match the given information.

### Launch

First, display the congruence statement \(\triangle ABC \cong \triangle A’B’C’\) for all to see. Tell students their job is to think of at least one thing they know must be true, one thing that could possibly be true, and one thing that definitely can’t be true about each statement or image. Give students 1 minute of quiet think time, and then 1 minute to discuss the things on their lists with their partner.

Then instruct students to view the applet. Give students an additional 1 minute of quiet think time to add to or change their lists based on what they saw.

### Student Facing

- What must be true?
- What could possibly be true?
- What definitely can’t be true?

### Student Response

For access, consult one of our IM Certified Partners.

### Launch

Arrange students in groups of 2. First, display the congruence statement, “Triangle \(ABC\) is congruent to triangle \(A'B'C'\).” for all to see. Tell students their job is to think of at least one thing they know must be true, one thing that could possibly be true, and one thing that definitely can’t be true about each statement or image. Give students 1 minute of quiet think time, and then 1 minute to discuss the things on their lists with their partner. Then display the image for all to see. Give students an additional 1 minute of quiet think time to add to or change their lists based on what they see.

### Student Facing

If triangle \(ABC\) is congruent to triangle \(A'B'C'\). . .

- What must be true?
- What could possibly be true?
- What definitely can’t be true?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share what they wrote down before they saw the image. Next, invite a student to mark congruent sides and angles of the image for all to see.

If students made claims like segment \(AC\) can’t possibly be congruent to segment \(B’C’\), ask for a counterexample. If you have access to the digital version of the curriculum, drag points on the applet to demonstrate the case in which triangle \(ABC\) is isosceles with segment \(AC\) congruent to segment \(BC\).

## 3.2: Invisible Triangles (20 minutes)

### Activity

The goal of this activity is for students to generalize the transformations that they have used in specific cases, to be able to assert that a certain sequence of rigid motions will always work to take a triangle onto a congruent triangle, no matter how the triangles are initially arranged.

If students complete the cards in order, they will build from 1 to 3 transformations required. For more challenge, consider shuffling the cards.

Monitor for groups that are trying different sequences of rigid motions each time, and for groups that have a clear routine.

### 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. You may choose to demonstrate one round of the game for the students, with you holding the triangle card, and the students working together to be the transformer.

For demonstration, you can either use this image or use the applet here, ggbm.at/cwtjcwvv, but do not show it to students until after the round is over.

Display the transformer sentence stems from the transformer card for students.

Tell students: \(\overline {JK} \cong \overline {PQ}, \overline {JL} \cong \overline {PR}, \overline {KL} \cong \overline {QR}, \angle J \cong \angle P, \angle K \cong \angle Q, \angle L \cong \angle R\).

Do not display the triangles in the figure. Instead, as students give you transformation instructions, use tracing paper to perform the requested transformations. Record each step as you go. After each transformation, tell students how many vertices coincide.

Once all three vertices coincide (or after a few minutes, if the struggle is unproductive), display the triangles for the students, with the intermediate transformations recorded.

As students play the game, support the transformers to use the language on the cards to give precise statements, and support the students with triangle cards to ask questions, accurately carry out their partner’s instructions, and record each step as they go.

*Conversing: MLR2 Collect and Display.*As students work on the Invisible Triangle activity, listen for and collect the language students use to describe the sequence of rigid motions that takes one of the triangles onto the other. Capture student language that suggests a systematic method for lining up the vertices of congruent triangles, regardless of starting orientation. Write the students’ words and phrases on a visual display. As students review the visual display, ask students to revise and improve how rigid transformations are communicated. Encourage students to use the sentence frames on the Transformer card. This will help students use the mathematical language necessary to precisely describe a sequence of rigid motions.

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

*Action and Expression: Internalize Executive Functions.*Begin with a whole-class demonstration and think aloud of the game to remind students how to use rigid transformations to show congruence. Keep the given congruence statements on display for students to reference as they work.

*Supports accessibility*

*for:*

*Memory; Conceptual processing*

### 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 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 vertices on your triangles.

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

Replay invisible triangles, but with a twist. This time, the transformer can only use reflections—the last 2 sentence frames on the transformer card. You may wish to include an additional sentence frame: Reflect _____ across the angle bisector of angle _____.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

If students are asking their partner, “Is it a rotation? Is it a reflection?”, instead, encourage them to use the sentence frames. Can they think of a transformation that will definitely line up at least one pair of corresponding points or segments?

If students are not clarifying whether they want to transform points, segments, or the whole triangle, direct them to the sentence frames. “What’s the goal?” (To take one triangle onto the other.) “So what goes in that first blank?” (A triangle.)

### Activity Synthesis

The goal is to make sure students have a set of transformations they feel confident using to take one triangle onto another congruent triangle, regardless of starting orientation.

Ask students to share what was difficult about the game. Invite a group that initially struggled but ultimately developed a systematic way to line up the vertices, to share their method and process. If no group had that experience, invite a group that was confident in their sequence of transformations from the beginning to share their strategy.

If many groups struggled with the game, have a group with more success teach their strategy to the class, and either play an additional round, or have the class play you.

## 3.3: Why Do They Coincide? (10 minutes)

### Activity

The goal of the activity is to develop informal reasoning about which properties of rigid transformations and which items of given information are useful in writing a triangle congruence proof. This activity presents a partially completed proof that a sequence of a translation, reflection, and rotation will always work to take one triangle to exactly coincide with another if all the pairs of corresponding parts are congruent.

### Launch

*Reading, Listening, Conversing: MLR6 Three Reads.* Use this routine to support reading comprehension of this word problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about (Noah and Priya were playing Invisible Triangles with Card 3.). Use the second read to identify geometric relationships. Listen for and amplify important relationships such as \(\overline{AB} \cong \overline{DE}\), \(\angle A \cong \angle D\), and \(\overline{AC} \cong \overline{DF}\). After the third read, ask students to brainstorm possible strategies to answer the first question, “Why do points \(B’’\) and \(E\) have to be in the exact same place?”. This routine will help students make sense of the language in the word problem and the reasoning needed to solve the problem.

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

### Student Facing

Noah and Priya were playing Invisible Triangles. For card 3, Priya told Noah that in triangles \(ABC\) and \(DEF\):

- \(\overline {AB} \cong \overline {DE}\)
- \(\overline {AC} \cong \overline {DF}\)
- \(\overline {BC} \cong \overline {EF}\)
- \(\angle A \cong \angle D\)
- \(\angle B \cong \angle E\)
- \(\angle C \cong \angle F\)

Here are the steps Noah had to tell Priya to do before all 3 vertices coincided:

- Translate triangle \(ABC\) by the directed line segment from \(A\) to \(D\).
- Rotate the image, triangle \(A’B’C’\), using \(D\) as the center, so that rays \(A’’B’’\) and \(DE\) line up.
- Reflect the image, triangle \(A’’B’’C’’\), across line \(DE\).

After those steps, the triangles were lined up perfectly. Now Noah and Priya are working on explaining why their steps worked, and they need some help. Answer their questions.

First, we translate triangle \(ABC\) by the directed line segment from \(A\) to \(D\). Point \(A’\) will coincide with \(D\) because we defined our transformation that way. Then, rotate the image, triangle \(A’B’C’\), by the angle \(B’DE\), so that rays \(A’’B’’\) and \(DE\) line up.

- We know that rays \(A’’B’’\) and \(DE\) line up because we said they had to, but why do points \(B’’\) and \(E\) have to be in the exact same place?
- Finally, reflect the image, triangle \(A’’B’’C’’\) across \(DE\).
- How do we know that now, the image of ray \(A’’C’’\) and ray \(DF\) will line up?
- How do we know that the image of point \(C’’\) and point \(F\) will line up exactly?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

If students struggle with the task, encourage them to find the triangle card from Invisible Triangles that required a translation, rotation, and reflection, and work through Noah and Priya’s steps. As they perform the transformations, why do points and rays coincide?

### Activity Synthesis

The goal of this discussion is to make sure all students understand why we can be sure that points and rays coincide after translating, rotating, and reflecting. Invite students to share their reasoning for each part.

“Why does Noah say ‘so that *rays* \(A''B''\) and \(DE\) line up’ rather than ‘so that *segments* \(A''B''\) and \(DE\) line up?’” (There are two steps to this part of the proof. First, use the angle to get the rays to coincide. Then, use the side length to get the points to coincide. You can‘t say the segments coincide until you‘ve given both reasons.)

## Lesson Synthesis

### Lesson Synthesis

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

If all pairs of corresponding sides and all pairs of corresponding angles are congruent, then the triangles must be congruent. (Theorem)

Invite students to share the big ideas of a transformation proof. (You do a transformation and then use what you know about a side length or angle to prove that corresponding parts coincide.)

Create a display of sentence frames for proofs. This display should be posted in the classroom for the remaining lessons within this unit. There is a blackline master with the final version; it will be built over several lessons. After this lesson, it should look something like:

Transformations:

- Translate \(\underline{\hspace{1in}}\) from \(\underline{\hspace{1in}}\) to \(\underline{\hspace{1in}}\).
- Rotate \(\underline{\hspace{1in}}\) using \(\underline{\hspace{1in}}\) as the center by angle \(\underline{\hspace{1in}}\).
- Rotate \(\underline{\hspace{1in}}\) using \(\underline{\hspace{1in}}\) as the center so that \(\underline{\hspace{1in}}\) coincides with \(\underline{\hspace{1in}}\).
- Reflect \(\underline{\hspace{1in}}\) across \(\underline{\hspace{1in}}\).
- Reflect \(\underline{\hspace{1in}}\) across the perpendicular bisector of \(\underline{\hspace{1in}}\).

Justifications:

- We know the image of \(\underline{\hspace{1in}}\) is congruent to \(\underline{\hspace{1in}}\) because rigid motions preserve measure.
- Points \(\underline{\hspace{1in}}\) and \(\underline{\hspace{1in}}\) coincide after translating because we defined our translation that way!
- Since points \(\underline{\hspace{1in}}\) and \(\underline{\hspace{1in}}\) are the same distance along the same ray from \(\underline{\hspace{1in}}\), they have to be in the same place.
- Rays \(\underline{\hspace{1in}}\) and \(\underline{\hspace{1in}}\) coincide after rotating because we defined our rotation that way!
- The image of \(\underline{\hspace{1in}}\)must be on ray \(\underline{\hspace{1in}}\) since both \(\underline{\hspace{1in}}\) and \(\underline{\hspace{1in}}\) are on the same side of \(\underline{\hspace{1in}}\) and make the same angle with it at \(\underline{\hspace{1in}}\).

## 3.4: Cool-down - Reflecting on Proof (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

If all corresponding parts of 2 triangles are congruent, then one triangle can be taken exactly onto the other triangle using a sequence of translations, rotations, and reflections. The congruence of corresponding parts justifies that the vertices of the triangles will line up exactly.

One of the most common ways to line up the vertices is through a translation to get one pair of vertices to line up, followed by a rotation to get a second pair of vertices to line up, and if needed, a reflection to get the third pair of vertices lined up. There are multiple ways to justify why the vertices must line up if the triangles are congruent, but here is one way to do it:

First, translate triangle \(ABC\) by the directed line segment from \(A\) to \(D\). Points \(A\) and \(D\) coincide after translating because we defined our translation that way! Then, rotate the image of triangle \(ABC\) using \(D\) as the center, so that rays \(A’B’\) and \(DE\) line up.

We know that rays \(A’B’\) and \(DE\) line up because that’s how we defined the rotation. The distance \(AB\) is the same as the distance \(DE\), because translations and rotations don’t change distances. Since points \(B’\) and \(E\) are the same distance along the same ray from \(D\), they have to be in the same place.

If necessary, reflect triangle \(A’B’C’\) across \(DE\) so that the image of \(C\) is on the same side of \(DE\) as \(F\). We know angle \(A\) is congruent to angle \(D\) because translation, rotation, and reflection don’t change angles.

\(C''\) must be on ray \(DF\) since both \(C''\) and \(F\) are on the same side of \(DE\) and make the same angle with it at \(D\). We know the distance \(AC\) is the same as the distance \(DF\), so that means \(C’’\) is the same distance from \(A’’\) as \(F\) is from \(D\) (because translations and rotations preserve distance). Since \(F\) and \(C’’\) are the same distance along the same ray from \(D\), they have to be in the same place.