# Lesson 2

Congruent Parts, Part 2

## 2.1: Math Talk: Which Are Congruent? (5 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for connecting corresponding parts to congruence. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to name corresponding parts accurately and use them in proofs. While participating in this activity, students need to be precise in their word choice and use of language (MP6).

### 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.

### Student Facing

Each pair of figures is congruent. Decide whether each congruence statement is true or false.

Triangle $$ABC$$ is congruent to triangle $$FED$$.

Quadrilateral $$PZJM$$ is congruent to quadrilateral $$LYXB$$.

Triangle $$JKL$$ is congruent to triangle $$QRS$$.

Pentagon $$ABCDE$$ is congruent to pentagon $$PQRST$$.

### Anticipated Misconceptions

If students struggle, encourage them to mentally “connect the dots,” or use their finger to trace around the figure in the order indicated by each set of letters. Students can also use a highlighter to highlight the first pair of corresponding sides (according to how the figures are named) the same color, then highlight the second pair with a second color, etc.

### Activity Synthesis

Ask students to share their strategies for each question. 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 reasoning but would explain it differently?”
• “Did anyone think of it in a different way?”
• “Does anyone want to add on to $$\underline{\hspace{.5in}}$$’s strategy?”
• “Do you agree or disagree? Why?”

Emphasize that yes, all the figures in the pictures are congruent, but we are asking about the figures named by the points. Invite a student to highlight for all to see the corresponding parts of triangles $$JKL$$ and $$QRS$$ as named (use colored pencils to draw over segments $$JK$$ and $$QR$$ in one color, segments $$KL$$ and $$RS$$ in a second color, and segments $$JL$$ and $$QS$$ in a third color). Ask students if the corresponding parts as named are congruent.

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)

## 2.2: Which Triangles Are Congruent? (15 minutes)

### Activity

In a previous lesson, students justified that two figures being congruent guarantees that all pairs of corresponding parts are congruent. In this activity, students explore a different direction. If any pair of corresponding parts is not congruent, then the two figures cannot be congruent.

Monitor for students who:

• focus on corresponding parts having the same measures
• try lining up the triangles (either physically, with tracing paper, or mentally)
• try to use the point-by-point method to prove it’s possible to use rigid transformations to take the figures onto one another

This activity previews the triangle congruence criteria. Here, they are given three pairs of congruent corresponding parts (two side lengths and one angle measurement), which is not enough information to be sure that all triangles with these measurements are congruent (this is an example of the Side-Side-Angle case).

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

### Launch

Consider instructing students how to draw triangles with specified angle and segment measurements using dynamic geometry software. This skill can be used in the activity to check conjectures.

This launch provides instructions for drawing a triangle with specified measurements using GeoGebra. If you choose to use different dynamic geometry software, you may need to provide alternate instructions. If you choose not to instruct students in this skill, you can skip the launch and proceed with the activity.

To draw a triangle with specified measurements using GeoGebra, open the Geometry App from the Math Tools or go to geogebra.org/geometry.

To begin with a measured angle, first construct a segment or a ray, say $$AB$$. Choose the Angle-With-Given-Size tool. Click on the point on the ray, point $$B$$, and then click on the endpoint of the ray, point $$A$$ (which will be the vertex of the angle).

You should see a pop-up window appear. Type the angle measure you need, and choose either clockwise or counter-clockwise. GeoGebra treats the initial ray like the positive $$x$$-axis of a unit circle or protractor; counter-clockwise is a positive turn, and clockwise is a negative turn. Point $$B'$$ will appear, and segment $$AB'$$ will be congruent to segment $$AB$$. It is probably simplest to hide $$B'$$ and $$AB'$$, since $$AB$$ was not a specified length.

To draw a measured side length, mark a distance from a vertex using the Circle-With-Center-and-Radius tool. Click on the vertex, point $$A$$, and a pop-up window appears. Enter the given measure as the radius of the circle.

Use the Intersection tool to mark one point of intersection of the angle and the circle. That point is the triangle’s second vertex. Continue with the appropriate tools to create more measured sides or angles. See ggbm.at/kaewvwyk for an example.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, invite students to draw triangle transformations (or use tracing paper or the applet) before asking them to identify which pair of triangles is congruent.
Supports accessibility for: Organization; Attention

### Student Facing

Here are three triangles.

1. Triangle $$PQR$$ is congruent to which triangle? Explain your reasoning.
2. Show a sequence of rigid transformations that takes $$PQR$$ to that triangle. Draw each step of the transformation.
3. Explain why there can’t be a rigid transformation to the other triangle.

### Launch

Remind students that images are drawn to scale in these materials.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, invite students to draw triangle transformations (or use tracing paper or the applet) before asking them to identify which pair of triangles is congruent.
Supports accessibility for: Organization; Attention

### Student Facing

Here are 3 triangles.

1. Triangle $$PQR$$ is congruent to which triangle? Explain your reasoning.
2. Show a sequence of rigid motions that takes triangle $$PQR$$ to that triangle. Draw each step of the transformation.
3. Explain why there can’t be a rigid motion from triangle $$PQR$$ to the other triangle.

### Anticipated Misconceptions

Students may believe all 3 triangles are congruent. If this happens, invite them to use available tools (tracing paper, compass, or ruler) to check.

### Activity Synthesis

The key point of this discussion is that triangle $$ACE$$ cannot be congruent to triangle $$PQR$$ because at least one of the corresponding parts is not congruent.

Select multiple students to explain their reasoning.

• students who answered either question in terms of corresponding parts
• students who say the triangles do or don’t line up
• students who used the point-by-point method to prove it’s not possible to use rigid transformations to take $$PQR$$ onto $$ACE$$

If no student used the point-by-point method, invite students to try to write a sequence of rigid motions to take $$PQR$$ onto $$ACE$$. Then, ask them which step they get stuck on. This builds toward the proofs they will write for triangle congruence theorems in subsequent lessons.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their explanations for the first question, present an incorrect answer and explanation. For example, “Triangle $$PQR$$ is congruent to triangle $$ACE$$ because angle $$P$$ is congruent to angle $$A$$, segment $$PQ$$ is congruent to segment $$AC$$, and segment $$QR$$ is congruent to segment $$CE$$.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who clarify the meaning of congruence. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to explain why triangle $$PQR$$ is not congruent to $$ACE$$. This will help students evaluate and improve upon the written mathematical arguments of others, as they clarify the meaning of congruence.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

## 2.3: Are These Parts Congruent? (15 minutes)

### Activity

The mathematical goal of this activity is to name corresponding parts, and recognize which parts correspond based on the congruence statement. The first figure is designed so students can see that the named angles do not correspond. The second figure is visually challenging, but the answer can be determined by looking at how the figures are named. The third figure has no diagram.

Monitor for students who see how to use the naming of figures to find the answer without a diagram.

### Launch

Speaking, Reading: MLR5 Co-Craft Questions. Before revealing the questions in this activity, display the first image along with the statement “Triangle $$ABD$$ is a rotation of triangle $$CDB$$ around point $$E$$ by $$180^\circ$$”, and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions involving the congruence of corresponding parts. Use this routine to help students interpret the naming of geometric figures and increase awareness of the language used to talk about corresponding parts.
Design Principle(s): Maximize meta-awareness; Support sense-making
Representation: Internalize Comprehension. Differentiate the degree of difficulty or complexity by beginning with an example with more accessible figures, such as a pair of segments. Highlight connections between representations by use of color to highlight connections between corresponding congruent segments.
Supports accessibility for: Conceptual processing

### Student Facing

1. Triangle $$ABD$$ is a rotation of triangle $$CDB$$ around point $$E$$ by $$180^{\circ}$$. Is angle $$ADB$$ congruent to angle $$CDB$$? If so, explain your reasoning. If not, which angle is $$ADB$$ congruent to?

2. Polygon $$HIJKL$$ is a reflection and translation of polygon $$GFONM$$. Is segment $$KJ$$ congruent to segment $$NM$$? If so, explain your reasoning. If not, which segment is $$NM$$ congruent to?

3. Quadrilateral $$PQRS$$ is a rotation of polygon $$VZYW$$. Is angle $$QRS$$ congruent to angle $$ZYW$$? If so, explain your reasoning. If not, which angle is $$QRS$$ congruent to?

### Student Facing

#### Are you ready for more?

Suppose quadrilateral $$PQRS$$ was both a rotation of quadrilateral $$VZYW$$ and also a reflection of quadrilateral $$YZVW$$. What can we conclude about the shape of our quadrilaterals? Explain why.

### Anticipated Misconceptions

Suggest that students who struggle more than is productive redraw each figure in the same orientation. Direct them to the order of the letters in the congruence statement to support them with labeling.

### Activity Synthesis

If most students struggled or used a diagram, invite a student to share how they used the naming of the figure to find the answer without a diagram. Tell students there is no one best strategy, but that this is an option.

## Lesson Synthesis

### Lesson Synthesis

Display the quadrilaterals for all to see. Also display the congruence statement: $$ABCD \cong JKLM$$.

Invite several students to explain why it’s not true to say $$ABCD \cong JKLM$$. Prompt students to include rigid transformations in their answer. (Segment $$AB$$ is not congruent to segment $$JK$$, so there’s no rigid motion that takes $$ABCD$$ to $$JKLM$$, so the congruence statement is false.)

Ask students to generate a true congruence statement about the figures. Invite several students to share. Discuss why there can be several true congruence statements. (Possible congruences: $$BCDA \cong MJKL, ADCB \cong LKJM$$. As long as the points are in order around the figure, it doesn’t matter which point you start with or whether you go clockwise or counterclockwise.)

Remind students they can use the order of the letters to see which points are supposed to correspond, so it is important to write congruence statements precisely.

## Student Lesson Summary

### Student Facing

Naming congruent figures so it’s clear from the name which parts correspond makes it easier to check whether 2 figures are congruent and to use corresponding parts. In this image, segment $$AB$$ appears to be congruent to segment $$DE$$. Also, segment $$EF$$ appears to be congruent to segment $$BC$$. So, it makes more sense to conjecture that triangle $$ABC$$ is congruent to triangle $$DEF$$ than to conjecture triangle $$ABC$$ is congruent to triangle $$FDE$$.

If we are told quadrilateral $$MATH$$ is congruent to quadrilateral $$LOVE$$, without even looking at the figures we know:

• Angle $$M$$ is congruent to angle $$L$$.
• Angle $$A$$ is congruent to angle $$O$$.
• Angle $$T$$ is congruent to angle $$V$$.
• Angle $$H$$ is congruent to angle $$E$$.
• Segments $$MA$$ and $$LO$$ are congruent.
• Segments $$AT$$ and $$OV$$ are congruent.
• Segments $$TH$$ and $$VE$$ are congruent.
• Segments $$HM$$ and $$EL$$ are congruent.

Quadrilaterals $$MATH$$ and $$LOVE$$ can be named in many different ways so that they still correspond—such as $$ATHM$$ is congruent to $$OVEL$$ or $$THMA$$ is congruent to $$VELO$$. But $$ATMH$$ is congruent to $$LOVE$$ means there are different corresponding parts. Note that quadrilateral $$MATH$$ refers to a different way of connecting the points than quadrilateral $$ATMH$$.