Lesson 1

Congruent Parts, Part 1

1.1: Notice and Wonder: Transformed Rectangles (5 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that corresponding points are connected in the same order after a transformation, which will be useful when students practice transformations in a later activity. While students may notice and wonder many things about these images, transformations and corresponding points are the important discussion points.

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?

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 you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the difference in labeling rectangle $$C$$ does not come up during the conversation, ask students to discuss this idea.

1.2: If We Know This, Then We Know That... (15 minutes)

Activity

The main goal of this activity is to establish that, in congruent triangles, corresponding parts must also be congruent. Students will justify this theorem by recognizing the same transformation that is used to show the triangles’ congruence can also be used to show the congruence of the parts. More problems like the extension can be found online by searching for “congruent halves.”

Launch

Start the synthesis as soon as students have had a chance to think about all the questions. They will have the opportunity to formalize their language and arguments during the discussion.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the sequence of rigid motions that takes triangle $$ABC$$ to triangle $$DEF$$. Give students time to meet with 2–3 partners to share and get feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What segment is the line of reflection?”, “What is the center of rotation?”, and “What is the angle of rotation?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students use precise language to describe the sequence of rigid motions that takes triangle $$ABC$$ to triangle $$DEF$$.
Design Principle(s): Optimize output (for explanation); 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 argument. For example, “I know _____ corresponds to _____ because…,” “I know _____ is congruent to _____ because….”
Supports accessibility for: Language; Social-emotional skills

Student Facing

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

1. Find a sequence of rigid motions that takes triangle $$ABC$$ to triangle $$DEF$$.
2. What is the image of segment $$BC$$ after that transformation?
3. Explain how you know those segments are congruent.
4. Justify that angle $$ABC$$ is congruent to angle $$DEF$$.

Student Facing

For each figure, draw additional line segments to divide the figure into 2 congruent polygons. Label any new vertices and identify the corresponding vertices of the congruent polygons.

Activity Synthesis

The focus of this discussion is that the corresponding parts of congruent figures are congruent precisely because the same transformation that established the figures’ congruence is guaranteed to establish the congruence of each corresponding part. Start by discussing segments $$BC$$ and $$EF$$.

Students should state that segment $$BC$$ is congruent to segment $$EF$$ because we know we can use rigid transformations to line up the triangles, and we know segments $$BC$$ and $$EF$$ will be lined up when the triangles are lined up, so the segments must be congruent. During this discussion, it’s not sufficient to say that they are corresponding parts so they must be congruent. After this discussion, it will be sufficient to say corresponding parts must be congruent because we’ve established why that is true here.

Arrange students in groups of 2. Invite them to make this specific argument more general—that is, turn the example into a proof of “If two figures are congruent, then corresponding segments of those figures must be congruent.” Give students 1 minute of quiet think time, followed by 2 minutes to write an outline of the proof with their partner. Then, invite pairs to share parts of their proof until the class has a complete proof. Here is a sample proof:

1. If figures are congruent, there is a rigid transformation that takes one figure to the other.
2. If there is a rigid transformation that takes one figure to another, it also takes one segment of the figure to a segment of the other, and we call these two corresponding segments.
3. Since there is a rigid transformation taking one segment to another, those segments are congruent.
4. Therefore, if figures are congruent, then the corresponding segments of those figures must also be congruent.

“Does this argument work for angles?” (Yes, if you replace the word segment with the word angle throughout the proof.)

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

If two figures are congruent, then corresponding parts of those figures must be congruent. (Theorem)

This will be the first new addition to the chart in Unit 2, but students should continue using the Unit 1 chart and this statement should be added in the next blank space. Resist the urge to shorten this to CPCTC, as students may forget what the abbreviation means if they are not reading and saying the words each time.

Activity

The purpose of this activity is to have students identify corresponding parts in congruent triangles.

Identify students who:

• make general statements such as “The lines look parallel.”
• make arguments based on properties of rotations
• make arguments based on alternate interior angles being congruent

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

Launch

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, mark congruent angles with the same color.
Supports accessibility for: Visual-spatial processing

Student Facing

1. Draw a triangle.
2. Find the midpoint of the longest side of your triangle.
3. Rotate your triangle $$180^{\circ}$$ using the midpoint of the longest side as the center of the rotation.
4. Label the corresponding parts and mark what must be congruent.
5. Make a conjecture and justify it.
1. What type of quadrilateral have you formed?
2. What is the definition of that quadrilateral type?
3. Why must the quadrilateral you have fit the definition?

Anticipated Misconceptions

If students are struggling to perform the rotation, encourage them to use whatever tools they are most comfortable with, including sketching.

Activity Synthesis

The purpose of this discussion is to share examples of convincing arguments.

Select previously identified students to share in this order:

• A student who made a general statement such as “The lines look parallel.”
• A student who used the fact that rotation by $$180^{\circ}$$ using a point not on the line as the center takes lines to parallel lines
• A student who used corresponding parts to establish that the lines were parallel because alternate interior angles are congruent

If no student used corresponding parts to establish alternate interior angles as congruent, ask students to look for structure by looking at the congruence marks and seeing whether they match any diagrams on the reference chart.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. As students share their justification for why the opposite sides must be parallel, 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 to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

This synthesis is brief to allow students plenty of time to write their proof in the cool down. Ask students to write down their reason for why corresponding parts of congruent figures must be congruent. (If the figures are congruent, then you can move one exactly on top of the other, which means all the vertices and edges and angles will line up. Since that’s what it means to be congruent, all those parts have to be congruent, too.)

Student Lesson Summary

Student Facing

If a part of the image matches up with a part of the original figure, we call them corresponding parts. The part could be an angle, point, or side. We can find corresponding angles, corresponding points, or corresponding sides.

If 2 figures are not congruent, then there is not a rigid transformation that takes one figure onto the other. If 2 figures are congruent, then there is a rigid transformation that takes one figure onto the other. The same rigid transformation can also be applied to individual parts of the figure, such as segments and angles, because rigid transformations move every point on the plane. Therefore, the corresponding parts of 2 congruent figures are congruent to each other.

Knowing that corresponding parts of congruent figures are congruent can help prove that 2 line segments or 2 angles are congruent, if they are corresponding parts of congruent figures.