# Lesson 15

Congruence for Quadrilaterals

## 15.1: True or . . . Sometimes True?: Parallelograms (5 minutes)

### Warm-up

In this activity, students list everything they remember about parallelograms. This creates a list they can use when generating parallelogram congruence criteria.

### Student Facing

Given that \(ABCD\) is a parallelogram.

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

### Student Response

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### Activity Synthesis

Invite students to share as many things as they can that must be true. Record these statements for all to see and leave them displayed throughout the lesson. Ask students to use their reference chart to justify what must be true.

## 15.2: Floppy Quadrilaterals (15 minutes)

### Activity

This activity introduces students to parallelogram congruence criteria. First, students explain why Side-Side-Side-Side is not a valid congruence criteria for parallelograms. Then, students prove the Side-Angle-Side Parallelogram Congruence Theorem. There are many ways students can reason informally about the Side-Angle-Side Parallelogram Congruence Theorem. For example, they might point out that since opposite sides of a parallelogram are congruent, knowing two adjacent side lengths tells you everything you need to know about side lengths. Side-Side-Side-Side Parallelogram Congruence doesn’t work because the angles can change, but fixing one angle in a parallelogram is enough to prevent it from “flopping.” Students might also explain how they know that knowing one angle is enough to figure out the measures of all four angles. Students do not need to produce a formal proof of this as they are working, but monitor for students who:

- try to define a sequence of rigid motions that takes one parallelogram onto the other
- decompose the parallelograms into triangles
- think about a rigid motion taking one congruent triangle onto the other
- explore what happens to the fourth point under the transformations they defined

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

### Launch

It is okay if, at the start of the synthesis, not every group has proved the Side-Angle-Side Parallelogram Congruence Theorem. Students should all have convinced themselves that it does work, though.

*Action and Expression: Develop Expression and Communication.*Maintain a display of important terms and vocabulary. During the launch take time to review the following terms from previous lessons that students will need to access for this activity: rigid motion transformations, auxiliary lines, triangle congruence theorems.

*Supports accessibility for: Memory; Language*

### Student Facing

Jada is learning about the triangle congruence theorems: Side-Side-Side, Angle-Side-Angle, and Side-Angle-Side. She wonders if there are any theorems like these for parallelograms.

- If 2 parallelograms have all 4 pairs of corresponding sides congruent, do the parallelograms have to be congruent? If so, explain your reasoning. If not, use the tools available to show that it doesn’t work.
- In parallelograms \(ABCD\) and \(EFGH\), segment \(AB\) is congruent to segment \(EF\), segment \(BC\) is congruent to segment \(FG\), and angle \(ABC\) is congruent to angle \(EFG\). Are \(ABCD\) and \(EFGH\) congruent? If so, explain your reasoning. If not, use the tools available to show that it doesn’t work.

### Student Response

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### Anticipated Misconceptions

If students struggle longer than is productive, invite them to use available tools (straightedge, compass, or card stock and metal fasteners) to make some examples.

### Activity Synthesis

Select students to share their thoughts about why the two parallelograms are congruent, focusing on students who:

- tried to define a sequence of rigid motions that takes one parallelogram onto the other
- decomposed the parallelograms into triangles
- used a rigid motion to take one congruent triangle onto the other
- explored what happens to the fourth point under the transformations they defined

The key point to bring out in the discussion is that there must be a sequence of rigid transformations that takes triangle \(ABC\) onto triangle \(EFG\) by the Side-Angle-Side Triangle Congruence Theorem. Here are some questions for discussion:

- If points \(A, B, C\), and \(E, F, G\) coincide, what happens to \(D\) and \(H\)? (It seems like they would coincide.)
- If three vertices of a parallelogram line up with three corresponding vertices of another parallelogram, can the fourth vertices not line up? (No. See student response for one argument which uses the parallel postulate and the claim that rigid motions take parallel lines to parallel lines. Other arguments could be based on rigid motions preserving distance or angle measure.)

After constructing this argument with students, explore the Side-Side-Side-Side Parallelogram Congruence case again. With so many known sides, can’t we use the same argument - line up two triangles, then ensure the fourth vertex lines up? (No, because we can’t guarantee that two corresponding triangles in the two parallelograms are congruent. In the image below, there’s no way to decompose the parallelograms into triangles such that you know corresponding triangles in the two parallelograms have three pairs of congruent corresponding sides.)

*Speaking: MLR8 Discussion Supports.*As students share their reasoning for why the Side-Angle-Side Parallelogram Congruence Theorem works, press for details by asking how they know that there is a rigid transformation that takes triangle \(ABC\) to triangle \(EFG\). Also, ask how they know that the image of \(D\) must be on the line through \(G\) parallel to \(EF\) and on the line through \(E\) parallel to \(FG\). Then ask students how they know that \(D\) and \(H\) coincide. This will help students justify their reasoning for why parallelogram \(ABCD\) and \(EFGH\) must be congruent.

*Design Principle(s): Support sense-making; Optimize output (for justification)*

## 15.3: Make Your Own Congruence Theorem (15 minutes)

### Activity

Exploring parallelogram congruence gives students an opportunity to test their understanding of why congruence theorems do or don’t work. Then they apply their understanding of proofs based on transformations, and engage in creative problem solving as they generate possible congruence conditions to test and prove.

Identify students who:

- test criteria that do not work using drawing or manipulatives
- justify that criteria they have found are equivalent to the Side-Angle-Side Parallelogram Congruence Theorem criteria
- use transformations that systematically line up corresponding, congruent parts to show that their congruence criteria work

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

### Launch

*Representation: Develop Language and Symbols.*Use virtual or concrete manipulatives to connect symbols to concrete objects or values. For example, straightedge, compass, card stock, and metal fasteners, or dynamic geometry virtual software.

*Supports accessibility for: Visual - Spatial Processing; Conceptual processing*

### Student Facing

Come up with another criteria that is enough to be sure that 2 parallelograms are congruent. Try to use as few measurements as you can. Be prepared to convince others that your shortcut works.

### Student Response

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### Student Facing

#### Are you ready for more?

- Will your rule work for any quadrilateral, not just parallelograms?
- If it does, justify your rule. If it doesn't, adjust your rule so it works for any quadrilateral and justify your new rule.

### Student Response

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### Activity Synthesis

First, select students to share some criteria that did not work, and explain how they figured out that it did not work. Remind students that using tools to look at examples is an important part of proof. Before we can prove something, we have to be convinced it works.

Second, invite all the groups to share congruence criteria that they have convinced themselves are valid. Display a list of their conjectures. Give students a chance to look for counterexamples or convince themselves that the criteria are probably valid.

Finally, invite groups that have developed proofs of their criteria to share their thinking. If possible, display a proof that doesn't use a shortcut and a proof that uses the Side-Angle-Side Parallelogram Congruence Theorem side by side. “Why didn't one group need to use transformations?” (The transformations were part of the shortcut so they didn't need to use transformations again.)

## Lesson Synthesis

### Lesson Synthesis

Draw students’ attention to the reference chart and display of sentence frames for proofs. Discuss which of the ideas were used in the parallelogram proofs. Encourage students to reflect back on all that they have learned about congruence. They know how to prove points, segments, triangles, and some quadrilaterals are congruent. Remind students they used those theorems to prove the properties of special triangles and quadrilaterals that they met along the way. Tell students they will continue learning and applying theorems throughout the course.

## 15.4: Cool-down - Rectangles are Special (5 minutes)

### Cool-Down

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