Lesson 15

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

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.

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).

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.

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.

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.)

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).

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.)