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 us everything we 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).

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.

Invite a group to present a draft proof of the Side-Angle-Side Parallelogram Congruence Theorem. Ask other groups to provide additional details that will help make the draft proof more convincing.

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

• If points , and coincide, what happens to and ? (It seems like they would coincide.)
• If three vertices of a parallelogram line up with three corresponding vertices of another parallelogram, can the fourth vertex not line up? (No. The rigid motion that takes one triangle to the other preserves distance and angle measure, so will have to land on since it’s the same distance along the same ray.)

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 we know corresponding triangles in the two parallelograms have three pairs of congruent corresponding sides.)

The purpose of this discussion is to walk through the entire proof process (making a conjecture, convincing ourselves, and writing a justification).

Invite previously selected students to share their work. Sequence the discussion of the shortcuts by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “What steps did you take before trying to prove a conjecture?” (I used the strips and fasteners to test if there was a unique parallelogram that fit my criteria.)
• “Are there any conjectures or proofs that you have questions about?”