The purpose of this discussion is to review in more detail the triangle congruence theorems. Invite one student to share their answers for each pair of triangles. Aim to elicit both key mathematical ideas and a variety of student voices, especially from students who haven’t shared recently. Pause to clear up any misconceptions as they arise, such as students assuming the last figure is a square.

The goal of this activity is for students to practice different strategies to prove theorems about parallelograms. This proof intentionally has multiple approaches, and students who complete a proof early are encouraged to make sense of the problem in a different way and persevere in attempting a different strategy (MP1). Monitor for groups who use different triangle congruence theorems. Here are some approaches students may use, ordered to start from the use of vertical angles, which is familiar from previous lessons, and then progress through other solution strategies to reveal different properties of parallelograms.

• Side-Angle-Side, using the two pairs of congruent sides created by the midpoint and using the vertical angles created by the intersection of the diagonals
• Side-Side-Side, using the two pairs of congruent sides created by the midpoint and the parallelogram property that opposite sides are congruent
• Angle-Side-Angle, using any pair of congruent sides and alternate interior angles formed by the sides of the parallelogram and the diagonals

The routine of Anticipate, Monitor, Select, Sequence, Connect (5 Practices) requires a balance of planning and flexibility. The anticipated approaches might not surface in every class, and there may be reason to change the order in which strategies are presented. While monitoring, keep in mind the learning goal and adjust the order to ensure all students have access to the first idea presented (whether that be a common misconception or a different approach).

If students are not sure how to start, consider asking:

• “What do you know about congruent sides or angles in this figure?”
• “Mark a pair of congruent sides or angles. How do you know they are congruent?”

The purpose of this discussion is to review how to use triangle congruence theorems in a proof.

Invite previously selected groups to share their proof. Sequence the discussion of the approaches in the order listed in the Activity Narrative, but do not feel the need to go through each approach, especially if students do not come up with the methods on their own. If possible, record and display the students’ work for all to see, marking congruent pieces on the given figure as they complete their proof.

If your students are showing diverse problem solving strategies, this might be a good opportunity for a Gallery Walk discussion. If students do not yet use formal proof language, edit their responses as they share to maintain content but polish the language, but do not spend much time on the formal language here. Students will have time to develop their proof-writing skills in future lessons.

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

• “What do the three proofs have in common?” (Each proof listed three pairs of congruent parts of the triangles. Each proof ended with a triangle congruence theorem.)
• “What was the process for completing each proof?” (First identify congruent parts, then match them up to a congruence theorem. The proof will have three statements justifying how we know those parts are congruent, and then we can use a triangle congruence theorem to prove the triangles are congruent.)
• “Could you have used the Angle-Side-Angle Triangle Congruence Theorem in a different way?” (Yes. We could have used different pairs of corresponding angles or a different pair of corresponding sides.)
• “Which proof(s) seemed the most obvious? Why do you think that is?” (Side-Angle-Side seemed most obvious because we recently worked with vertical angle pairs.)