In this activity, students consider how to adapt the Side-Angle-Side Triangle Congruence proof they wrote in a previous lesson to the case in which the given information is two pairs of congruent corresponding angles and a pair of congruent corresponding sides between the angles. There is less structure provided in this activity than in previous proofs, so monitor for students who need support.

Identify students who provide varying levels of precision in their explanation that the final pair of vertices coincide.

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

• “How is the information you have now different from the information you had earlier?”
• “How could you and your partner change the proofs you used earlier to match the Angle-Side-Angle information given?”

Display this image for all to see and ask, “Could the situation look like this after we reflect across ? How do you know?”

Invite multiple students to share their reasoning for why vertices and have to coincide. Questions to help students make their reasoning more precise include:

• If students assumed that is the same distance along the ray as is: “We don’t know how far is from and , or how far is from and . Is there another way you could prove they were in the same place?”
• If students assert that two rays can only intersect in one place but don’t say anything about how we know the points are on the same rays: “How do you know that the rays that point is on are the same as the rays that point  is on?”
• If students say, “because they are on the same rays” but don’t mention the intersection: “Why can’t those rays intersect in more than one place?”

After multiple students have shared and clarified their reasoning, point out that this is a new reason why vertices have to coincide. Add this to the display of sentence frames for proofs. This display should be posted in the classroom for the remaining lessons within this unit. More will be added to the display throughout the unit. An example template is provided with the blackline master for this lesson.

Justifications:

• Points  and  coincide because they are both at the intersection of the same lines, and two distinct lines can only intersect in one place.

Ask students, "To use Angle-Side-Angle Triangle Congruence, the congruent side must be between the 2 angles. What if the congruent side is not between the 2 angles? What else do you know about angles in a triangle that could help you prove congruence for Angle-Angle-Side (or Side-Angle-Angle)?" (Because the angles in a triangle have a sum of 180 degrees and we know 2 of the angles are congruent, the third angle must also be congruent. Then we know both angles next to the congruent side are congruent and we can use the Angle-Side-Angle Triangle Congruence Theorem.)

Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

Angle-Side-Angle Triangle Congruence Theorem: In two triangles, if two pairs of corresponding angles are congruent and the pair of corresponding sides between the angles is congruent, then the triangles must be congruent. (Theorem)

In preparation for subsequent activities, this optional activity reminds students that alternate interior angles formed by parallel lines and a transversal are congruent.

Direct students' attention to the reference created using Collect and Display.

Ask students about their reasoning for the relationship between different pairs of angles in this order:

• and (the angles form a linear pair)
• and (the angles are vertical angles)
• and (the angles are corresponding angles)
• and (the angles are alternate interior angles)

Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. 

If students describe how they found and by referring back to them both being equal to , first congratulate them on an accurate use of the transitive property, then ask students if they recall the name of the angle pair that the angles with measures and form. (alternate interior angles) Having a single statement about the angles with measure  and makes later proofs that rely on their relationship much simpler to describe.

The goal of this activity is for students to see the usefulness of the triangle congruence theorems in proving other results. In this proof, students are expected to draw an auxiliary line that allows them to use the Angle-Side-Angle Triangle Congruence Theorem. When writing the proof, students have an opportunity to attend to precision in the language they use to describe how each conclusion follows from the previous information (MP6).

Students may also choose to prove the triangles are congruent because they are rotations of one another, which is fine. If students use any technique other than the Angle-Side-Angle Triangle Congruence Theorem, discuss that the Angle-Side-Angle Triangle Congruence Theorem is a shortcut to use without having to write out the transformations and reasons each time.

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

If students struggle longer than is productive, direct them to their reference charts. What do they know about parallel lines? (Alternate interior angles are congruent.)

Focus the discussion on the process of writing a proof. Work toward identifying these components of using triangle congruence criteria in proofs:

• We need to find triangles that are congruent.
• We might need to draw an auxiliary line.
• We need to find three parts of the triangles that are congruent (Side-Angle-Side or Angle-Side-Angle).
• We might need to use something like alternate interior angles—why the angles or sides are congruent is not always given to us.
• Once we find the right three parts, then we know that the triangles are congruent and that any of their pairs of corresponding parts are congruent.

Add this theorem to the class reference chart, and ask students to add it to their reference charts:

In a parallelogram, pairs of opposite sides are congruent. (Theorem)