The main goal of this activity is to establish that, in congruent triangles, corresponding parts must also be congruent. This theorem is intentionally not shortened to CPCTC, as students may forget what the abbreviation means if they are not reading and saying the words each time. Students will justify this theorem by recognizing that the same transformation that is used to show the triangles’ congruence can also be used to show the congruence of the parts.

This activity includes the first new addition to the reference chart in this unit. Students should continue using the reference chart from a previous unit, and this statement should be added in the next blank space. 

More problems like the extension can be found online by searching for “congruent halves.”

Invite students to share how they know segments and are congruent.

If students say that the triangles are congruent, so the segments must be congruent, tell them that is what they are about to prove. Continue the discussion until students understand that segment is congruent to segment because we know we can use rigid transformations to line up the triangles. That means we know segments and will be lined up when the triangles are lined up, so the segments must be congruent. 

Arrange students in groups of 2. Invite them to make this specific argument more general—that is, turn the example into a proof of “If two figures are congruent, then corresponding segments of those figures must be congruent.” Give students 1 minute of quiet think time followed by 2 minutes to write an outline of the proof with their partner. Then invite pairs to share parts of their proof until the class has a complete proof. Here is a sample proof:

1. If figures are congruent, there is a rigid transformation that takes one figure to the other.
2. If there is a rigid transformation that takes one figure to another, it also takes one segment of the figure to a segment of the other, and we call these two “corresponding segments.”
3. Since there is a rigid transformation taking one segment to another, those segments are congruent.
4. Therefore, if figures are congruent, then the corresponding segments of those figures must also be congruent.

Ask, “Does this argument work for angles?” (Yes, if you replace the word “segment” with the word “angle” throughout the proof.)

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

If two figures are congruent, then corresponding parts of those figures must be congruent. (Theorem)

The purpose of this activity is to have students identify corresponding parts in congruent triangles.

Monitor for students who:

• Make arguments based on properties of rotations.
• Make arguments based on alternate interior angles being congruent.

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

The purpose of this discussion is to share examples of convincing arguments.

Display 2–3 approaches from previously selected students for all to see. If time allows, invite students to briefly describe their approach, then use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion: 

• “What did the approaches have in common? How were they different?”
• “Which theorems or definitions in your reference chart does each approach rely on?”

If no student used corresponding parts to establish alternate interior angles as congruent, ask students to look for structure by looking at the congruence marks and seeing whether they match any diagrams on the reference chart.