The purpose of this Warm-up is to elicit the idea that not every set of three pairs of congruent corresponding parts will guarantee triangle congruence, which will be useful when students explore Side-Side-Angle Triangle Congruence in a later activity. While students may notice and wonder many things about these statements and images, the fact that the triangles are not congruent despite having so many corresponding congruent parts is the important discussion point.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If students don’t express surprise that there are so many congruent parts yet the image shows triangles that are not congruent, ask students to discuss this idea.

In this activity, students are arranged in groups of 3 and given side lengths and an angle measure. They are encouraged to make triangles that do not look like one another’s. Given an angle and two sides that are not both adjacent to the given angle, there are three possible triangles that students can make.

As students negotiate how they will make the triangles, listen for different strategies students have for making different triangles. Monitor for students who complete the task in these ways, from least to most efficient:

• Build different triangles by trial and error.
• Draw the arc for the nonadjacent side to see if there are multiple triangles possible.

In the previous activity, students may have noticed that once they decided which side would be adjacent to the given angle, they could sometimes make two triangles and sometimes make one triangle. This may lead to questions about whether knowing an angle, a side, and another side (moving clockwise around the triangle) could be enough information to guarantee we can make a congruent copy of the triangle. This activity formalizes that question and answer by giving students instructions to make different triangles and by exploring which instructions always lead to a single unique triangle and which are ambiguous.

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