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.

Invite previously selected students to share their strategies. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see. If no student drew the arc, demonstrate this strategy to show the two possible triangles.

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

• “Is it possible to prove two triangles are congruent if all you know is two pairs of corresponding sides and one pair of corresponding angles are congruent?” (No, we have three different triangles that fit those criteria.)
• “Is one of these diagrams more convincing?” (No, they all provide counterexamples. Yes, the arcs diagram shows why it’s possible to make different triangles with the same measurements.)

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).

Display the following prompt: “When it are given that two pairs of corresponding sides are congruent and a pair of corresponding angles that are not between the sides are congruent, that is enough to guarantee triangle congruence if ​, but not enough information if .”

Invite students to do a gallery walk and determine how to fill in the blanks. (the longer side is opposite the angle; the shorter side is opposite the given angle)

Note that triangles and have the same side lengths and angle measures, just with a different ordering. Comparing these two cases might help students who are struggling to see the difference.

“Only one triangle can be made—and triangle congruence is guaranteed—when we know that the longer of the two given sides is opposite the given angle.” Add this theorem to the display of triangle congruence theorems.

If a group of students decides it is only possible to make one or two different triangles, encourage them to list all the possible ways to order the given pieces and check that they have tried all of them. (angle, short side, long side or angle, long side, short side)