In this activity, students examine different orientations of triangles that all share 2 sides lengths and one angle measure. They recognize that some of these triangles are identical copies and others are different triangles (not identical copies).

In the coming lessons, students are asked to draw their own triangles. On their own, students often have trouble thinking about triangles where the three given conditions are not necessarily intended to be placed adjacent to one another. For example, when given two sides and an angle, many students will immediately think of putting the given angle between the two sides, but struggle with visualizing putting the angle anywhere else. This task is important for helping students view this as a viable option.

Select students to share the similarities and differences between the triangles in the set.

Trace a few of the triangles from the set and show how you can turn, flip, or move some of them to line up while others cannot be lined up. Ask students what this means about all the triangles in the set (they are not all identical to each other). Explain that, “While there are certainly times when the position of a triangle is important (‘I wouldn’t want my roof upside down!’), for this unit in geometry, we will consider shapes to be the same if they are identical copies.”

To highlight the differences among the triangles, ask students: 

• “Is there only one possible triangle that could be created from the given conditions?” (No, there were 4.)
• “How would you explain what is different about these four triangles?” (Some have the angle between the two sides of known length and others have the  angle next to the side of unknown length.)

Explain to students that it seems that the order in which the conditions are included in the triangle (for example, is the angle between the two sides or not?) matters in creating different triangles. Emphasize that the three required pieces (2 sides and 1 angle) do not have to all be put next to one another. When they are asked to draw triangles with three or more conditions, they should consider the way in which the conditions are arranged in their drawing. For example, think about whether the given angle must go between the two sides or not.

In this activity, students are asked to draw as many different triangles as they can with the given conditions. The purpose of this activity is to provide an opportunity for students to see the three main results for this unit: a situation in which only a unique triangle can be made, a situation in which it is impossible to create a triangle from the given conditions, and a situation in which multiple triangles can be created from the conditions. Students make use of the structure of triangle side lengths and angles as they explore these conditions (MP7).

Students are not expected to remember which conditions lead to which results, but should become more familiar with some methods for attempting to create different triangles. They will practice including various conditions into the triangles, including the conditions in different combinations, and practice recognizing when the resulting triangles are identical copies or not.

In the digital version of the activity, students use an applet to create triangles with given conditions. The applet allows students to change the conditions dynamically. 

Ask students to indicate how many different triangles (triangles that are not identical copies) they could draw for each set of conditions. Select students to share their drawings and reasoning about the uniqueness of each problem. Discuss the methods that students used as they thought about other triangles that might fit the conditions.

Consider asking some of the following questions:

• “Which conditions produced a unique triangle?” (the first set of conditions)
• “Were there conditions that produced more than one triangle?” (the third set of conditions)
• “Were there conditions you could not draw a triangle for?” (the second set of conditions)
• “Why could you not draw a triangle for the second set of conditions?” (because two sides are parallel and will never intersect)

If not mentioned by students, explain to students that for the third set of conditions it is possible that all students drew identical copies using the 4-cm length as the side between the and angles. Consider asking them to think of the previous activity and to try to draw the triangle the way Lin would.

In grade 7, students do not need to know that the angle measures within triangle have a sum of . Tell them that next year they will learn more about why these different conditions determine different numbers of triangles.