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.

Some students may say that there are 9 different triangles, because they do not recognize that some of them are identical copies oriented differently. Prompt them to use tracing paper to compare the triangles.

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.

This activity builds from previous work with determining unique triangles with certain conditions. In this activity, the conditions given are 2 angles and 1 side.

In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).

Some students may say there are only 2 different triangles in this set, because they do not notice the slight size difference between the smaller two groups of triangles. Prompt them to look at where the angle is located in comparison to the 6-unit side.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to “How many different triangles are there? Explain or show your reasoning” by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:
  “There are 2 different sets of triangles. One set is the triangles where the side that is 6 units is between the 2 angles. The other set is the triangles where the side that is 6 units is not between the 2 angles.” 
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

If time permits, consider asking students to use a protractor to measure the unlabeled angle from each of the three different triangles. Discuss what they notice about the third angle. (It’s the same size in every triangle.)

Explain to students that here we see another example of different triangles that can be made using the same conditions (2 angles and 1 side) in different orders (side between the two angles, side next to the 40-degree angle, and side next to the 80-degree angle). Tell them that in upcoming lessons we will continue to investigate what they noticed here, with the addition of drawing the different triangles.