If students struggle to write the Triangle Inequality Theorem, consider asking:

• “Write any inequality and tell me more about what it means, in everyday language.”
• “Using the list of side lengths that do form triangles, how might writing the inequality with numbers help you write the inequality in general?”

The goal of this discussion is to formalize and name the Triangle Inequality Theorem. Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

Triangle Inequality Theorem: If a triangle has side lengths , , and , then .

(Theorem)

Ask students, “Given our new theorem, why don’t the side lengths 4, 6, 2 and 4, 6, 10 work?” (In the first set, it is true that and , but , so those three lengths cannot form a triangle. A similar argument can be used to show that the side lengths 4, 6, 10 do not form a triangle.)

To help students think about the triangle inequality from a different perspective, pose the question, “Did anyone think of the Triangle Inequality Theorem using the difference of sides? For example, starting with sides 4 and 6, how do we know 2 isn’t a possible third side length?” Invite students to share their thinking about the idea that the length of any one side in a triangle must be greater than the difference of the other two sides’ lengths. If no one thought about using the difference of two sides, invite students to consider a few examples from their lists of side lengths that do and do not form triangles. Ask students, “What do you notice if you pick two sides of the triangle and take the difference of their lengths?” (The third side’s length is always greater than that difference.) Use examples from the list of side lengths that create triangles to build consensus on the following idea:

In a triangle, any one side’s length must be greater than the difference of the other two sides’ lengths.

To help students think about the Triangle Inequality Theorem in this way, ask “Given side lengths 2 and 7, what is the range of possibilities for the third side length?” (Let be the third side length. Since the inequality must be true for all sides,  and and . Since a side length must be positive, the first inequality gives us and the last inequality gives us . The third side must be between 5 and 9.)

Explain to students that, not only have they learned a new theorem today, they have practiced justifying their answers. In later lessons, we will build on these skills to write formal proofs.

The purpose of this discussion is to apply the Triangle Inequality Theorem and review how different pieces of information further constrain how a triangle might be drawn. Students will have more opportunities to develop their understanding of triangle congruence criteria in later lessons. There is no need to slow down or add additional work to review this concept at this time if students don’t bring up the triangle congruence theorems in this activity.

Invite selected students to share their partner’s ideas. Remind students that, since we can draw two different triangles with the given information, SSA is not a method to prove two triangles are congruent.