Lesson 9

This warm-up invites students to construct a triangle given only three side lengths. This builds students’ confidence that three side lengths are enough to uniquely determine a triangle, so it’s worth trying to prove that they are sufficient triangle congruence criteria in subsequent activities.

Collect students’ tracing paper. Overlay them one on top of the other so students can see that all the triangles constructed are congruent. Ask students, “Do you think we will be able to prove that, if all we know about a pair of triangles is that all the pairs of corresponding sides are congruent, then we will be able to use transformations to take one onto the other and prove that all the vertices coincide?” (Yes.)

The proof that two triangles are congruent if all three pairs of corresponding sides are congruent uses a new line of argument: two points coincide after reflection if they are endpoints of a segment that is perpendicularly bisected by the line of reflection. Students will study the proof, provide key phrases, and write a two-sentence summary of the proof that demonstrates that students understand the argument.

Ask students for predictions of steps or reasons that might be in the proof. (We will translate, rotate, and reflect. We will show that points that are the same distance along the same ray coincide. We won’t be able to use any angle pairs, so that might be hard to show that rays coincide.)

The goal of this discussion is to come to a consensus on the summary of Priya’s proof. It’s okay to begin summarizing before every student has their own complete summary so long as all students have done some work to make sense of the proof. 

Invite students to share important ideas they noticed in the proof. 

• Priya used a rigid motion to line up congruent segments, just like the other proofs.
• She defined a perpendicular bisector, and reflected over the perpendicular bisector to line up the third vertex, proving the triangles congruent.
• Each pair of corresponding congruent sides is used in the proof, and no angle measures are needed.

Point out that using perpendicular bisectors is a new reason why vertices have to coincide. Add this to the display of sentence frames for proofs. This display should be posted in the classroom for the remaining lessons within this unit. This is the final statement to add to the display. An example template is provided in the blackline masters for this lesson.

Justifications:

• \(\underline{\hspace{1in}}\) is the perpendicular bisector of the segment connecting \(\underline{\hspace{1in}}\) and \(\underline{\hspace{1in}}\), because the perpendicular bisector is determined by two points that are both equidistant from the endpoints of a segment.

Ask students to add the Side-Side-Side Triangle Congruence Theorem to their reference charts as you ceremoniously add it to the class reference chart:

Side-Side-Side Triangle Congruence Theorem:
In two triangles, if all three pairs of corresponding sides are congruent, then the triangles must be congruent. (Theorem)

\(\overline{HU} \cong \overline{HJ}, \overline{UG} \cong \overline{JG}, \overline{HG} \cong \overline {HG}\), so \(\triangle HUG \cong \triangle HJG\)

The goal of this activity is for students to see the usefulness of the triangle congruence theorems in proving other results. In this proof, students are expected to draw an auxiliary line, find two congruent triangles, determine that they are congruent by the Side-Side-Side Triangle Congruence Theorem, and then use the fact that corresponding parts of congruent triangles are congruent to finish the proof.

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

Arrange students in groups of 2. Remind students of the tips for writing proofs using triangle congruence theorems that are on display.

If students struggle longer than is productive, direct them to their reference charts. What do they know about parallelograms? (Opposite sides are congruent.)

Select partners who used the same approach and whose work is at similar levels of clarity to work together in groups of 4.

Instruct students to read the other group’s proof and decide if they agree with it, and if it could be improved to convince a skeptic. Invite each partnership to say one thing they noticed and liked about the proof, and one thing a skeptic might wonder about the proof.

If needed, brainstorm some things skeptics might be wondering about, such as:

• When you did that rotation, how did you know these points had to line up?
• How did you know those would be the same line?
• How did you know those lines could only intersect there?
• How did you know the Side-Side-Side Triangle Congruence Theorem would apply to these triangles? Did you list all three reasons?
• How did you know those sides were congruent?
• How does knowing the triangles are congruent help you prove the angles are congruent?
• Did you use any information that wasn’t given or that you didn’t have a reason for?
• Did what you proved match the problem?