# Lesson 9

Side-Side-Side Triangle Congruence

## 9.1: Dare to Be Different (5 minutes)

### Warm-up

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.

### Student Facing

Construct a triangle with the given side lengths using technology.

Side lengths:

• 2 cm
• 1.5 cm
• 2.4 cm

Can you make a triangle that doesn’t look like anyone else’s?

### Student Facing

Construct a triangle with the given side lengths on tracing paper.

Can you make a triangle that doesn’t look like anyone else’s?

### Activity Synthesis

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.)

## 9.2: Proving the Side-Side-Side Triangle Congruence Theorem (15 minutes)

### Activity

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.

### Launch

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.)

Representation: Access for Perception. Read the Student Task Statement aloud, including the steps of the proof. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language

### Student Facing

Priya was given this task to complete:

Use a sequence of rigid motions to take $$STU$$ onto $$GHJ$$. Given that segment $$ST$$ is congruent to segment $$GH$$, segment $$TU$$ is congruent to segment $$HJ$$, and segment $$SU$$ is congruent to segment $$GJ$$. For each step, explain how you know that one or more vertices will line up.

Help her finish the missing steps in her proof:

1. $$ST$$ is the same length as $$\underline{\hspace{0.5in}\hspace{0.5in}}$$, so they are congruent. Therefore, there is a rigid motion that takes $$ST$$ to $$\underline{\hspace{0.5in}\hspace{0.5in}}$$.

2. Apply this rigid motion to triangle $$STU$$. The image of $$T$$ will coincide with $$\underline{\hspace{0.5in}\hspace{0.5in}}$$ , and the image of $$S$$ will coincide with $$\underline{\hspace{0.5in}\hspace{0.5in}}$$.

3. We cannot be sure that the image of $$U$$, which we will call $$U’$$, coincides with $$\underline{\hspace{0.5in}\hspace{0.5in}}$$ yet. If it does, then our rigid motion takes $$STU$$ to $$GHJ$$, proving that triangle $$STU$$ is congruent to triangle $$GHJ$$. If it does not, then we continue as follows.

4. $$HJ$$ is congruent to the image of $$\underline{\hspace{0.5in}\hspace{0.5in}}$$, because rigid motions preserve distance.

5. Therefore, $$H$$ is equidistant from  $$U’$$ and $$\underline{\hspace{0.5in}\hspace{0.5in}}$$.

6. A similar argument shows that $$G$$ is equidistant from $$U’$$ and $$\underline{\hspace{0.5in}\hspace{0.5in}}$$ .

7. $$GH$$ is the $$\underline{\hspace{0.5in}\hspace{0.5in}}$$ of the segment connecting $$U’$$ and $$J$$, because the $$\underline{\hspace{0.5in}\hspace{0.5in}}$$ is determined by 2 points that are both equidistant from the endpoints of a segment.

8. Reflection across the $$\underline{\hspace{0.5in}\hspace{0.5in}}$$ of $$U’J$$, takes $$\underline{\hspace{0.5in}\hspace{0.5in}}$$  to $$\underline{\hspace{0.5in}\hspace{0.5in}}$$.

9. Therefore, after the reflection, all 3 pairs of vertices coincide, proving triangles $$\underline{\hspace{0.5in}\hspace{0.5in}}$$ and  $$\underline{\hspace{0.5in}\hspace{0.5in}}$$ are congruent.

Now, help Priya by finishing a few-sentence summary of her proof. “To prove 2 triangles must be congruent if all 3 pairs of corresponding sides are congruent . . . .”

### Student Facing

#### Are you ready for more?

It follows from the Side-Side-Side Triangle Congruence Theorem that, if the lengths of 3 sides of a triangle are known, then the measures of all the angles must also be determined. Suppose a triangle has two sides of length 4 cm.

1. Use a ruler and protractor to make triangles and find the measure of the angle between those sides if the third side has these other measurements.

Side Length of Third Side Angle Between First Two Sides
1 cm
2 cm
3 cm
4 cm
5 cm
6 cm
7 cm
2. Do the side length and angle measures exhibit a linear relationship?

### Activity Synthesis

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)

Speaking: MLR8 Discussion Supports. As students share their summary of Priya’s proof, press for details by asking how they know that $$GH$$ is the perpendicular bisector of the segment connecting $$U’$$ and $$J$$. Also, ask how they know that $$U’$$ coincides with $$J$$ after a reflection across the perpendicular bisector of $$U’J$$. Listen for students who reference the Perpendicular Bisector Theorem from the previous lesson to justify why $$GH$$ is the perpendicular bisector of $$U’J$$. To help students identify the important ideas of the proof, provide sentence frames such as: “Priya’s proof is similar to other proofs because _____” and “Priya’s proof is different from other proofs because _____.”
Design Principle(s): Support sense-making; Optimize output (for explanation)

## 9.3: What Else Do We Know For Sure About Parallelograms? (15 minutes)

### Activity

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).

### Launch

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

Representing, Conversing: MLR7 Compare and Connect. Use this routine to help students develop the mathematical language of geometric proofs. After partners revise their drafts into a clear proof that angle $$B$$ is congruent to angle $$D$$, invite them to create a visual display of their proof. Then ask partners to quietly circulate and read at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their proofs. Next, ask partners to discuss what they noticed. Listen for students who reference the proof from a previous lesson that opposite sides of a parallelogram are congruent. Also, listen for and amplify the language students use to justify why triangle $$ABC$$ is congruent to triangle $$CDA$$ and angle $$B$$ is congruent to angle $$D$$
Design Principle(s): Cultivate conversation
Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as dynamic geometry software. Some students may benefit from a checklist or list of steps to be able to use the software.
Supports accessibility for: Organization; Conceptual processing; Attention

### Student Facing

Quadrilateral $$ABCD$$ is a parallelogram. By definition, that means that segment $$AB$$ is parallel to segment $$CD$$, and segment $$BC$$ is parallel to segment $$AD$$.

Prove that angle $$B$$ is congruent to angle $$D$$.

1. Work on your own to make a diagram and write a rough draft of a proof.
2. With your partner, discuss each other’s drafts.
• What revision would help them move forward?
3. Work together to revise your drafts into a clear proof that everyone in your class could follow and agree with.

### Anticipated Misconceptions

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

### Activity Synthesis

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?

## Lesson Synthesis

### Lesson Synthesis

Remind students of the original, very specific prompt from the previous activity. “Quadrilateral $$ABCD$$ is a parallelogram. By definition, that means that segment $$AB$$ is parallel to segment $$CD$$, and segment $$BC$$ is parallel to segment $$AD$$. Prove that angle $$B$$ is congruent to angle $$D$$.” Ask students to generalize: what is always true about parallelograms, based on what we just proved? (A pair of opposite angles is congruent.)

Invite multiple students to explain why we can conclude that both pairs of opposite angles in parallelograms are congruent. (You could have used either diagonal and the proof would have been the same.)

Ask students to add this theorem to their reference charts as you add it to the class reference chart:

In a parallelogram, opposite angles are congruent. (Theorem)