Lesson 9
SideSideSide Triangle Congruence
9.1: Dare to Be Different (5 minutes)
Warmup
This warmup 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 Response
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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?
Student Response
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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 SideSideSide 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 twosentence 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.)
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:

\(ST\) 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 \(\underline{\hspace{0.5in}\hspace{0.5in}}\).

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 \(\underline{\hspace{0.5in}\hspace{0.5in}}\).

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.

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

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

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

\(GH\) 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.

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

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 fewsentence summary of her proof. “To prove 2 triangles must be congruent if all 3 pairs of corresponding sides are congruent . . . .”
Student Response
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Student Facing
Are you ready for more?
It follows from the SideSideSide 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.

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 
Do the side length and angle measures exhibit a linear relationship?
Student Response
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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 SideSideSide Triangle Congruence Theorem to their reference charts as you ceremoniously add it to the class reference chart:
SideSideSide Triangle Congruence Theorem:
In two triangles, if all three pairs of corresponding sides are congruent, then the triangles must be congruent. (Theorem)
Design Principle(s): Support sensemaking; 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 SideSideSide 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.
Design Principle(s): Cultivate conversation
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 \(D\).
 Work on your own to make a diagram and write a rough draft of a proof.
 With your partner, discuss each other’s drafts.
 What do you notice your partner understands about the problem?
 What revision would help them move forward?
 Work together to revise your drafts into a clear proof that everyone in your class could follow and agree with.
Student Response
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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 SideSideSide 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)
9.4: Cooldown  Practice Seeing Shortcuts (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
So far, we‘ve learned the SideAngleSide and AngleSideAngle Triangle Congruence Theorems. Sometimes, we don’t have any information about corresponding pairs of angle measures in triangles. In this case, use the SideSideSide Triangle Congruence Theorem: In 2 triangles, if all 3 pairs of corresponding sides are congruent, then the triangles must be congruent.
To prove that 2 triangles are congruent, look at the diagram and given information and think about whether it will be easier to find pairs of corresponding angles that are congruent or pairs of corresponding sides that are congruent. Then, check to see if all the information matches the AngleSideAngle, SideAngleSide, or SideSideSide Triangle Congruence Theorem.