Lesson 6
SideAngleSide Triangle Congruence
6.1: Information Overload? (5 minutes)
Warmup
Students are familiar with the idea that only some information is needed to uniquely determine a triangle, and this warmup emphasizes that not every piece of information you can measure about two triangles is needed to prove the triangles are congruent. This activity gives students a chance to read a complete proof and makes a great point of reference as they write their own proofs.
Launch
Invite students to use highlighters or colored pencils to match the given information to the proof statement.
Student Facing
Highlight each piece of given information that is used in the proof, and each line in the proof where that piece of information is used.
Given:
 \(\overline {AB} \cong \overline {DE}\)
 \(\overline {AC} \cong \overline {DF}\)
 \(\overline {BC} \cong \overline {EF}\)
 \(\angle A \cong \angle D\)
 \(\angle B \cong \angle E\)
 \(\angle C \cong \angle F\)
Proof:

Segments \(AB\) and \(DE\) are the same length so they are congruent. Therefore, there is a rigid motion that takes \(AB\) to \(DE\).

Apply that rigid motion to triangle \(ABC\). The image of \(A\) will coincide with \(D\), and the image of \(B\) will coincide with \(E\).

We cannot be sure that the image of \(C\) coincides with \(F\) yet. If necessary, reflect the image of triangle \(ABC\) across \(DE\) to be sure the image of \(C\), which we will call \(C’\), is on the same side of \(DE\) as \(F\). (This reflection does not change the image of \(A\) or \(B\).)

We know the image of angle \(A\) is congruent to angle \(D\) because rigid motions don’t change the size of angles.

\(C’\) must be on ray \(DF\) since both \(C’\) and \(F\) are on the same side of \(DE\), and make the same angle with it at \(D\).

Segment \(DC’\) is the image of \(AC\) and rigid motions preserve distance, so they must have the same length.

We also know \(AC\) has the same length as \(DF\). So \(DC’\) and \(DF\) must be the same length.

Since \(C’\) and \(F\) are the same distance along the same ray from \(D\), they have to be in the same place.

We have shown that a rigid motion takes \(A\) to \(D\), \(B\) to \(E\), and \(C\) to \(F\); therefore, triangle \(ABC\) is congruent to triangle \(DEF\).
Student Response
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Activity Synthesis
The goal of this discussion is for students to be clear that not every piece of given information was used in the proof, and so some information wasn’t needed to be sure that the triangles are congruent. This discussion gives another opportunity to connect these ideas:
 Not every piece of information you can measure about two triangles is needed to prove the triangles are congruent.
 Not every piece of information you can measure about one triangle is needed to make an exact copy of that triangle.
6.2: Proving the SideAngleSide Triangle Congruence Theorem (15 minutes)
Activity
In this activity, students prove the SideAngleSide Triangle Congruence Theorem. They are encouraged to use an example of a proof that they can repurpose for this situation. This puts the focus on looking for and making use of structure (MP7), rather than coming up with the right words to justify that their sequence of transformations works.
Because students are asked to draw their own triangles, some students will use different transformations than other students. Monitor for students whose proofs do and do not require reflection steps.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Design Principle(s): Cultivate conversation
Supports accessibility for: Language; Organization
Student Facing
 Two triangles have 2 pairs of corresponding sides congruent, and the corresponding angles between those sides are congruent. Sketch 2 triangles that fit this description and label them \(LMN\) and \(PQR\), so that:
 Segment \(LM\) is congruent to segment \(PQ\)
 Segment \(LN\) is congruent to segment \(PR\)
 Angle \(L\) is congruent to angle \(P\)
 Use a sequence of rigid motions to take \(LMN\) onto \(PQR\). For each step, explain how you know that one or more vertices will line up.
 Look back at the congruent triangle proofs you’ve read and written. Do you have enough information here to use a proof that is like one you saw earlier? Use one of those proofs to guide you in writing a proof for this situation.
Student Response
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Student Facing
Are you ready for more?
It follows from the SideAngleSide Triangle Congruence Theorem that if the lengths of 2 sides of a triangle are known, and the measure of the angle between those 2 sides is known, there can only be one possible length for the third side.
Suppose a triangle has sides of lengths of 5 cm and 12 cm.
 What is the longest the third side could be? What is the shortest it could be?
 How long would the third side be if the angle between the two sides measured 90 degrees?
Student Response
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Anticipated Misconceptions
If students are searching too far back, point students toward the proof in the warmup activity Information Overload. The goal is for students to understand and adapt that proof to this situation, so help students find the proof relatively quickly so they can have time to engage in productive struggle as they try to understand and adapt it.
Activity Synthesis
The goal of this discussion is to continue to emphasize that proofs using transformations are generalized statements that work for all triangles that match the given criteria, rather than just one specific drawing.
Select students whose triangles require translation and rotation but not reflection to share their drawings and the steps in their transformations. Record a proof these triangles are congruent.
Then, select a student whose triangles also require reflection. Rather than writing a separate proof for that case, ask students what we could add or change in our existing proof to account for that case. Add an ‘if needed’ reflection step and explain when you might need to reflect and why that reflection will work.
Explain that we now have a general theorem that any triangles in which you know two pairs of corresponding sides are congruent and the corresponding angles between them are congruent must be congruent, and that we don’t have to show the transformations anymore if we can just show that we’ve met the SideAngleSide Triangle Congruence Theorem.
Point out the concluding statement. 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. You will add to the display throughout the unit. An example template is provided in the blackline masters for this lesson.
Conclusion Statement:
 We have shown that a rigid motion takes \(\underline{\hspace{1in}}\) to \(\underline{\hspace{1in}}\), \(\underline{\hspace{1in}}\) to \(\underline{\hspace{1in}}\), and \(\underline{\hspace{1in}}\) to \(\underline{\hspace{1in}}\); therefore, triangle \(\underline{\hspace{1in}}\) is congruent to triangle \(\underline{\hspace{1in}}\).
Ask students to add the SideAngleSide Triangle Congruence Theorem to their reference charts as you ceremoniously add it to the class reference chart:
SideAngleSide Triangle Congruence Theorem: In two triangles, if two pairs of congruent corresponding sides and the pair of corresponding angles between the sides are congruent, then the two triangles are congruent. (Theorem)
6.3: What Do We Know For Sure About Isosceles Triangles? (15 minutes)
Activity
A major component of looking for and making use of structure (MP7) in geometry is drawing and defining auxiliary lines. In this lesson, a dialogue introduces students to the thinking that mathematicians do when they add lines to figures. In this proof, the line of symmetry is defined as the bisector of angle \(P\). Because the line of symmetry of an isosceles triangle has so many properties, students may want to define it as the perpendicular bisector of \(AB\), or as the line from \(P\) to the midpoint of \(AB\), which are all the same auxiliary line. The case of the perpendicular bisector will appear in the cooldown, and students haven‘t proven a necessary property of perpendicular bisectors yet. Students don‘t have the SideSideSide Triangle Congruence Theorem, so the midpoint option won‘t result in a complete proof, either.
As students work, listen for students who have chosen those different definitions, and how they think about using the SideAngleSide Triangle Congruence Theorem in each case.
Launch
Display an isosceles triangle with base angles \(A\) and \(B\), and other vertex \(P\).
Invite two students to act out this dialogue:
Kiran: I’m stumped on this proof.
Mai: What are you trying to prove?
Kiran: I’m trying to prove that in an isosceles triangle, the two base angles are congruent. So in this case, that angle \(A\) is congruent to angle \(B\).
Mai: Let’s think of what geometry ideas we already know are true.
Kiran: We know if two pairs of corresponding sides and the corresponding angles between the sides are congruent, then the triangles must be congruent.
Mai: Yes, and we also know that we can use reflections, rotations, and translations to prove congruence and symmetry . . . . The isosceles triangle you’ve drawn makes me think of symmetry. If you draw a line down the middle of it, I wonder if that could help us prove that the angles are the same? [Mai draws the line of symmetry of the triangle and labels the intersection of AB and the line of symmetry Q].
Kiran: Wait, when you draw the line, it breaks the triangle into two smaller triangles. I wonder if I could prove those triangles are congruent using SideAngleSide Triangle Congruence?
Mai: It’s an isosceles triangle, so we know that one pair of corresponding sides is congruent. [Mai marks the congruent sides].
Kiran: And this segment in the middle here is part of both triangles, so it has to be the same length for both. Look. [Kiran draws the two halves of the isosceles triangle and marks the shared sides as congruent].
Mai: So we have two pairs of corresponding sides that are congruent. How do we know the angles between them are congruent?
Kiran: I’m not sure. Maybe it has to do with how we drew that line of symmetry?
Tell students: Mathematicians call additional lines auxiliary lines, because auxiliary means “providing additional help or support.” Ask students what properties Mai might have used to draw the line of symmetry. (It’s perpendicular to the base. It bisects the base. It bisects angle \(P\).) If no student mentions it, draw their attention to the bisected angle. If students want to use a different property, invite them to attempt to write their own proof rather than fill in the blanks (there are several ways to get close to a proof but then get stuck since they have limited theorems available to prove something).
Supports accessibility for: Language; Memory
Student Facing
Mai and Kiran want to prove that in an isosceles triangle, the 2 base angles are congruent. Finish the proof that they started. Draw the auxiliary line and define it so that you can use the SideAngleSide Triangle Congruence Theorem to complete each statement in the proof.
Draw \(\underline{\hspace{1in}}\).
Segment \(PA\) is congruent to segment \(PB\) because of the definition of isosceles triangle.
Angle \(\underline{\hspace{1in}}\) is congruent to angle \(\underline{\hspace{1in}}\) because \(\underline{\hspace{1in}}\).
Segment \(PQ\) is congruent to itself.
Therefore, triangle \(APQ\) is congruent to triangle \(BPQ\) by the SideAngleSide Triangle Congruence Theorem.
Therefore, \(\underline{\hspace{1in}}\).
Student Response
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Activity Synthesis
The goal of this discussion is to have students explain why, in order to use the SideAngleSide Triangle Congruence, it was helpful to define the auxiliary line as an angle bisector.
 Why did Kiran and Mai use the angle bisector as their auxiliary line? (So they would have congruent angles at point \(P\) between two congruent sides.)
 Could they have used a different definition for the auxiliary line? (The perpendicular bisector of \(AB\) almost works, but we would have to prove that point \(P\) is on that line before we could use the SideAngleSide Triangle Congruence Theorem.)
Ask students to add the Isosceles Triangle Theorem to their reference charts as you add it to the class reference chart:
Isosceles Triangle Theorem: In an isosceles triangle, the base angles are congruent. (Theorem)
Design Principle(s): Support sensemaking; Optimize output (for explanation)
Lesson Synthesis
Lesson Synthesis
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a wholeclass discussion. The purpose of this Math Talk is to elicit strategies and understandings students have related to adding auxiliary lines, and to do some auxiliary line error analysis.
“What auxiliary line would you add to the diagram to help you use the SideAngleSide Triangle Congruence Theorem to prove that, in a square, the diagonals form \(45^{\circ}\) angles with the sides?” (Draw diagonal \(HF\). Then you can use the SideAngleSide Triangle Congruence Theorem because all the sides are equal and all the right angles are equal. Then because the two triangles are congruent, the angles formed on each side of the diagonal must be equal, so they must be half of \(90^{\circ}\), which is \(45^{\circ}\).)
“What auxiliary line would you add to the diagram to help you use the SideAngleSide Triangle Congruence Theorem to prove that, in a quadrilateral with both pairs of opposite sides congruent and one pair of opposite angles congruent, opposite sides are parallel?” (Draw segment \(BC\), creating two triangles \(ABC\) and \(DCB\) that are congruent by the SideAngleSide Triangle Congruence Theorem. \(BC\) is also a transversal for lines \(BA\) and \(DC\), and the congruent triangles make congruent alternate interior angles, so the lines are parallel.)
“This is an example of how not to draw an auxiliary line.
Conjecture: Given any two points on a circle and any point outside the circle, the two points on the circle are the same distance to the exterior point. Tyler says, “I know that radii of a circle are congruent, and \(LI\) is congruent to itself. So then I drew the angle bisector from \(I\) through \(L\) and now I can use the SideAngleSide Triangle Congruence Theorem to prove that triangles \(ILK\) and \(ILJ\) are congruent.” Can that really be true? If not, what went wrong?” (Tyler says to draw the angle bisector from \(I\) through \(L\), but how does he know that angle bisector of angle \(JIK\) will go through \(L\)?)
6.4: Cooldown  Revisiting Perpendicular Bisectors (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
If all pairs of corresponding sides and angles in 2 triangles are congruent, then it is possible to find a rigid transformation that takes corresponding vertices onto one another. This proves that if 2 triangles have all pairs of corresponding sides and angles congruent, then the triangles must be congruent. But, justifying that the vertices must line up does not require knowing all the pairs of corresponding sides and angles are congruent. We can justify that the triangles must be congruent if all we know is that 2 pairs of corresponding sides and the pair of corresponding angles between the sides are congruent. This is called the SideAngleSide Triangle Congruence Theorem.
To find out if 2 triangles, or 2 parts of triangles, are congruent, see if the given information or the diagram indicates that 2 pairs of corresponding sides and the pair of corresponding angles between the sides are congruent. If that is the case, we don’t need to show and justify all the transformations that take one triangle onto the other triangle. Instead, we can explain how we know the pairs of corresponding sides and angles are congruent and say that the 2 triangles must be congruent because of the SideAngleSide Triangle Congruence Theorem.
Sometimes, to find congruent triangles, we may need to add more lines to the diagram. We can decide what properties those lines have based on how we construct the lines (An angle bisector? A perpendicular bisector? A line connecting 2 given points?). Mathematicians call these additional lines auxiliary lines because auxiliary means “providing additional help or support.” These are lines that give us extra help in seeing hidden triangle structures.