Lesson 6

Students are familiar with the idea that only some information is needed to uniquely determine a triangle, and this warm-up 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.

Invite students to use highlighters or colored pencils to match the given information to the proof statement.

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.

In this activity, students prove the Side-Angle-Side 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).

If students are searching too far back, point students toward the proof in the warm-up 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.

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

Side-Angle-Side 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)

\(\overline{AB} \cong \overline{GB}, \overline{BC} \cong \overline{BC}, \angle ABC \cong \angle GBC\) so \(\triangle ABC \cong \triangle GBC\)

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 cool-down, and students haven‘t proven a necessary property of perpendicular bisectors yet. Students don‘t have the Side-Side-Side 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 Side-Angle-Side Triangle Congruence Theorem in each case.

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 Side-Angle-Side 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).

The goal of this discussion is to have students explain why, in order to use the Side-Angle-Side 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 Side-Angle-Side 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)

\(\overline{AP} \cong \overline{PB}\) so \(\angle A \cong \angle B\)