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Unit 2, Lesson 6

Side-Angle-Side Triangle Congruence

Geometry

6.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Highlighters

Activity Narrative

Students are familiar with the idea that only some information is needed to uniquely determine a triangle. This Warm-up emphasizes a related fact: only some measures of two triangles are needed to prove they 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.

Activity

None

Highlight each piece of given information that is used in the proof, and highlight each line in the proof where that piece of information is used.

Given:

Segment is congruent to segment .
Segment is congruent to segment .
Segment is congruent to segment .
Angle is congruent to angle .
Angle is congruent to angle .
Angle is congruent to angle .

Proof:

Segments and are the same length, so they are congruent. Therefore, there is a rigid motion that takes to .
Apply that rigid motion to triangle . The image of will coincide with , and the image of will coincide with .
We cannot be sure that the image of coincides with yet. If necessary, reflect the image of triangle across to be sure the image of , which we will call , is on the same side of as . (This reflection does not change the image of or .)
We know the image of angle is congruent to angle because rigid motions don’t change the size of angles.
must be on ray since both and are on the same side of , and make the same angle with it at .
Segment is the image of , and rigid motions preserve distance, so they must have the same length.
We also know has the same length as . So and must be the same length.
Since and are the same distance along the same ray from , they have to be in the same place.
We have shown that a rigid motion takes to , to , and to ; therefore, triangle is congruent to triangle .

Student Response

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Building on Student Thinking

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, 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 that can be measured on two triangles is needed to prove the triangles are congruent.
Not every piece of information that can be measured on one triangle is needed to make an exact copy of that triangle.

6.2

Activity

Instructional Routines

Draw It MLR7: Compare and Connect

Materials

To Copy (from Blackline Masters)

Triangle Transformation Proof Template Handout

Activity Narrative

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 transformations:

Do not require a reflection step.
Do require a reflection step.

Comparing these two examples will ensure every student ends with a complete proof that works in either case.

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

This activity uses the Compare and Connect math language routine to advance mathematically precise language in discussion.

6.3

Activity

Instructional Routines

None

Materials

To Copy (from Blackline Masters)

What Do We Know About Isosceles Triangles Handout

Activity Narrative

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. A copy of the script is provided with the blackline masters for this lesson. In this proof, the line of symmetry is defined as the bisector of angle . Because the line of symmetry of an isosceles triangle has so many properties, students may want to define it as the perpendicular bisector of , or as the line from to the midpoint of , 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.

<p>Isoceles triangle A B P with vertical dotted line drawn from P to A B.<br>
</p>

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.

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.

Ask, “What auxiliary line would you add to the diagram to help you use the Side-Angle-Side Triangle Congruence Theorem to prove that, in a square, the diagonals form angles with the sides?” (Draw diagonal . Then the Side-Angle-Side Triangle Congruence Theorem can be used 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 , which is .)

“What auxiliary line would you add to the diagram to help you use the Side-Angle-Side 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 , creating two triangles and that are congruent by the Side-Angle-Side Triangle Congruence Theorem. is also a transversal for lines and , and the congruent triangles make congruent alternate interior angles, so the lines are parallel.)

<p>Circle Center I, Radii segments I J and I K, and line segment L I drawn.<br>
</p>

“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 is congruent to itself. So then I drew the angle bisector from through and now I can use the Side-Angle-Side Triangle Congruence Theorem to prove that triangles and are congruent.’ Can that really be true? If not, what went wrong?” (Tyler says to draw the angle bisector from through , but how does he know that angle bisector of angle will go through ?)

to access Cool-down.

Student Lesson Summary

If all pairs of corresponding sides and angles in two triangles are congruent, then it is possible to find a rigid transformation that takes corresponding vertices onto one another. This proves that if two 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 two pairs of corresponding sides and the pair of corresponding angles between the sides are congruent. This is called the Side-Angle-Side Triangle Congruence Theorem.

<p>Two congruent triangles. On each triangle, the long diagonal is marked with 1 tic mark, the bottom short side is marked with 2 tic marks, and the angle between the 2 sides is marked with 1 tic mark.<br>
</p>

To find out if two triangles, or two 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 two triangles must be congruent because of the Side-Angle-Side 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 two 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.

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 that these triangles are congruent.

Then select a student whose triangles also require reflection. Display their drawings as well. Use Compare and Connect to help students compare, contrast, and connect the different representations. Here are some questions for discussion:

“What do the sequences of transformations have in common? How are they different?”
“What can we add or change in our existing proof to account for the reflection?”

Conclude by adding a step to the recorded proof that begins, “If necessary, reflect the image of triangle . . . .” Encourage students to borrow language from the proof in the Warm-up.

Explain that we now have a general theorem that for any triangles in which it is known that two pairs of corresponding sides are congruent and the corresponding angles between them are congruent, they must be congruent. This means 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 with the blackline master for this lesson.

Conclusion Statement:

We have shown that a rigid motion takes to , to , and to ; therefore, triangle is congruent to triangle .

Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

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)

<p>Side-Angle-Side triangle congruence.</p> — , and , so .

Launch

Display an isosceles triangle with base angles and , and other vertex .

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 is congruent to angle .

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 and the line of symmetry .]

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 .) 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).

Representation: Access for Perception. Display or provide students with a physical copy of the acted out dialogue. Check for understanding by inviting students to rephrase dialogue in their own words. Consider keeping the display of dialogue visible throughout the activity.
Supports accessibility for: Language, Memory

Activity

None

Mai and Kiran want to prove that in an isosceles triangle, the two base angles are congruent. Finish the proof that they started. Draw the auxiliary line, and define it so that you can use the Side-Angle-Side Triangle Congruence Theorem to complete each statement in the proof.

<p>Isosceles triangle A P B. A B is the bottom side length. Point P points upwards.<br>
</p>

Draw .

Segment is congruent to segment because of the definition of isosceles triangle.

Angle is congruent to angle because .

Segment is congruent to itself.

Therefore, triangle is congruent to triangle by the Side-Angle-Side Triangle Congruence Theorem.

Therefore, .

Student Response

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Lesson 6

Side-Angle-Side Triangle Congruence

Warm-up

Standards Alignment

Building On

HSG-CO.B.7

Addressing

HSG-CO.B.8

Building Toward

HSG-CO.C.10

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

Activity Narrative

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

HSG-CO.C.11

Standards Alignment

Building On

HSG-CO.A.5

HSG-CO.B.7

Addressing

HSG-CO.B.8

HSG-CO.C.10

Building Toward

Standards Alignment

Building On

HSG-CO.B.8

Addressing

HSG-CO.C.10

Building Toward