# Lesson 4Congruent TrianglesSolidify Understanding

## Jump Start

Examine the diagram.

Is congruent to triangle ? List all of the things you would need to know in order to know for sure that the two triangles are congruent.

## Learning Focus

Explore and justify triangle congruence criteria using rigid transformations.

What do I need to know about two triangles before I can say that they are congruent?

How do I verify that a set of criteria that seems to imply triangles are congruent will always work?

## Open Up the Math: Launch, Explore, Discuss

We know that two triangles are congruent if all pairs of corresponding sides are congruent and all pairs of corresponding angles are congruent. We may wonder if knowing less information about the triangles would still guarantee that they are congruent.

For example, we may wonder if knowing that two angles and the included side of one triangle are congruent to the corresponding two angles and the included side of another triangle—a set of criteria we will refer to as ASA—is enough to know that the two triangles are congruent. And, if we think that this is enough information, how might we justify that this is true?

Here is a diagram illustrating ASA criteria for triangles:

### 1.

Based on the diagram, which angles are congruent? Which sides are congruent?

### 2.

To convince ourselves that these two triangles are congruent, what else would we need to know?

### 3.

Use tracing paper to find a sequence of transformations that will show whether or not these two triangles are congruent.

### 4.

Your sequence of transformations is enough to show that these two triangles are congruent, but how can we guarantee that all pairs of triangles that share ASA criteria are congruent?

Perhaps your sequence of transformations looked like this:

• translate point until it coincides with point .

• rotate about point until it coincides with .

• reflect across .

We can use the word coincides when we want to say that two points or line segments occupy the same position on the plane. When making arguments using transformations, we will use the word a lot.

Now the question is, how do we know that point has to land on point after the reflection, making all of the sides and angles coincide?

### 5.

Answer this problem as best you can to justify why ASA criteria guarantee that two triangles are congruent. It may be helpful to think about how you know point can’t land anywhere else in the plane except on top of .

Using tracing paper, experiment with these additional pairs of triangles. Try to determine if you can find a sequence of transformations that will show whether the triangles are congruent. Be careful; there may be some that aren’t. If the triangles appear to be congruent based on your experimentation, write an argument to explain how you know that this type of criterion will always work. That is, what guarantees that the unmarked sides or angles must also coincide?

### 6.

Given criterion:

Are the triangles congruent?

List your transformations in the order performed:

If the triangles are congruent, justify why this will always be true based on this criterion:

### 7.

Given criterion:

Are the triangles congruent?

List your transformations in the order performed:

If the triangles are congruent, justify why this will always be true based on this criterion:

### 8.

Given criterion:

Are the triangles congruent?

List your transformations in the order performed:

If the triangles are congruent, justify why this will always be true based on this criterion:

### 9.

Given criterion:

Are the triangles congruent?

List your transformations in the order performed:

If the triangles are congruent, justify why this will always be true based on this criterion:

### 10.

Based on these experiments and your justifications, what criteria or conditions seem to guarantee that two triangles will be congruent? List as many examples as you can. Make sure you include ASA from the triangles we worked with first.

### 11.

Your friend wants to add AAS to your list, even though you haven’t experimented with this particular case. What do you think? Should AAS be added or not? What convinces you that you are correct?

### 12.

Your friend also wants to add HL (hypotenuse-leg) to your list, even though you haven’t experimented with right triangles at all, and you know that SSA doesn’t work in general from problem 8. What do you think? Should HL for right triangles be added or not? What convinces you that you are correct?

Sione and Zac are working on their precision in language as they critique each other’s thinking on the first problem of this lesson.

Sione has been watching Zac experiment with the following pair of triangles that have three corresponding parts which are congruent, and he has a concern. “How do we know the conditions you found that make congruent triangles in your experiments will work for all triangles, and not just for these triangles?”

Zac’s argument:

“I know what I did for the first set of triangles,” says Zac, “and I think I can do the same for any pair of triangles that have ASA criteria marked.”

“We can translate point until it coincides with point , then rotate about point until it coincides with . Finally, we can reflect across , and then everything coincides so that the triangles are congruent.” (Zac and Sione’s teacher has suggested they use the word coincides when they want to say that two points or line segments occupy the same position on the plane. They like the word, so they plan to use it a lot.)

What do you think about Zac’s argument? Does it convince you that the two triangles are congruent? Does it leave out any essential ideas that you think need to be included? Reflect on these questions as you read Sione and Zac’s discussion.

Sione isn’t sure that Zac’s argument is really convincing. He asks Zac, “How do you know point coincides with point after you reflect the triangle?”

### 1.

How do you think Zac might answer Sione’s question?

While Zac is trying to think of an answer to Sione’s question, he adds this comment, “And you really didn’t use all of the information about the corresponding congruent parts of the two triangles.”

“What do you mean?” asks Zac.

Sione replies, “You started using the fact that when you translated , so that vertex coincides with vertex . And you used the fact that when you rotated to coincide with , but where did you use the fact that ?

“Yeah, and what does it really mean to say that two angles are congruent?” Zac adds. “Angles are more than just their vertex points.”

### 2.

How might thinking about Zac and Sione’s questions help improve Zac’s argument?

Sione’s argument:

“I would start the same way you did, by translating point until it coincides with point , rotating about point until it coincides with , and then reflecting across ,” Sione says. “But then I would want to convince myself that points and coincide. I know that an angle is made up of two rays that share a common endpoint. Since I know that coincides with and , that means that coincides with . Likewise, I know that coincides with and , so must coincide with . Since and intersect at point , and and intersect at point , points and must also coincide, because the corresponding rays coincide. Therefore, , , and , because both angles are made up of rays that coincide!”

At first, Zac is confused by Sione’s argument, but he draws diagrams and carefully marks and sketches out each of his statements until it starts to slowly make sense.

### 3.

Do the same kind of work that Zac did to make sense of Sione’s argument. What ideas did sketching out the words of his proof help you to clarify? What parts of his argument are unclear to you?

## Takeaways

When working with transformations, we use words like coincide, superimposed, or carried onto to refer to

One way we can justify a claim is to use a proof by contradiction method, in which

Sufficient criteria to guarantee that two triangles are congruent:

Criteria that may not guarantee that triangles are congruent (along with observations about what may happen under these conditions):

## Adding Notation, Vocabulary, and Conventions

We use the following notation to indicate that corresponding parts of two triangles are congruent.

Given two triangles:

SAS (side-angle-side) means:

ASA (angle-side-angle) means:

SSS (side-side-side) means:

## Lesson Summary

In this lesson, we learned that it is not necessary to know that all pairs of corresponding angles and sides are congruent before we can claim that two triangles are congruent. There are several conditions where three pieces of information about corresponding congruent parts of the two triangles are sufficient to guarantee congruence of the triangles. We were able to justify the triangle congruence criteria by relying on the properties of rigid transformations to preserve distance and angle measures.

## Retrieval

### 1.

#### a.

Mark the corresponding sides and angles on the figures.

#### b.

List the eight congruency statements about line segments and angles.

#### c.

Indicate whether carrying one quadrilateral onto the other using rigid transformations would require a reflection. Justify your answer.

### 2.

Solve the equation for :