Lesson 4 Congruent Triangles Solidify Understanding

Examine the diagram.

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:

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?

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?

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?”

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.”

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.