Lesson 2 You Say It’s the Same Solidify Understanding

Learning Focus

Justify the triangle congruence criteria using reasoning based on rigid transformations.

Is there a “best” sequence of transformations for showing that two figures are congruent to each other?

What is the minimum information needed to prove that triangles are congruent?

Open Up the Math: Launch, Explore, Discuss

In previous courses, you have learned about congruence and defined it in terms of rigid transformations. That is, two figures are congruent to each other if the second can be obtained from the first by a sequence of rotations, reflections, and translations. And, if you know that two figures are congruent, then there will exist a sequence of rotations, reflections, and translations that will carry one onto another.

When trying to find a sequence of transformations that will carry a figure onto a congruent figure, some questions arise:

Is there a sequence of rigid transformations that is more logically efficient than others?
Can we find a sequence of transformation that does not require the creation of points or lines that are not already given in the figures provided?
How many parts of a figure need to be congruent to corresponding parts of another figure in order to get the two figures to coincide with one another?

Part I

1.

The two given figures are congruent. Find a sequence of transformations that will carry one of the figures onto the other.

2.

Is there more than one sequence of transformations that can be applied to the figures provided? Are the sequences significantly different? What similarities are there?

3.

Do you always have to perform all three types of rigid transformations to get two congruent images to coincide? Would you ever need to perform the same transformation more than once?

4.

When looking for a sequence of transformations between two congruent figures, which transformation, of the three rigid transformations, can most easily be used first, without the creation of additional points or lines? Why do you think this is?

5.

Which transformation can most easily be used second in your sequence of transformations without having to create additional points or lines? Why is this the case?

Part II

6.

When considering the components of triangles, what information about congruence of corresponding sides and angles guarantee the triangles are congruent, and which do not? All possible combinations of three corresponding congruent parts are provided. Sort them below based on what you remember from your work in prior courses.

AAA, ASA, SSA, SAS, AAS, SSS

Guarantee Triangle Congruence	Don’t Guarantee Triangle Congruence

7.

Below are examples of triangles with different combinations of congruent sides and angles. Use rigid transformations to reason logically about why the triangles are or are not congruent. Justify which of the triangles can be considered congruent and which cannot using transformations. For each, state whether the triangles are congruent or not congruent. If they are congruent, provide the specific sequence of transformations used to justify the congruence and the logical reasoning that guarantees the other parts of the triangles are congruent. (Because it looks like it, or patty paper is not logical justification.)

AAA:	ASA:
SSA:	SAS:
AAS:	SSS:

My claims and justifications:

AAA:

ASA:

SSA:

SAS:

AAS:

SSS:

Ready for More?

Prove SSA can be used to justify two right triangles are congruent. This is also known as the HL triangle congruence criteria for right triangles.

Given: $△ A B C$ and $△ D E F$ are right triangles with congruent hypotenuses: $\overset{―}{A B} ≅ \overset{―}{D E}$ , and a pair of congruent legs: $\overset{―}{A C} ≅ \overset{―}{D F}$

Prove: $△ A B C ≅ △ D E F$ , thus proving HL (or SSA) works for right triangles.

Takeaways

The proof -process involves several stages:

Today, we used the proof-process to justify conditions under which two triangles are guaranteed to be congruent:

Lesson Summary

In this lesson, we reviewed triangle congruence criteria that guarantee two triangles are congruent without having to know that all corresponding sides and corresponding angles are congruent. We justified each set of triangle congruence criteria using rigid transformations and reviewed what it means to justify a claim.

Retrieval

1.

Solve for $x$ . Provide the justifications for each step.

$6 x - 8 = 4 x + 10$	Justification

2.

The definition of a parallelogram is that it has two pairs of parallel sides. List as many other attributes of a parallelogram as you can.

3.

List all of the combinations of congruent sides and angles that will prove triangle congruence.