Lesson 3 Can You Get There From Here? Develop Understanding

Jump Start

Notice and Wonder

Record at least two things you notice and one thing you wonder about the diagram and description given:

Given: $A B C D ≅ Q R S T$

What I noticed:

What I am wondering:

Learning Focus

Show two figures are congruent based on an efficient and consistent sequence of rigid transformations.

How do I know if two images, such as the frog and lizard images of previous tasks, are congruent?

Is there a “best” sequence of transformations for showing that two figures are congruent to each other? What features of the figures themselves support this work?

Is there a sequence of transformations that would by easy to replicate every time?

Open Up the Math: Launch, Explore, Discuss

The two quadrilaterals shown, quadrilateral $A B C D$ and quadrilateral $Q R S T$ , are congruent, with corresponding congruent parts marked in the diagrams.

Describe a sequence of rigid-motion transformations that will carry quadrilateral $A B C D$ onto quadrilateral $Q R S T$ . Be very specific in describing the sequence and types of transformations you will use so that someone else could perform the same series of transformations.

Once you have a written description, exchange it with another student or two and see if each of you can follow the other’s strategy to show that the two figures are congruent. You should ask questions of each other and suggest edits that will help each of you refine your descriptions.

Here are some questions for reflection that will help:

• Look carefully at the words and diagrams. Do they make sense?

• Do we see a mistake that needs to be fixed?

• Did we leave something out that needs to be put in?

• Do we know something more that needs to be added?

• Why do we want to make the changes we are proposing?

1.

Ready for More?

In the Ready for More? for Unit 1 Lesson 4, we learned that reflecting an image consecutively over two intersecting lines produces a rotation. (Try this out if you didn’t do the Ready for More? in Unit 1 Lesson 4.)

a.

What happens if we reflect an image consecutively over two parallel lines? (Try out several examples until you can explain what happens when an image is reflected consecutively over two parallel lines.)

b.

Based on these observations about what happens when a figure is reflected consecutively over two intersecting lines, or over two parallel lines, can you carry one of two congruent figures onto the other using only reflections? Do you think this is always possible? Why do you think so?

Takeaways

Congruent figures are the same size and shape.

To prove that two figures are congruent, I need to .

A good strategy for doing this is:

Once I have shown that two figures are congruent, I know that corresponding segments and angles in the two figures are congruent because .

Vocabulary

corresponding angles
corresponding points / sides
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we explored a sequence of rigid transformations that could be used to demonstrate that one geometric figure is congruent to another. While many sequences can be found, one particular sequence was identified as being more consistent and easy to replicate each time we need to show that two figures are congruent.

Retrieval

1.

Reflect triangle $H M S$ across the $y = x$ line (Shown as dotted line).

Label the new image as $H^{'} M^{'} S^{'}$ .

2.

Graph each equation on the same set of axes. Make a list of the things you notice about the two graphs.

$y = \frac{3}{2} x - 2$

$y = - \frac{3}{2} x + 2$