Lesson 4

In middle school, students studied many examples of dilations and verified experimentally that dilations preserve angle measure. In this activity, students confirm this claim for one example, and recall what they know about dilations from middle school.

If students struggle to come up with a conjecture ask, "What stays the same? What changes?"

Invite students to share their conjectures.

Once students have shared several conjectures, ask students to generalize a claim about angles under dilation. (Dilations keep the measures of angles constant.) They may recall that dilations preserve angle measure from middle school. They may also connect the preservation of angle to the idea that dilations don’t distort the shape of a figure, it just changes its size.

Tell students that we aren’t going to prove that dilations preserve angle measure in this class, we’re going to assert that it is true.

Ask students to add this assertion to their reference charts as you add it to the class reference chart:

If a figure is dilated, then corresponding angles are congruent. (Assertion)

\(\triangle A’B’C’\) is a dilation of \(\triangle ABC\) so \(\angle B \cong \angle B'\)

The purpose of this activity is for students to verify experimentally that dilations take lines through the center of dilation to the same line, even though specific points on the line get farther from or closer to the center of dilation according to the same ratio given by the scale factor. Students first dilate points on given lines, and then they are asked to describe what happens to the lines when they are dilated.

For the sake of time invite students to estimate rather than measure precisely.

If students are distracted by all the other points on the diagram, suggest they use tracing paper to trace only the relevant points. Repeat for each question. Then transfer all the points back onto the original diagram before the synthesis.

The goal of the synthesis is for students to connect dilating points on a line and dilating the line itself. Specifically, what happens if the line goes through the center of the dilation?

Invite students to share how the definition of dilation can help them answer these questions. Students should have the opportunity to hear and articulate that because dilations, by definition, take points along lines through the center, then dilating a line through the center will take all the points to points on that same line, so the line doesn’t move. It may be hard for students to put into words that the points are dilated but due to the nature of infinity, the line is not changed, so invite several students to put their explanation into their own words. In the next synthesis, students will state and record a theorem about lines that do and do not pass through the center of the dilation, so it’s useful for students to be clear about why this is true.

In this activity, students figure out that because dilations preserve angle measures, we can prove that dilations take lines to parallel lines. They draw on the many proofs they did in previous units that use congruent angles to prove that lines are parallel. For students who struggled with proofs in prior units, make sure that they have their reference charts and proof writing sentence frames available. 

In the previous lesson you collected many dilations of triangles from students, drawn on tracing paper. Display several of these examples for all to see. Superimpose the examples so that the center of dilation and original figure are lined up. Ask students what they notice about the angles in the problem. (Corresponding angles in the image and original figure are congruent.) Ask students what they notice about the lines in the problem. (They are longer or shorter according to the scale factor. They are parallel.)

If students don’t mention that the lines are parallel, use a highlighter or colored pencil to draw their attention to two corresponding line segments on lines that don’t go through the center. Ask them what they notice about those lines specifically.

For the proof it might be easier to look at one pair of corresponding segments rather than the whole triangle. Recommend students look at their reference chart and proof writing template.

If students are stuck on the proof, encourage them to draw the rays that show how the points in the image were dilated, and to focus on just one pair of corresponding segments at a time (perhaps using colored pencils to highlight the segment of interest).

The goal of this synthesis is to conclude if two figures are dilations of one another, then any distinct corresponding lines must be parallel. In a later lesson in this unit, students will need to use this result to prove that lines are parallel. Students will get more opportunities to draw conclusions about lines in dilated figures in the cool-down and lesson synthesis. 

Invite students to share what helped them get started on the proof. (Drawing auxiliary lines. Looking at the reference chart.) Ask students to contribute ideas to the proof until everyone understands this chain of reasoning:

• because we know the figures are dilations, we know that corresponding angles are congruent
• because we know that corresponding angles are congruent, we know that the lines cut by the transversal are parallel

Ask students to add this theorem to their reference charts as you add it to the class reference chart:

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. (Theorem)

Dilate using center \(C\). \(\overleftrightarrow{DE} \parallel \overleftrightarrow{D'E'}\)