# Lesson 9

Moves in Parallel

Let’s transform some lines.

### 9.1: Line Moves

For each diagram, describe a translation, rotation, or reflection that takes line $$\ell$$ to line $$\ell’$$. Then plot and label $$A’$$ and $$B’$$, the images of $$A$$ and $$B$$.

### 9.2: Parallel Lines

Use a piece of tracing paper to trace lines $$a$$ and $$b$$ and point $$K$$. Then use that tracing paper to draw the images of the lines under the three different transformations listed.

As you perform each transformation, think about the question:

What is the image of two parallel lines under a rigid transformation?

1. Translate lines $$a$$ and $$b$$ 3 units up and 2 units to the right.

1. What do you notice about the changes that occur to lines $$a$$ and $$b$$ after the translation?
2. What is the same in the original and the image?
2. Rotate lines $$a$$ and $$b$$ counterclockwise 180 degrees using $$K$$ as the center of rotation.

1. What do you notice about the changes that occur to lines $$a$$ and $$b$$ after the rotation?
2. What is the same in the original and the image?

3. Reflect lines $$a$$ and $$b$$ across line $$h$$.

1. What do you notice about the changes that occur to lines $$a$$ and $$b$$ after the reflection?
2. What is the same in the original and the image?

When you rotate two parallel lines, sometimes the two original lines intersect their images and form a quadrilateral. What is the most specific thing you can say about this quadrilateral? Can it be a square? A rhombus? A rectangle that isn’t a square? Explain your reasoning.

### 9.3: Let’s Do Some 180’s

1. The diagram shows a line with points labeled $$A$$, $$C$$, $$D$$, and $$B$$
1. On the diagram, draw the image of the line and points $$A$$, $$C$$, and $$B$$ after the line has been rotated 180 degrees around point $$D$$.

2. Label the images of the points $$A’$$, $$B’$$, and $$C’$$.

3. What is the order of all seven points? Explain or show your reasoning.

2. The diagram shows a line with points $$A$$ and $$C$$ on the line and a segment $$AD$$ where $$D$$ is not on the line.
1. Rotate the figure 180 degrees about point $$C$$. Label the image of $$A$$ as $$A’$$ and the image of $$D$$ as $$D’$$.

2. What do you know about the relationship between angle $$CAD$$ and angle $$CA’D’$$? Explain or show your reasoning.

3. The diagram shows two lines $$\ell$$ and $$m$$ that intersect at a point $$O$$ with point $$A$$ on $$\ell$$ and point $$D$$ on $$m$$.
1. Rotate the figure 180 degrees around $$O$$. Label the image of $$A$$ as $$A’$$ and the image of $$D$$ as $$D’$$.

2. What do you know about the relationship between the angles in the figure? Explain or show your reasoning.

### Summary

Rigid transformations have the following properties:

• A rigid transformation of a line is a line.

• A rigid transformation of two parallel lines results in two parallel lines that are the same distance apart as the original two lines.

• Sometimes, a rigid transformation takes a line to itself. For example:

• A translation parallel to the line. The arrow shows a translation of line $$m$$ that will take $$m$$ to itself.

• A rotation by $$180^\circ$$ around any point on the line. A $$180^\circ$$ rotation of line $$m$$ around point $$F$$ will take $$m$$ to itself.

• A reflection across any line perpendicular to the line. A reflection of line $$m$$ across the dashed line will take $$m$$ to itself.

These facts let us make an important conclusion. If two lines intersect at a point, which we’ll call $$O$$, then a $$180^\circ$$ rotation of the lines with center $$O$$ shows that vertical angles are congruent. Here is an example:

Rotating both lines by $$180^\circ$$ around $$O$$ sends angle $$AOC$$ to angle $$A’OC’$$, proving that they have the same measure. The rotation also sends angle $$AOC’$$ to angle $$A’OC$$.

### Glossary Entries

• corresponding

When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.

For example, point $$B$$ in the first triangle corresponds to point $$E$$ in the second triangle. Segment $$AC$$ corresponds to segment $$DF$$.

• rigid transformation

A rigid transformation is a move that does not change any measurements of a figure. Translations, rotations, and reflections are rigid transformations, as is any sequence of these.

• vertical angles

Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal.

For example, angles $$AEC$$ and $$DEB$$ are vertical angles. If angle $$AEC$$ measures $$120^\circ$$, then angle $$DEB$$ must also measure $$120^\circ$$.

Angles $$AED$$ and $$BEC$$ are another pair of vertical angles.