Lesson 9Moves in Parallel
Learning Goal
Let’s transform some lines.
Learning Targets
I can describe the effects of a rigid transformation on a pair of parallel lines.
If I have a pair of vertical angles and know the angle measure of one of them, I can find the angle measure of the other.
Lesson Terms
- corresponding
- rigid transformation
- vertical angles
Warm Up: Line Moves
Problem 1
For each diagram, describe a translation, rotation, or reflection that takes line
Activity 1: Parallel Lines
Use a piece of tracing paper to trace lines
As you perform each transformation, think about the question:
What is the image of two parallel lines under a rigid transformation?
Problem 1
Translate lines
and 3 units up and 2 units to the right. What do you notice about the changes that occur to lines
and after the translation? What is the same in the original and the image?
Problem 2
Rotate lines
and counterclockwise 180 degrees using as the center of rotation. What do you notice about the changes that occur to lines
and after the rotation? What is the same in the original and the image?
Problem 3
Reflect lines
and across line . What do you notice about the changes that occur to lines
and after the reflection? What is the same in the original and the image?
Are you ready for more?
Problem 1
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.
Activity 2: Let’s Do Some 180’s
Problem 1
The diagram shows a line with points labeled
On the diagram, draw the image of the line and points
, , and after the line has been rotated 180 degrees around point . Label the images of the points
, , and . What is the order of all seven points? Explain or show your reasoning.
Problem 2
The diagram shows a line with points
Rotate the figure 180 degrees about point
. Label the image of as and the image of as . What do you know about the relationship between angle
and angle ? Explain or show your reasoning.
Problem 3
The diagram shows two lines
Rotate the figure 180 degrees around
. Label the image of as and the image of as . What do you know about the relationship between the angles in the figure? Explain or show your reasoning.
Lesson 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
that will take to itself. A rotation by
around any point on the line. A rotation of line around point will take to itself. A reflection across any line perpendicular to the line. A reflection of line
across the dashed line will take to itself.
These facts let us make an important conclusion. If two lines intersect at a point, which we’ll call
Rotating both lines by