Lesson 5 Parallelism Preserved and Protected Solidify Understanding
Jump Start
How would you complete the following statement?
Two lines in a plane are parallel if .
Try to complete the statement in as many different ways as you can.
Learning Focus
Determine when a line will be parallel to its pre-image after a translation, rotation, or reflection.
Build a system of geometry based on definitions, postulates, and theorems.
What kinds of transformations produce lines that are parallel to their pre-images?
How are definitions, theorems, and postulates different? Why do we need all three?
Open Up the Math: Launch, Explore, Discuss
In a previous lesson, “How Do You Know That,” you might have wondered how you knew this figure, which was formed by rotating a triangle about the midpoint of one of its sides, was a parallelogram.
You may have found it difficult to explain how you knew that sides of the original triangle and its rotated image were parallel to each other except to say, “It just has to be so.” There are always some statements we have to accept as true in order to convince ourselves that other things are true. We try to keep this list of statements as small as possible, and as intuitively obvious as possible. For example, in our work with transformations we have agreed that distance and angle measures are preserved by rigid transformations since our experience with these transformations suggest that translating, rotating, and reflecting figures do not distort the images in any way. Likewise, parallelism within a figure is preserved by rigid transformations: for example, if we reflect a parallelogram, the image is still a parallelogram—the opposite sides of the new quadrilateral are still parallel.
Mathematicians call statements that we accept as true without proof postulates. Statements that are supported by justification and proof are called theorems.
Knowing that lines or line segments in a diagram are parallel is often a good place from which to start a chain of reasoning. Almost all descriptions of geometry include a parallel postulate among the list of statements that are accepted as true. In this task we develop some parallel postulates for rigid transformations.
1.
Translations
a.
Under what conditions are the corresponding line segments in an image and its pre-image parallel after a translation? That is, which word best completes this statement?
After a translation, corresponding line segments in an image and its pre-image are parallel.
A.
never
B.
sometimes
C.
always
b.
Give reasons for your answer. If you choose “sometimes,” be very clear in your explanation how to tell when the corresponding line segments before and after the translation are parallel and when they are not.
c.
Based on this reasoning, write a parallel postulate for translation:
2.
Rotations
a.
Under what conditions are the corresponding line segments in an image and its pre-image parallel after a rotation? That is, which word best completes this statement?
After a rotation, corresponding line segments in an image and its pre-image are parallel.
A.
never
B.
sometimes
C.
always
b.
Give reasons for your answer. If you choose “sometimes,” be very clear in your explanation how to tell when the corresponding line segments before and after the rotation are parallel and when they are not.
c.
Based on this reasoning, write a parallel postulate for rotation:
3.
Reflections
a.
Under what conditions are the corresponding line segments in an image and its pre-image parallel after a reflection? That is, which word best completes this statement?
After a reflection, corresponding line segments in an image and its pre-image are parallel.
A.
never
B.
sometimes
C.
always
b.
Give reasons for your answer. If you choose “sometimes,” be very clear in your explanation how to tell when the corresponding line segments before and after the reflection are parallel and when they are not.
c.
Based on this reasoning, write a parallel postulate for reflection:
Ready for More?
We have developed three parallel postulates for the three rigid transformations based on experimentation, but it is possible for them to be formally proven and treated as theorems. Create arguments to prove each of the parallel postulates explored in this task.
Takeaways
By experimentation, we accepted the following observations as true:
Parallel Postulate for Translations
Parallel Postulate for Rotations
Parallel Postulate for Reflections
We can reclassify these statements as theorems, if/since:
Vocabulary
- definition
- equidistant
- postulate
- theorem
- Bold terms are new in this lesson.
Lesson Summary
In this lesson, we learned that a system of geometry includes definitions, postulates, and theorems. We developed postulates for parallelism for each of the rigid transformations based on experiments that helped us determine under what conditions corresponding pre-image and image lines would always be parallel following a translation, a rotation, or a reflection.
1.
What are complementary angles?
2.
Find the complement for each of the given angles.
a.
b.
c.
3.
What are supplementary angles?
4.
Find the supplement for each of the given angles.
a.
b.
c.
5.
Do all angles have both a complementary angle and a supplementary angle that pair with them? Explain.