This document provides instruction on reflecting images over parallel lines. It explains how to use the three steps of the Two Reflections Theorem for Translations to determine the point of reflection between two parallel lines. These steps are to find the starting point, determine the sign based on the direction of the first line, and write the distance between the lines twice. An example works through applying the three steps to identify the point of reflection.

1. Composing Reflections
Over Parallel Lines
By: Bradley, Jake, and Rocco

2. Goals
By the end of this lesson, you should be
able to:
✦ Reflect images over parallel lines
✦ Define composite, composition,
translation. You should also be able to
determine the magnitude and direction of
a translation.
✦ Demonstrate the use of the three steps
needed from the Two Reflections Theorem
For Translations.

3. Creating a Reflection of
a Reflection
If you have t wo parallel lines and you
need to find the point of the reflection.

4. Composite
When one transformation is followed
by a second, we say that the
transformations have been composed.

5. Composition
The operation denoted by a small circle. ( )

6. Translation
The composite of t wo reflections over
parallel lines.

7. Properties of
Translations
A translation is fully described only
when it’s magnitude and direction are
known.

8. Point of Reflection
If you have t wo parallel lines and you need to find the
point of reflection...

9. Point of Reflection
If you have t wo parallel lines and you need to find the
point of reflection...
rm (I) The I is your starting point
Identify the point by a
single letter
rm (I)

10. Point of Reflection
If you have t wo parallel lines and you need to find the
point of reflection...
rm (I) The I is your starting point
Draw t wo lines
that connect
point I to line m.
Identify the point by a
single letter
rm (I)

11. Point of Reflection
If you have t wo parallel lines and you need to find the
point of reflection...
rm (I) The I is your starting point
Draw t wo lines
that connect
point I to line m.
Make one
reflection of the
image drawn
Identify the point by a
single letter
rm (I)

12. Point of Reflection
If you have t wo parallel lines and you need to find the
point of reflection...
rm (I) The I is your starting point
Draw t wo lines
that connect
point I to line m.
Make one
reflection of the
image drawn
Identify the point by a
single letter
rm (I) Your answer is C

13. Now you try...
Identify the point by a
single letter
rn(E)

14. How’d you do?

15. How’d you do?
rn (E) The E is your starting point

16. How’d you do?
rn (E) The E is your starting point
Draw t wo lines
that connect
point E to line n.

17. How’d you do?
rn (E) The E is your starting point
Draw t wo lines
that connect
point E to line n.
Make one
reflection of the
image drawn

18. How’d you do?
rn (E) The E is your starting point
Draw t wo lines
that connect
point E to line n.
Make one
reflection of the
image drawn
Your answer is A

19. Learning the Two Reflections Theorem for
Translations with 3 steps
1.

20. Learning the Two Reflections Theorem for
Translations with 3 steps
1.

21. Learning the Two Reflections Theorem for
Translations with 3 steps
1. Find the value of your
starting point
rc rb (E)
Starting point

22. Learning the Two Reflections Theorem for
Translations with 3 steps
1. Find the value of your 1
starting point
rc rb (E)
Starting point

23. Learning the Two Reflections Theorem for
Translations with 3 steps
1. Find the value of your 1
starting point
2. If the line given first is +
on the right then put
t wo plus signs (if going +
to left put t wo minus
signs).
rc rb (E)
Starting point
first given line

24. Learning the Two Reflections Theorem for
Translations with 3 steps
1. Find the value of your 1
starting point
distance=6
2. If the line given first is +
on the right then put
t wo plus signs (if going +
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rc rb (E) and write it t wice
Starting point
first given line

25. Learning the Two Reflections Theorem for
Translations with 3 steps
1. Find the value of your 1
starting point
distance=6
2. If the line given first is + 6
on the right then put
t wo plus signs (if going +
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rc rb (E) and write it t wice
Starting point
first given line

26. Learning the Two Reflections Theorem for
Translations with 3 steps
1. Find the value of your 1
starting point
distance=6
2. If the line given first is + 6
on the right then put
t wo plus signs (if going + 6
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rc rb (E) and write it t wice
Starting point
first given line

27. Learning the Two Reflections Theorem for
Translations with 3 steps
1. Find the value of your 1
starting point
distance=6
2. If the line given first is + 6
on the right then put
t wo plus signs (if going + 6
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rc rb (E) and write it t wice
Starting point 13
first given line

28. Now You Try...
Find the value of your
starting point
rb rc (D)

29. How’d You Do?
1. Find the value of your
starting point
distance=6
2. If the line given first is
on the right then put
t wo plus signs (if going
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rb rc (D) and write it t wice
Starting point
first given line

30. How’d You Do?
1. Find the value of your -2
starting point
distance=6
2. If the line given first is
on the right then put
t wo plus signs (if going
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rb rc (D) and write it t wice
Starting point
first given line

31. How’d You Do?
1. Find the value of your -2
starting point
distance=6
2. If the line given first is -
on the right then put
t wo plus signs (if going
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rb rc (D) and write it t wice
Starting point
first given line

32. How’d You Do?
1. Find the value of your -2
starting point
distance=6
2. If the line given first is -
on the right then put
t wo plus signs (if going -
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rb rc (D) and write it t wice
Starting point
first given line

33. How’d You Do?
1. Find the value of your -2
starting point
distance=6
2. If the line given first is - 6
on the right then put
t wo plus signs (if going -
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rb rc (D) and write it t wice
Starting point
first given line

34. How’d You Do?
1. Find the value of your -2
starting point
distance=6
2. If the line given first is - 6
on the right then put
t wo plus signs (if going - 6
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rb rc (D) and write it t wice
Starting point
first given line

35. How’d You Do?
1. Find the value of your -2
starting point
distance=6
2. If the line given first is - 6
on the right then put
t wo plus signs (if going - 6
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rb rc (D) and write it t wice
Starting point -14
first given line

36. How’d You Do?
1. Find the value of your -2
starting point
distance=6
2. If the line given first is - 6
on the right then put
t wo plus signs (if going - 6
to left put t wo minus
signs).
3. Find the distance
bet ween the t wo lines
rb rc (D) and write it t wice
Starting point Answer = -14
first given line