Lesson 9Moves in Parallel

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.

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 .

  1. Two parallel lines. One is labelled L and has points A and B labelled on it. The other line is labelled L prime.
  2. Two lines, one labelled L, on labelled L prime. They intersect at a point A. another point, labelled B is on line L.

9.2 Parallel Lines

Three parallel lines. One is labelled A. One is labelled with B. There is a point on line B labelled with a K. The third line is labelled H.

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?

Are you ready for more?

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.

    Four points on a line, labelled in order: A, C, D, B.
  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.

    A line with points A and C on the line and a segment A D where D is not on the line
  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.

    The diagram shows two lines L and M that intersect at a point O with point A on L and point D on M

Lesson 9 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 line, labelled M. Points A, B, F, B prime and A prime are labelled on the line. A line of reflection intersects the line at point F and is perpendicular to the line M.
    • 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:

A pair of lines that intersect at point O. Two pairs of congruent vertical angles are labelled.

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 Terms

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 measure 120^\circ , then angle DEB must also measure 120^\circ .

Angles AED and BEC are another pair of vertical angles.

Lesson 9 Practice Problems

    1. Draw parallel lines AB and CD .
    2. Pick any point E . Rotate AB 90 degrees clockwise around E .
    3. Rotate CD 90 degrees clockwise around E .
    4. What do you notice?
  1. Use the diagram to find the measures of each angle. Explain your reasoning.

    1. m{\angle ABC}
    2. m{\angle EBD}
    3. m{\angle ABE}
  2. Points P and Q are plotted on a line.

    A line that slants upward and to the right with two plots labeled P and Q pointed on it. Point P is below point Q.
    1. Find a point R so that a 180-degree rotation with center R sends P to Q and Q to P .
    2. Is there more than one point R that works for part a?
  3. In the picture triangle A’B’C’ is an image of triangle ABC after a rotation. The center of rotation is D .

    Triangle A prime, B prime, C prime is an image of triangle A, B, C after rotation around another point, D. Side B, C has length 4. Angle B prime has measure 52 degrees. Angle C prime has measure 50 degrees.
    1. What is the length of side B’C’ ? Explain how you know.
    2. What is the measure of angle B ? Explain how you know.
    3. What is the measure of angle C ? Explain how you know.
  4. The point (\text-4,1) is rotated 180 degrees counterclockwise using center (0,0) . What are the coordinates of the image?

    1. (\text-1,\text-4)
    2. (\text-1,4)
    3. (4,1)
    4. (4,\text-1)