Lesson 9

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 $B$.

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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?

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.

   

   

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 $A’OC$.