What does congruence have to do with rigid transformations?
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Guidance
When a figure is transformed with one or more rigid transformations, an image is created that is congruent to the original figure. In other words, two figures are congruent if a sequence of rigid transformations will carry the first figure to the second figure. In the picture below, trapezoid
Recall that rigid transformations preserve distance and angles. This means that congruent figures will have corresponding angles and sides that are the same measure and length.
In order to determine if two shapes are congruent, you can:
 Carefully describe the sequence of rigid transformations necessary to carry the first figure to the second. AND/OR
 Verify that all corresponding pairs of sides and all corresponding pairs of angles are congruent.
Example A
Are the two rectangles congruent? Explain.
Solution: One way to determine whether or not the rectangles are congruent is to consider if transformations to rectangle
Because a rigid transformation on rectangle
Example B
Give another explanation for why the two rectangles from Example A are congruent.
Solution: To verify that the rectangles are congruent, you could also verify that all corresponding angles and sides are congruent. Notice that the slopes of each line segment making up the rectangles is either +1 or 1. All adjacent sides have opposite reciprocal slopes and are therefore perpendicular. This means that all angles are

AD=BC=FI=GH=12+12−−−−−−√=2√ , soAD¯¯¯¯¯¯≅FI¯¯¯¯ andBC¯¯¯¯¯≅GH¯¯¯¯¯¯ 
CD=BA=FG=IH=22+22−−−−−−√=22√ , soCD¯¯¯¯¯≅HI¯¯¯¯¯ andAB¯¯¯¯¯≅FG¯¯¯¯¯
Because all corresponding pairs of sides are congruent and all corresponding pairs of angles are congruent, the rectangles are congruent.
Example C
The triangles below are congruent. What does that tell you about
Solution: Because the triangles are congruent, corresponding sides and angles are congruent. By looking at the sides, you can see that
Concept Problem Revisited
Rigid transformations create congruent figures. You might think of congruent figures as shapes that “look exactly the same”, but congruent figures can always be linked to rigid transformations as well. If two figures are congruent, you will always be able to perform a sequence of rigid transformations on one to create the other.
Vocabulary
Rigid transformations are transformations that preserve distance and angles. The rigid transformations are reflections, rotations, and translations.
Two figures are congruent if a sequence of rigid transformations will carry one figure to the other. Congruent figures will always have corresponding angles and sides that are congruent as well.
Guided Practice
1. Are the two triangles congruent? Explain.
2. Give another explanation for why the two triangles from #1 are congruent.
3. The symbol for congruence is
Answers:
1.
2. You can see that
3. Remember that to denote that two sides are congruent, you can either mark them as being the same length (e.g., each 7 units), or use corresponding tick marks. It works the same way with angles. Corresponding angle markings mean congruent angles.
Practice
Use the triangles below for #1  #3.
1. Explain why the triangles are congruent in terms of rigid transformations.
2. Explain why the triangles are congruent in terms of corresponding angles and sides.
3. Use notation like \begin{align*}\Delta CAT \cong \Delta DOG\end{align*} to state how the triangles are congruent. Note that there are multiple correct ways to write this!
Use the parallelograms below for #4  #6.
4. Explain why the parallelograms are congruent in terms of rigid transformations.
5. Explain why the parallelograms are congruent in terms of corresponding angles and sides.
6. Use notation like \begin{align*}ABCD \cong A^\prime B^\prime C^\prime D^\prime\end{align*} to state how the parallelograms are congruent. Note that there are multiple correct ways to write this!
\begin{align*}\Delta MRG \cong \Delta KPS\end{align*}
7. Draw a picture that matches this situation.
8. \begin{align*}\angle R \cong \angle \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}\overline{RG} \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}\overline{SK} \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}m \angle M=60^\circ\end{align*} and \begin{align*}m \angle S=20^\circ\end{align*}. What does this tell you about \begin{align*}m \angle R\end{align*}?
12. \begin{align*}\Delta DLP\end{align*} is reflected across the \begin{align*}xaxis\end{align*}, then rotated \begin{align*}90^\circ\end{align*} clockwise to create \begin{align*}\Delta MRK\end{align*}. How are the two triangles related?
13. Why will rigid transformations always produce congruent figures? Could nonrigid transformations also produce congruent figures?
14. If you know that all pairs of corresponding angles for two triangles are congruent, must the triangles be congruent? Explain and provide a counterexample if relevant.
15. If you know that two pairs of corresponding angles and all pairs of corresponding sides for two triangles are congruent, must the triangles be congruent? Explain and provide a counterexample if relevant.