What does congruence have to do with rigid transformations?

### Congruence

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 \begin{align*}ABCD\end{align*} has been reflected, then rotated, and then translated. All four trapezoids are congruent to one another.

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.

#### Determining Congruence

1. Are the two rectangles congruent? Explain.

One way to determine whether or not the rectangles are congruent is to consider if transformations to rectangle \begin{align*}ABCD\end{align*} would produce rectangle \begin{align*}FGHI\end{align*}. Just from looking at the rectangles it appears that if rectangle \begin{align*}ABCD\end{align*} were rotated \begin{align*}90^\circ\end{align*} counterclockwise about the origin it would produce rectangle \begin{align*}FGHI\end{align*}. To verify this, you can check the points and notice that \begin{align*}(x, y) \rightarrow (-y, x)\end{align*} for rectangle \begin{align*}ABCD\end{align*} to rectangle \begin{align*}FGHI\end{align*}, so this is in fact a \begin{align*}90^\circ\end{align*} counterclockwise rotation about the origin.

Because a rigid transformation on rectangle \begin{align*}ABCD\end{align*} produces rectangle \begin{align*}FGHI\end{align*}, the two rectangles are congruent.

2. Give another explanation for why the two rectangles from #1 are congruent.

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 \begin{align*}90^\circ\end{align*}. All pairs of angles are congruent since all angles are \begin{align*}90^\circ\end{align*}. To find the length of the line segments, you can use the Pythagorean Theorem (which is the same as the distance formula).

- \begin{align*}AD=BC=FI=GH=\sqrt{1^2+1^2}=\sqrt{2}\end{align*}, so \begin{align*}\overline{AD} \cong \overline{FI}\end{align*} and \begin{align*}\overline{BC} \cong \overline{GH}\end{align*}
- \begin{align*}CD=BA=FG=IH=\sqrt{2^2+2^2}=2 \sqrt{2}\end{align*}, so \begin{align*}\overline{CD} \cong \overline{HI}\end{align*} and \begin{align*}\overline{AB} \cong \overline{FG}\end{align*}

Because all corresponding pairs of sides are congruent and all corresponding pairs of angles are congruent, the rectangles are congruent.

#### Drawing Conclusions

The triangles below are congruent. What does that tell you about \begin{align*}\angle A\end{align*}?

Because the triangles are congruent, corresponding sides and angles are congruent. By looking at the sides, you can see that \begin{align*}\angle A\end{align*} corresponds to \begin{align*}\angle D\end{align*}, because both of these angles are in between the sides of lengths 4 and 7. Since \begin{align*}\angle D\end{align*} is \begin{align*}24^\circ\end{align*}, \begin{align*}\angle A\end{align*} must also be \begin{align*}24^\circ\end{align*}.

**Examples**

**Example 1**

Earlier, you were asked what does congruence have to do with rigid transformation.

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.

#### Example 2

Are the two triangles congruent? Explain.

\begin{align*}\Delta ABC\end{align*} can be reflected across the \begin{align*}y\end{align*}-axis and then translated over one unit to the right and down four units to create \begin{align*}\Delta EFG\end{align*}. Therefore, the triangles are congruent.

#### Example 3

Give another explanation for why the two triangles from #1 are congruent.

You can see that \begin{align*}\angle A \cong \angle F\end{align*}, \begin{align*}\angle C \cong \angle G\end{align*}, \begin{align*}\angle B \cong \angle E\end{align*}. You can also see that from \begin{align*}A\end{align*} to \begin{align*}B\end{align*} is 3 units and from \begin{align*}E\end{align*} to \begin{align*}F\end{align*} is 3 units so \begin{align*}\overline{AB} \cong \overline{EF}\end{align*}. Similarly, from \begin{align*}A\end{align*} to \begin{align*}C\end{align*} is 4 units and from \begin{align*}F\end{align*} to \begin{align*}G\end{align*} is 4 units so \begin{align*}\overline{AC} \cong \overline{FG}\end{align*}. Using the 3, 4, 5 Pythagorean triple you know that both \begin{align*}\overline{BC}\end{align*} and \begin{align*}\overline{EG}\end{align*} must be 5 units, so \begin{align*}\overline{BC} \cong \overline{EG}\end{align*}. Because all pairs of corresponding angles and sides are congruent, the triangles are congruent.

#### Example 4

The symbol for congruence is \begin{align*}\cong\end{align*}. \begin{align*}\Delta ABC \cong \Delta DEF\end{align*} means “triangle \begin{align*}ABC\end{align*} is congruent to triangle \begin{align*}DEF\end{align*}”. The order of the letters matters. When you say \begin{align*}\Delta ABC \cong \Delta DEF\end{align*} it means that \begin{align*}\angle A \cong \angle D\end{align*}, \begin{align*}\angle B \cong \angle E\end{align*}, and \begin{align*}\angle C \cong \angle F\end{align*}. Suppose \begin{align*}\Delta CAT \cong \Delta DOG\end{align*}. Draw a picture that matches this situation.

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.

### Review

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*}x-axis\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 non-rigid 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.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.1.