What does congruence have to do with rigid transformations?
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 has been reflected, then rotated, and then translated. All four trapezoids are congruent to one another.
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.
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 would produce rectangle . Just from looking at the rectangles it appears that if rectangle were rotated counterclockwise about the origin it would produce rectangle . To verify this, you can check the points and notice that for rectangle to rectangle , so this is in fact a counterclockwise rotation about the origin.
Because a rigid transformation on rectangle produces rectangle , the two rectangles are congruent.
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 . All pairs of angles are congruent since all angles are . To find the length of the line segments, you can use the Pythagorean Theorem (which is the same as the distance formula).
- , so and
- , so and
Because all corresponding pairs of sides are congruent and all corresponding pairs of angles are congruent, the rectangles are congruent.
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 corresponds to , because both of these angles are in between the sides of lengths 4 and 7. Since is , must also be .
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.
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.
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 . means “triangle is congruent to triangle ”. The order of the letters matters. When you say it means that , , and . Suppose . Draw a picture that matches this situation.
1. can be reflected across the -axis and then translated over one unit to the right and down four units to create . Therefore, the triangles are congruent.
2. You can see that , , . You can also see that from to is 3 units and from to is 3 units so . Similarly, from to is 4 units and from to is 4 units so . Using the 3, 4, 5 Pythagorean triple you know that both and must be 5 units, so . Because all pairs of corresponding angles and sides are congruent, the triangles are congruent.
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.
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 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 to state how the parallelograms are congruent. Note that there are multiple correct ways to write this!
7. Draw a picture that matches this situation.
11. and . What does this tell you about ?
12. is reflected across the , then rotated clockwise to create . 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.