Applications of Congruence in Terms of Rigid Motions

New York State Common Core Math Geometry, Module 1, Lesson 20

Worksheets for Geometry

Student Outcomes

• Students will understand that a congruence between figures gives rise to a correspondence between parts such that corresponding parts are congruent, and they will be able to state the correspondence that arises from a given congruence.
• Students will recognize that correspondences may be set up even in cases where no congruence is present, they will know how to describe and notate all the possible correspondences between two triangles or two quadrilaterals and they will know how to state a correspondence between two polygons.

Applications of Congruence in Terms of Rigid Motions

Classwork

Opening

Every congruence gives rise to a correspondence.

Under our definition of congruence, when we say that one figure is congruent to another, we mean that there is a rigid motion that maps the first onto the second. That rigid motion is called a congruence.

Recall the Grade 7 definition: A correspondence between two triangles is a pairing of each vertex of one triangle with one and only one vertex of the other triangle. When reasoning about figures, it is useful to be able to refer to corresponding parts (e.g., sides and angles) of the two figures. We look at one part of the first figure and compare it to the corresponding part of the other. Where does a correspondence come from? We might be told by someone how to make the vertices correspond. Conversely, we might make our own correspondence by matching the parts of one triangle with the parts of another triangle based on appearance. Finally, if we have a congruence between two figures, the congruence gives rise to a correspondence.

A rigid motion 𝐹 always produces a one-to-one correspondence between the points in a figure (the pre-image) and points in its image. If 𝑃 is a point in the figure, then the corresponding point in the image is 𝐹(𝑃). A rigid motion also maps each part of the figure to a corresponding part of the image. As a result, corresponding parts of congruent figures are congruent since the very same rigid motion that makes a congruence between the figures also makes a congruence between each part of the figure and the corresponding part of the image.

In proofs, we frequently refer to the fact that corresponding angles, sides, or parts of congruent triangles are congruent. This is simply a repetition of the definition of congruence. If △ 𝐴𝐵𝐶 is congruent to △ 𝐷𝐸𝐺 because there is a rigid motion 𝐹 such that 𝐹(𝐴) = 𝐷, 𝐹(𝐵) = 𝐸, and 𝐹(𝐶) = 𝐺, then 𝐴𝐵 is congruent to 𝐷𝐸, △ 𝐴𝐵𝐶 is congruent to △ 𝐷𝐸𝐺, and so forth because the rigid motion 𝐹 takes 𝐴𝐵 to 𝐷𝐸 and ∠𝐵𝐴𝐶 to ∠𝐸𝐷𝐹.

There are correspondences that do not come from congruences.

The sides (and angles) of two figures might be compared even when the figures are not congruent. For example, a carpenter might want to know if two windows in an old house are the same, so the screen for one could be interchanged with the screen for the other. He might list the parts of the first window and the analogous parts of the second, thus making a correspondence between the parts of the two windows. Checking part by part, he might find that the angles in the frame of one window are slightly different from the angles in the frame of the other, possibly because the house has tilted slightly as it aged. He has used a correspondence to help describe the differences between the windows not to describe a congruence.

In general, given any two triangles, one could make a table with two columns and three rows and then list the vertices of the first triangle in the first column and the vertices of the second triangle in the second column in a random way. This would create a correspondence between the triangles, though generally not a very useful one. No one would expect a random correspondence to be very useful, but it is a correspondence nonetheless.

Later, when we study similarity, we find that it is very useful to be able to set up correspondences between triangles despite the fact that the triangles are not congruent. Correspondences help us to keep track of which part of one figure we are comparing to that of another. It makes the rules for associating part to part explicit and systematic so that other people can plainly see what parts go together.

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Discussion

Let’s review function notation for rigid motions.
a. To name a translation, we use the symbol 𝑇_𝐴𝐵 . We use the letter 𝑇 to signify that we are referring to a translation and the letters 𝐴 and 𝐵 to indicate the translation that moves each point in the direction from 𝐴 to 𝐵 along a line parallel to line 𝐴𝐵 by distance 𝐴𝐵. The image of a point 𝑃 is denoted 𝑇_𝐴𝐵(𝑃). Specifically, 𝑇_𝐴𝐵(𝐴) = 𝐵.
b. To name a reflection, we use the symbol 𝑟𝑙 , where 𝑙 is the line of reflection. The image of a point 𝑃 is denoted 𝑟_𝑙(𝑃). In particular, if 𝐴 is a point on 𝑙, 𝑟𝑙 (𝐴) = 𝐴. For any point 𝑃, line 𝑙 is the perpendicular bisector of segment 𝑃_𝑟𝑙(𝑃).
c. To name a rotation, we use the symbol 𝑅_𝐶,𝑥° to remind us of the word rotation. 𝐶 is the center point of the rotation, and 𝑥 represents the degree of the rotation counterclockwise around the center point. Note that a positive degree measure refers to a counterclockwise rotation, while a negative degree measure refers to a clockwise rotation.

Example 1

In each figure below, the triangle on the left has been mapped to the one on the right by a 240° rotation about 𝑃. Identify all six pairs of corresponding parts (vertices and sides).

What rigid motion mapped △ 𝐴𝐵𝐶 onto △ 𝑋𝑌𝑍? Write the transformation in function notation.

Example 2

Given a triangle with vertices 𝐴, 𝐵, and 𝐶, list all the possible correspondences of the triangle with itself.

Example 3

Give an example of two quadrilaterals and a correspondence between their vertices such that (a) corresponding sides are congruent, but (b) corresponding angles are not congruent.

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