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Congruence and Rigid Motion

Determining Congruence

Each type of geometric object comes in different features and sizes. However, if they are the same size and shape they're congruent to each other.
#REDIRECT Reference:Congruence

Congruence Transformation

All rigid motions create congruent figures. Therefore, a rigid motion or a composite rigid motion is sometimes referred to as a congruence transformation. This includes translation, rotation, reflection, or a combination of two or more of them.

Identifying Congruence

Two congruent figures can be mapped onto each other using rigid motions. Since rigid motions preserve length and angle measures, corresponding parts of congruent figures are also congruent. Thus, if the corresponding parts of two figures are congruent, there exists a rigid motion or a composite rigid motion that maps one figure onto the other.

Congruent Figures

Congruence occurs when one object has the same shape and size as another object. Line segments are congruent when they have the same length, congruent angles have the same angle measurement, and polygons are congruent when all of their sides and all of their angles are congruent.

Use rigid motion to determine if the two triangles are congruent.

Show Solution

If the triangles are congruent, that means that there exist one or several rigid motions that maps them onto each other. Let's see if we can find it. Both triangles are isosceles, so the equal sides must correspond to each other. Rotating the blue triangle clockwise around its center will orient it the same way as the red triangle.

Now, they have the same orientation. We can translate the blue triangle, by translating the vertex up units and to the right units.

Notice that, although one vertex has been mapped onto another, the other vertices do not align. This is because one pair of corresponding sides are not the same length. Thus, the triangles are not congruent.

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