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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.
Concept

Congruence

Geometric figures are said to be congruent if they can be mapped onto each other using one or more rigid motions.
Transform

Because rigid motions preserve angle and length measurements, congruent figures have the same angles measures and side lengths. The quadrilaterals below are all congruent to each other.

Polygon BB has been reflected and rotated, compared to Polygon A,A, while Polygon CC has been rotated and translated. Congruence is denoted with the sign .\cong\hspace{-4pt}. Since the quadrilaterals above are congruent, it can be written that ABC.A \cong B \cong C.
Concept

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.
Rule

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.
Concept

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.
Exercise

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

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 9090^\circ 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 (-2,-1)(\text{-} 2, \text{-} 1) up 44 units and to the right 44 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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