How Do You Know if Triangles Are Congruent?

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Theory

Congruence

You have congruence when a shape is an exact copy of another—that is to say, they are the same size and have the same shape.

Another way to say this is that two shapes are congruent if they are similar with the ratio $n = 1$ . The symbol for congruence is $≅$

Congruent triangles

Triangles are congruent when one of these requirements is fulfilled:

Rule

Requirements for Two Triangles to be Congruent

All three sides are equal (SSS).
Two sides, plus the angle between them, are equal (SAS).
Two angles, and a corresponding side, are equal (AAcorrS)
One right angle, the hypothenuse, and a corresponding side are equal (RHS)

Example 1

You are working with this figure:

Example of congruent triangles in parallelogram 1

The quadrilateral $□ A B C D$ is a parallelogram, where $∠ B E C = ∠ A E D$ and $∠ A E B = ∠ D E C$ . You have four triangles in the parallelogram. The cross between the diagonals creates vertical angles, which are in pairs of equal size.

Because $□ A B C D$ is a parallelogram, it must be true that $A D = B C$ , and they are parallel to each other. Because they are parallel, $∠ B C E$ and $∠ D A E$ are corresponding angles, so they are equal. The two angles and the corresponding side that are equal in $△ B C E$ and $△ A D E$ are marked on the figure:

Example of congruent triangles in parallelogram 2

Thus, $△ B C E$ $≅$ $△ A D E$ by the AAcorrS requirement given in the list above.

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What Is the Definition of Similar Figures?

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