Congruent Triangles

Definition: If all corresponding edges and interior angles match, the triangles match. The triangles are of the same shape and size, but one can be a different mirror

In the simple case below, the two triangles PQR and LMN are congruent because each corresponding edge has the same length and each corresponding angle has the same metric. The angle of P has the same scale (degree) as the angle of L, and the side PQ is the same length as the side LM

In the figure above, the triangles are drawn next to each other. Obviously they are the same. However, it is possible to rotate, invert (reflect) a triangle, or share two common edges with triangles. These incidents will be further discussed on other pages:

One way to consider the consistency of the triangle is to imagine that they are made of cardboard. They are consistent if they can be slid, rotated, flipped in different ways and stacked to fit closely together.

Each triangle is defined by six metrics (three sides, three angles). However, you do not need to know that all of this indicates that the two triangles are matched. All three groups do this. The triangles do not contradict in the following cases.

If the angles of the corresponding triangles are all the same, the triangles have the same shape, but they do not necessarily have the same size. For details, see why AAA does not work.

Assuming that there are no two sides and angles, we can draw two different triangles that satisfy these values. Therefore, proving consistency is not enough. See why SSA does not work

Another way to think about the above is to ask if you can make a unique triangle based on your knowledge. For example, if you specify both side lengths and angles (SAS), only one possible triangle can be drawn. If you draw two, they will be the same shape and size - a matched definition. For details of construction, please see the construction overview.

If two triangles are congruent, each part of the triangle (edge ​​or corner) will match the corresponding part of the other triangle. This is the true value of this concept and if we prove that the two triangles are matched we can find one of them angles or edges from another triangle.

To remember this important concept, it turned out to be convenient to use CPCTC which is the abbreviation "congruent triangle equivalents are congruent."

All other properties of the triangle, such as area, perimeter, center position, circle, etc., except for sides and angles are the same.

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The two triangle parts that have the same measurement (matching) are called corresponding parts. This means that the corresponding parts of the congruent triangle are congruent (CPCTC). Name the congruent triangle by listing the vertices in the proper order. In the figure, ΔBAT ≈ ΔICE

Triangles of exactly the same size and shape are called congruent triangles. The joint symbol is ≈. If three sides and three corners of one triangle have the same measurements as the three sides of the other triangle and the three corners, the two triangles are congruent. The triangle in Figure 1 is a congruent triangle. The two triangle parts that have the same measurement (matching) are called corresponding parts. This means that the corresponding parts of the congruent triangle are congruent (CPCTC). Name the congruent triangle by listing the vertices in the proper order. In the figure, ΔBAT ≈ ΔICE

We already know that they are congruent if the angles are the same as the corresponding sides of the triangle. However, there are too many requirements to fulfill in order to apply this. In this section you will learn two hypotheses that prove that the triangles are consistent and that there is little information. These assumptions are useful because they only require that three corresponding triangular parts match (instead of six corresponding parts like CPCTC). Let's see the first hypothesis