Triangle : Congruency and Similarity

Definition: Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.

In the simple case below, the two triangles PQR and LMN are congruent because every corresponding side has the same length, and every corresponding angle has the same measure. The angle at P has the same measure (in degrees) as the angle at L, the side PQ is the same length as the side LM etc.

In the diagram above, the triangles are drawn next to each other and it is obvious they are identical. However, one triangle may be rotated, flipped over (reflected), or the two triangles may share a common side.

Imagine the triangles are cardboard

One way to think about triangle congruence is to imagine they are made of cardboard. They are congruent if you can slide them around, rotate them, and flip them over in various ways so they make a pile where they exactly fit over each other.

How to tell if triangles are congruent

Any triangle is defined by six measures (three sides, three angles). But you don’t need to know all of them to show that two triangles are congruent. Various groups of three will do. Triangles are congruent if:

1. SSS (side side side)
   All three corresponding sides are equal in length.
   See Triangle Congruence (side side side).
2. SAS (side angle side)
   A pair of corresponding sides and the included angle are equal.
   See Triangle Congruence (side angle side).
3. ASA (angle side angle)
   A pair of corresponding angles and the included side are equal.
   See Triangle Congruence (angle side angle).
4. AAS (angle angle side)
   A pair of corresponding angles and a non-included side are equal.
   See Triangle Congruence (angle angle side).
5. HL (hypotenuse leg of a right triangle)
   Two right triangles are congruent if the hypotenuse and one leg are equal.
   See Triangle Congruence (hypotenuse leg).

AAA does not work.

If all the corresponding angles of a triangle are the same, the triangles will be the same shape, but not necessarily the same size. For more on this see Why AAA doesn’t work.

They are called similar triangles.

SSA does not work.

Given two sides and a non-included angle, it is possible to draw two different triangles that satisfy the the values. It is therefore not sufficient to prove congruence.

Constructions

Another way to think about the above is to ask if it is possible to construct a unique triangle given what you know. For example, If you were given the lengths of two sides and the included angle (SAS), there is only one possible triangle you could draw. If you drew two of them, they would be the same shape and size – the definition of congruent.

Properties of Congruent Triangles

If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle. This is the true value of the concept; once you have proved two triangles are congruent, you can find the angles or sides of one of them from the other.

To remember this important idea, some find it helpful to use the acronym CPCTC, which stands for “Corresponding Parts of Congruent Triangles are Congruent”.

In addition to sides and angles, all other properties of the triangle are the same also, such as area, perimeter, location of centers, circles etc.

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