| Math Open Reference | search | Home Index About Contact |
Congruent Triangles
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 they are identical; every corresponding side and angle is identical. The angle at P has the same measure as the angle at L, the side PQ is the same length as the side LM etc. Try this Drag any orange dot at P,Q,R. The other triangle LMN will change to remain congruent to it. (If there is no image below, see support page.)RotationOne triangle can be rotated, but as long as they are otherwise identical, the triangles are still congruent. In the figure below, the triangle LMN is congruent to PQR even though it rotated anti-clockwise about 30°Try this In the figure below, drag any orange dot at P,Q, R and the other triangle will change to remain congruent to it. (If there is no image below, see support page.) ReflectionOne triangle can be a mirror image of the other, but as long as they are otherwise identical, the triangles are still congruent. It can be reflected in any direction, up down, left, right or anything in between. In the figure below, triangle LMN is a mirror image of PQR, but is still considered congruent to it.Try this Drag any orange dot at P,Q or R. The other triangle LMN will change to remain congruent to it. (If there is no image below, see support page.) They can have common partsCongruent triangles can also have a common side or vertex that is shared by both triangles. For example in the figure below, the two triangles have the side QR in common and are congruent. Note that they are also reflections (mirror images) of each other.Try thisDrag any orange dot at P,Q,R,S. The other triangle, which shares a side PQ with it, will change to remain congruent to it. (If there is no image below, see support page.) Imagine the triangles are cardboardOne 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 congruentAny 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:
Careful! 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.
In the figure on the right, the triangles have the same angles and look the same shape,
but are not congruent because they are not the same size.
They are called similar triangles (See Similar Triangles).
ConstructionsAnother 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 conguent. For more on constructions, see Introduction to ConstructionsProperties of Congruent TrianglesIf two triangles are congruent, then each side or angle is also congruent to the corresponding side or angle 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.Related topicsCongruent Triangles
Congruent Polygons
(C) 2007 Copyright John Page
|