What Is A Congruence Statement?

When it comes to the study of geometry, precision and specificity is key. It should come as no surprise, then, that determining whether or not two items are the same shape and size is crucial. Congruence statements express the fact that two figures have the same size and shape.

Objects that have the same shape and size are said to be congruent. Congruence statements are used in certain mathematical studies — such as geometry — to express that two or more objects are the same size and shape.

Nearly any geometric shape — including lines, circles and polygons — can be congruent. When it comes to congruence statements, however, the examination of triangles is especially common.

Altogether, there are six congruence statements that can be used to determine if two triangles are, indeed, congruent. Abbreviations summarizing the statements are often used, with S standing for side length and A standing for angle. A triangle with three sides that are each equal in length to those of another triangle, for example, are congruent. This statement can be abbreviated as SSS. Two triangles that feature two equal sides and one equal angle between them, SAS, are also congruent. If two triangles have two equal angles and a side of equal length, either ASA or AAS, they will be congruent. Right triangles are congruent if the hypotenuse and one side length, HL, or the hypotenuse and one acute angle, HA, are equivalent. Of course, HA is the same as AAS, since one side, the hypotenuse, and two angles, the right angle and the acute angle, are known.

When making the actual congruence statement– that is, for example, the statement that triangle ABC is congruent to triangle DEF– the order of the points is very important. If triangle ABC is congruent to triangle DEF, and they are not equilateral triangles, then the statement, "ABC is congruent to FED" is incorrect– that would be saying that line AB is equal to line FE, when in fact line AB is equal to line DE. The correct statement must be: "ABC is congruent to DEF".