Geometric Congruence
Two geometric figures are congruent if they have exactly the same shape and size, meaning one can be transformed into the other through rigid motions such as translation, rotation, or reflection without any stretching or scaling. Congruence is denoted by the symbol ≅ and is a foundational concept for proving geometric theorems and properties. It is distinct from similarity, which allows size differences while preserving shape.
Triangle Congruence Criteria (Postulates)
| Criterion | Full Name | Required Elements | Example Use |
|---|---|---|---|
| SSS | Side-Side-Side | All 3 sides equal | Comparing two triangles with known side lengths |
| SAS | Side-Angle-Side | 2 sides and included angle equal | Two sides and the angle between them match |
| ASA | Angle-Side-Angle | 2 angles and included side equal | Two angles and the side between them match |
| AAS | Angle-Angle-Side | 2 angles and non-included side equal | Two angles and a corresponding side match |
| RHS / HL | Right-Hypotenuse-Side | Right angle, hypotenuse, one side equal | Only for right triangles |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Geometric Similarity
Two geometric figures are similar if they have the same shape but not necessarily the same size, meaning one can be obtained from the other by a combination of rigid motions and a uniform scaling (dilation). Similar figures have equal corresponding angles and proportional corresponding sides. Similarity is denoted by the symbol ~ and is the basis for scale drawings, maps, and many real-world applications including shadow calculations and indirect measurement.
Triangle
A triangle is a polygon with exactly three sides, three angles, and three vertices. The sum of the interior angles of any triangle always equals 180°, making it the simplest closed polygon. Triangles are the most rigid of all polygons and are widely used in engineering structures, architecture, and navigation due to their inherent stability.
Pythagorean Theorem
The Pythagorean Theorem states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides (the legs). It is one of the most famous and widely applied theorems in mathematics, used in distance calculations, navigation, construction, and virtually every branch of science and engineering.
From Latin "congruentia" meaning agreement or harmony, derived from "congruere" meaning to come together, agree, or correspond. "Con-" means together and "gruere" relates to falling or moving together. The mathematical usage was established by Euclid's translators in the Renaissance.