geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

s. In these tests, each

side Side or Sides may refer to: Geometry * Edge (geometry) of a polygon (two-dimensional shape) * Face (geometry) of a polyhedron (three-dimensional shape) Places * Side (Ainis), a town of Ainis, ancient Thessaly, Greece * Side (Caria), a town of an ...

and each

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

in one polygon is paired with a side or angle in the second polygon, taking care to preserve the order of adjacency. For example, if one polygon has sequential sides , , , , and and the other has sequential sides , , , , and , and if and are corresponding sides, then side (adjacent to ) must correspond to either or (both adjacent to ). If and correspond to each other, then corresponds to , corresponds to , and corresponds to ; hence the th element of the sequence corresponds to the th element of the sequence for On the other hand, if in addition to corresponding to we have corresponding to , then the th element of corresponds to the th element of the reverse sequence . Congruence tests look for all pairs of corresponding sides to be equal in length, though except in the case of the

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

this is not sufficient to establish congruence (as exemplified by a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

and a

rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...

that have the same side length). Similarity tests look at whether the

ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

s of the lengths of each pair of corresponding sides are equal, though again this is not sufficient. In either case equality of corresponding angles is also necessary; equality (or proportionality) of corresponding sides combined with equality of corresponding angles is necessary and sufficient for congruence (or similarity). The corresponding angles as well as the corresponding sides are defined as appearing in the same sequence, so for example if in a polygon with the side sequence and another with the corresponding side sequence we have vertex angle appearing between sides and then its corresponding vertex angle must appear between sides and .

References

{{reflist Geometry