Congruence Basics

Congruence refers to the exact equivalence of size and shape between two or more geometric figures.
If two geometric figures are congruent, they can be perfectly overlaid on top of one another.
When two shapes are congruent, all corresponding sides and angles are congruent (identically sized and shaped).
The symbol for congruence is ‘≅’.

Congruence in Shapes

Two triangles are congruent if three sides and three angles of one triangle are respectively equal to the three sides and three angles of the other triangle.
Two circles are congruent if their radii are equal.
Two rectangles are congruent if their corresponding sides are equal.
Two squares are congruent if their sides are equal.
Two polygons are congruent if their matching sides are equal in length and their matching angles are equal in measure.

Congruence Tests for Triangles

Side-Side-Side (SSS): If all three sides of one triangle are equal to the corresponding sides of another, then the triangles are congruent.
Side-Angle-Side (SAS): If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another, then the triangles are congruent.
Angle-Side-Angle (ASA): If two angles and the included side of one triangle are equal to the corresponding angles and side of another, then the triangles are congruent.
Angle-Angle-Side (AAS): If two angles and a non-included side of one triangle are equal to the corresponding angles and side of another, then the triangles are congruent.
Right angle-Hypotenuse-Side (RHS): If the hypotenuse and one side of a right triangle are equal to the corresponding hypotenuse and side of another right triangle, then the triangles are congruent.

Transformations and Congruence

A transformation is a change in position, size, or shape of a geometric figure.
The main types of transformations that can result in congruent shapes are translations (moving), rotations (turning), and reflections (fliping). These are called rigid transformations, because they do not change the size or shape of the figure.
A dilation or scaling transformation changes the size but not the shape. It results in similar but not congruent figures.

Remember, understanding congruence is key in geometric proofs and problem-solving. All these tests and facts are tools to determine congruence or to prove that two figures are congruent.