Geometry
Congruence
Shapes are congruent if they have identical angles and sides. They must have the same:
- number of angles
- angle values
- number of sides
- side lengths
Congruence is preserved if a shape is:
- translated
- rotated
- reflected
However, congruency is not preserved if the shape is scaled because the side lengths would not be equal.
Congruent Triangles
Only 3 pieces of information are required to prove congruence or solve a triangle. Despite being true in most cases, there are some situations where this statement does not appear to be true.
| Rule | Valid | Description |
|---|---|---|
| SSS | check | All 3 sides are known |
| SAS | check | Known angle is between 2 known sides |
| ASS | close | Opposite and adjacent sides of the known angle is known |
| SAA | check | Opposite and adjacent angles of the known side is known |
| ASA | check | Known side is between 2 known angles |
| AAA | close | All 3 angles are known |
AAA congruence does not always work because the side lengths could be different. ASS congruence does not always work because it is possible for the triangle to have 2 solutions. That being said, there are cases where only 1 solution is possible using ASS congruence when:
- angle ≥ 90
- opposite ≥ adjacent
- opposite = adjacent × sine of angle (right triangle)