Euclidean GeometryGeometric Definitions
Before we can write any proofs, we need some common terminology that will make it easier to talk about geometric objects. These are not particularly exciting, but you should already know most of them:
In Mathigon, large, solid dots indicate interactive points you can move around, while smaller, outlined dots indicate fixed points which you can’t move.
Lines are labeled using lower-case letters. We can also refer to them using two points that lie on the line, for example
When labelling rays, the arrow shows the direction where it extends to infinity, for example
The two shapes on the right basically look identical. They have the same size and shape, and we could turn and slide one of them to exactly match up with the other. In geometry, we say that the two shapes are
The symbol for congruence is
Here are a few different geometric objects – connect all pairs that are congruent to each other. Remember that more than two shapes might be congruent, and some shapes might not be congruent to any others:
Two line segments are congruent if they
Note the that “congruent” does not mean “equal”. For example, congruent lines and angles don’t have to point in the same direction. Still, congruence has many of the same properties of equality:
- Congruence is symmetric: if
X ≅ Ythen also Y ≅ X.
- Congruence is reflexive: any shape is congruent to itself. For example,
A ≅ A.
- Congruence is transitive: if
X ≅ Yand Y ≅ Zthen also X ≅ Z.
Parallel and Perpendicular
Two straight lines that never intersect are called
A good example of parallel lines in real life are railroad tracks. But note that more than two lines can be parallel to each other!
In diagrams, we denote parallel lines by adding one or more small arrows. In this example,
The opposite of parallel is two lines meeting at a 90° angle (right angle). These lines are called
In this example, we would write a