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Euclidean GeometryGeometric Definitions

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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:

A point is a specific location in space. Points describe a position, but have no size or shape themselves. They are labelled using capital letters.

In Mathigon, large, solid dots indicate interactive points you can move around, while smaller, outlined dots indicate fixed points which you can’t move.

A line is a set of infinitely many points that extend forever in both directions. Lines are always straight and, just like points, they don’t take up any space – they have no width.

Lines are labeled using lower-case letters like a or b. We can also refer to them using two points that lie on the line, for example PQ or QP. The order of the points does not matter.

A line segment is the part of a line between two points, without extending to infinity. We can label them just like lines, but without arrows on the bar above: AB or BA. Like, before the order of the points does not matter.

A ray is something in between a line and a line segment: it only extends to infinity on one side. You can think of it like sunrays: they start at a point (the sun) and then keep going forever.

When labelling rays, the arrow shows the direction where it extends to infinity, for example AB. This time, the order of the points does matter.

A circle is the collection of points that all have the same distance from a point in the center. This distance is called the radius.


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 congruent.

The symbol for congruence is , so we would say that AB.

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 have the same lengthintersect. Two angles are congruent if they have the same sizemeet at a point (in degrees).

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 XY then also YX.
  • Congruence is reflexive: any shape is congruent to itself. For example, AA.
  • Congruence is transitive: if XY and YZ then also XZ.

Parallel and Perpendicular

Two straight lines that never intersect are called parallel. They point into the same direction, and the distance between them is always the sameincreasingdecreasing.

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, abc and de. The symbol simply means “is parallel to”.

The opposite of parallel is two lines meeting at a 90° angle (right angle). These lines are called perpendicular.

In this example, we would write a b. The symbol simply means “is perpendicular to”.