Terms used in geometry fall into two categories. The first category contains those terms that cannot be defined by simpler terms. The second category contains those that can be defined by simpler terms. Although some terms cannot be technically defined, these can be described in everyday language.
There are three terms used in geometry that cannot be defined: points, lines, and planes.
Point. A point in geometry is a location. It has no size i.e. no width, no length, and no depth. A point is shown by a dot and named by an uppercase letter.
Line. A line is a set of points that extends infinitely in two directions. It has length but no width. Points that are on the same line are called collinear points.
A line is identified by two points and is written as shown above with an arrowhead at each end. The line above would be designated line or line f. Two lines that meet in a point are called intersecting lines.
A plane extends infinitely in two dimensions. It has no thickness. An example of a plane is a coordinate plane. A plane is named by three points in the plane that are not on the same line. Shown below is an example of a plane: plane ABC.
A plane extends infinitely in all directions and is a set of all points in three dimensions.
These are terms that are defined by undefined terms, previously defined terms, and words used in everyday conversation.
A line segment is a part of a line that has defined endpoints. A line segment as the segment between C and D below is written as:
A line segment is named by its two endpoints. The shorthand for a line segment is to write the line segments two endpoints and draw a dash above them, like .
A ray can be defined as a part of a line that has a fixed starting point but no end point. It can extend infinitely in one direction. Two rays form the sides of an angle. The vertex of the angles is the starting point of the rays.
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 a plane.
An angle can be identified in two ways.
- Like this: ∠ABC
The angle symbol, followed by three points that define the angle, with the middle letter being the vertex, and the other two on the legs. In the figure above the angle would be ∠ABC or ∠CBA. So long as the vertex is the middle letter, the order is not important. As a shorthand we can use the ‘angle’ symbol. For example, ‘∠ABC’ would be read as ‘the angle ABC’.
- Or like this: ∠B
Just by the vertex, so long as it is not ambiguous. In the figure above the angle could also be called simply ‘∠B’
A postulate is a statement, sometimes called an axiom, which is taken to be true without proof. Postulates are the basic structure from which theorems are derived. The whole of Euclidean geometry, for example, is based on five postulates known as Euclid’s postulates.
Examples of Postulates
Two points determine a line segment.
A line segment can be extended indefinitely along a line.
A circle can be drawn with a center and any radius.
All right angles are congruent
If two lines intersect, they intersect in exactly one point.
Example of two lines intersecting at a point.
A theorem is a true statement that can be proven. Here are some examples:
- Angles on one side of a straight line always sum to 180°.
- Vertical angles are equal.
- In any triangle, the sum of two interior angles is less than two right angles.
- If two lines are intersected by a transversal, and if the alternate angles are equal, then the two lines are parallel.
Below is an example of interior alternate angles forming parallel lines.
A corollary is a statement that follows with little or no proof required from an already proven statement. For example, it is a theorem in geometry that the angles opposite two congruent sides of a triangle are also congruent. A corollary to that statement is that an equilateral triangle is also equiangular. Shown below is an equilateral triangle. In this triangle all the sides are congruent, and all the angles are also congruent.
Vertical angles are congruent.
We show that
from both sides, we have
Therefore, vertical angles are congruent.
If two lines are intersected by a transversal, and if the alternate angles are equal, then the two lines are parallel.
Solve the following problems.
- Vertical Angles
2. Alternate Interior Angles