Geometry Theorems and Postulates List with Examples

Many geometric problems require a strong knowledge of geometry theorems and postulates. That’s why I’ve put together this handy geometry theorems and postulates list with examples to help you dig into the most important ones!

What are Geometry Theorems?

In the study of geometry in general, a theorem is a statement that can be proven by using definitions, postulates, or other proven theorems. Simply put, we can prove geometry theorems by using other known geometry facts. In Euclidean geometry, we use a mathematical system that relies on theorems to help us solve a variety of different geometric problems.

There are many examples of common geometry theorems that you have likely explored already! For example, the angle sum theorem tells us that the sum of the measures of the angles in a triangle will add up to 180 degrees.

What are Geometry Postulates?

So if geometry theorems are proven using postulates, what are geometry postulates? We can define postulates as statements that we assume are true without proving them. Notice the difference between theorems and postulates here!

In most cases, postulates don’t actually need a proof. For example, the parallel postulate tells us that for a given point not on a line, there is only one line passing through that point that is parallel to the other line.

It is hard to argue this postulate, isn’t it? You can’t make the line pass through the point in a different way that is still parallel to the other line. And there isn’t a geometry proof that can show this postulate to be false (unless you are in non-Euclidean space, but that is a topic for another day!).

Geometry Theorems and Postulates List with Examples

Now that you understand the difference between geometry theorems and postulates, you are ready to dig into this geometry theorems and postulates list with examples.

I have included a few examples and diagrams because I think this is an important part of learning geometry! Many times, these postulates sound so confusing that you need a diagram!

Euclid’s 5 Postulates

Euclid was a Greek mathematician who’s postulates and axioms form the basis of much of the geometry that we study today. Euclid has five famous postulates that are some of the most important in the study of geometry. The first four postulates are quite simple without the need for much mathematical reasoning:

• A straight line may be drawn between any two points.
• A line segment may be extended indefinitely.
• A circle may be drawn with a center and any radius.
• All right angles are congruent.

Euclid’s fifth postulate, however, is know for its length, wordiness, and confusing nature:

• Euclid’s Fifth Postulate: If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

Pretty confusing, right? To help you understand Euclid’s fifth postulate, I have included an important diagram below. In the diagram, we have two lines: L1 and L2. There is a third line that intersects both of these lines. Angles A and B are each less than 90 degrees. The postulate tells us that the only way L1 and L2 will intersect is on the side where the angles are each less than 90 degrees.

This postulate’s confusing nature led to the study of non-Euclidean geometry, which is a much more complex topic that is interesting to explore if you are curious!

Check out this short video for a visual example of each of these postulates!

Angle Theorems and Postulates

There are also many angle theorems and postulates that are useful in studies of geometry and trigonometry. For example, angles of elevation and depression word problems require the use of the alternate interior angles theorem.

The following are examples of angle theorems and postulates:

• Linear Pair Theorem: If two angles form a linear pair (ie. a straight angle), then the angles are supplementary angles.
• Corresponding angles postulate: If two parallel lines are intersected by a transversal, then the corresponding angles have equal measure.
• Alternate Interior Angles Theorem: If two parallel lines are intersected by a transversal, then alternate interior angles are equal in measure.
• Alternate Exterior Angles Theorem: If two parallel lines are intersected by a transversal, then alternate exterior angles are equal in measure.
• Angle Addition Postulate: the sum of two angle measures that are joined by a common ray will be equal to the measure of the angle they form.

• If two parallel lines are intersected by a transversal, then the interior angles on the same side of the transversal are are supplementary.
• If two congruent angles are supplementary, then each is a right angle.

Parallelogram Theorems

There are four key theorems that allow us to classify a given quadrilateral as a parallelogram.

1. If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram.
2. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
3. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
4. If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.

Line and Plane Theorems

Next up are a couple important theorems that help us work with lines and planes.

• Perpendicular lines theorem: In a coordinate plane, two nonvertical lines are perpendicular lines if and only if the product of their slopes is -1.
• Through any three noncollinear points there is exactly one plane containing them.
• If a point lies outside a line, then exactly one plane contains both the line and the point.

Triangle Postulates and Theorems

Lastly, there are many postulates and theorems that help us understand relationships between side lengths and angles of a triangle. For example:

• If the three sides of a triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar.
• Angle-angle-side Congruence Theorem (AAS theorem): If two angles and a non-included side of one triangle are equal in measure to the corresponding angles and side of another triangle, then the triangles are congruent.
• Each angle of an equilateral triangle is the same and measures 60 degrees each.

Using Geometry Theorems and Postulates

Geometry is a subject that depends heavily upon understanding its theorems and postulates. Having a good understanding of some of the key ones will help you in many ways as you study mathematics.

Whether you are learning how to complete your first proof or you are looking to explore deeper mathematical questions about the very foundations of geometry, understanding geometric theorems is important.

This geometry theorems and postulates list with examples will get you started as you dig into the wonderful world of geometry!

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