The **main difference** between postulates and theorems is that **postulates are taken to be true without proof , while theorems can and must be proven to be true** .

Theorems and postulates are two concepts that you will find in geometry. In fact, these are statements of geometric truth. Postulates are the ideas that are believed to be obviously true that do not require proof. Theorems are mathematical statements that we can / must prove to be true.

### Key areas covered

**1. What are postulates** *- definition, properties* **2. What are theorems** *- definition, properties* **3. Relationship between postulates and theorems** *- overview of the similarities* **4. Difference between postulates and theorems** *- comparison of the main differences*

### key terms

*Postulates, theorems *

## What are postulates?

Postulates are the mathematical statements that we hold to be true without evidence. They are ideas that we find so obviously true that they do not require proof. For example, the statement that a line contains at least two points is a postulate. This is so obvious and widely accepted that we usually don't need any evidence to accept it as true. We build theorems and lemmas based on theorems. In fact, it is possible to create a sentence from one or more postulates.

### Basic properties of postulates

- A postulate must be easy to understand - for example; it shouldn't contain many words that are difficult to understand.
- They should be consistent when combined with other postulates.
- We should be able to use them independently.

However, it is also important to note that some postulates are not always correct. A postulate can turn out to be false after a new discovery. For example, Einstein's postulate that the universe is homogeneous is no longer accepted as correct.

## What are theorems?

A theorem is a mathematical statement that we can prove to be true. We can prove them using logical reasoning or other theorems already proven to be true. In fact, a proposition that has to be proved in order to prove another proposition is called a lemma. Postulates are the foundation on which we build both lemmas and theorems. The four-color theorem, the Pythagorean theorem and Fermat's last theorem are some examples of sentences.

### Pythagorean theorem

According to the Pythagorean Theorem, if a triangle is at right angles (90 °) and each of the three sides of the triangle forms squares, the largest square has the same area as the other two squares combined. In other words, the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. We can state this with the equation a ^{2} + b ^{2} = c ^{2} .

## Relationship between postulates and theorems

- Theorems and postulates are statements about geometric truth.
- We create theorems based on postulates.

## Difference Between Postulates and Theorems

### definition

Postulates are the mathematical statements that we assume are true without proof, while theorems are mathematical statements that we can or must prove to be true.

### Proof

Postulates are taken to be true without proof, while theorems can be proven to be true.

### Need for proof

Also, we don't have to prove postulates because they say the obvious, but theorems are not that obvious and can be proved by reasoning or by using lemmas.

### diploma

Postulates are the mathematical statements that we assume are true without proof, while theorems are mathematical statements that we can or must prove to be true. Hence the main difference between postulates and theorems is their proof.

##### Reference:

1. “Postulates and Theorems.” CliffNotes, Available here .

##### Image courtesy:

1. “Parallel Postulates” by Alecmconroy in the English language Wikipedia (CC BY-SA 3.0) via Commons Wikimedia 2. “Pythagorean Proof (3)” by Brews ohare - Own work (CC BY-SA 3.0) via Commons Wikimedia