# Proving congruent with Euclidean axioms

• B
• ali PMPAINT
In summary, Euclidean axioms are fundamental principles used to prove geometric constructions and theorems in traditional Euclidean geometry. To prove congruence using these axioms, one must demonstrate that all corresponding sides and angles of two figures are equal. This can be done without specific measurements, as the axioms rely on definitions and properties of geometric figures. Congruence differs from similarity in that congruent figures have the same size and shape, while similar figures may differ in size. Proving congruence is important in geometry as it allows for confidence in stating that two figures are exactly the same, which has practical applications in fields such as construction. It also serves as a foundation for more complex theorems and constructions.
ali PMPAINT
TL;DR Summary
Proving SAS, SSS, ASA and that in a triangle, the bigger the opposite angle, the bigger the side is and in an Isosceles triangle, the two angles are ‌equal.
So, given one, you can prove the others, but I don't know how to prove one with using the five axioms.

#### Attachments

• euclid-sas.jpg
17.8 KB · Views: 310
ali PMPAINT
stevendaryl said:
According to http://www.math.uga.edu/sites/default/files/inline-files/10.pdf, these rules are not proved using Euclid's axioms, but are proved using unstated rules about moving triangles around.
Thank you, could you introduce me another one but this time for real numbers' axioms and basic proofs?

## 1. What are Euclidean axioms?

Euclidean axioms are a set of five basic assumptions that form the foundation of Euclidean geometry. They include the notions of points, lines, and planes, as well as the concepts of congruence, parallelism, and perpendicularity.

## 2. How do you prove congruence using Euclidean axioms?

Congruence can be proven using Euclidean axioms by showing that two geometric figures have the same shape and size. This can be done by demonstrating that all corresponding angles and sides are equal.

## 3. What is the difference between Euclidean and non-Euclidean geometry?

Euclidean geometry follows the five basic axioms, while non-Euclidean geometry does not. Non-Euclidean geometries include hyperbolic and elliptic geometries, which have different rules for parallel lines and angles.

## 4. Can Euclidean axioms be used to prove all geometric theorems?

No, Euclidean axioms are not sufficient to prove all geometric theorems. In fact, some theorems can only be proven using non-Euclidean geometries.

## 5. How do Euclidean axioms relate to real-world applications?

Euclidean axioms form the basis for traditional geometry and are used in various fields such as architecture, engineering, and navigation. They help us understand and describe the physical world around us.

• General Math
Replies
12
Views
1K
• General Math
Replies
1
Views
871
• General Math
Replies
8
Views
1K
• General Math
Replies
72
Views
5K
• General Math
Replies
1
Views
758
• General Math
Replies
1
Views
1K
• General Math
Replies
1
Views
751
• General Math
Replies
10
Views
2K
• General Math
Replies
1
Views
1K
• General Math
Replies
3
Views
1K