Euclid is well known for being the Father of Geometry. He didn't discover Geometry, but revolutionized the way Geometry was used and added postulates, axioms, theorems, and proofs to the world of Geometry.
Euclid was a Greek mathematician, who was best known for his contributions to Geometry. He wrote the Elements ; it was a volume of books which consisted of the basic foundation in Geometry. The foundation included five postulates, or statements that are accepted true without proof, which became the fundamentals of Geometry. The postulates were as follows:
- A straight line segment can be drawn joining any two points.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight lines segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All Right Angles are congruent.
- If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two Right Angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the Parallel Postulate.
Click on the link to watch the TED-Ed video about proofs:
A proof is a way of rationalizing your thinking. The different types of proofs are:
- Arithmetic Proofs
- Geometric Proofs
Geometric proofs are proofs related to any special component in Geometry, such as segment pairs and angle pairs. They usually consist of postulates, axioms, and theorems. Below are a couple of Geometric proofs:
- Protractor Postulate: Given and a point O on , all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180.
- Angle Addition Postulate: If S is in the interior of ∠PQR, then m∠PQS + m∠SQR = m∠PQR.
- Linear Pair Theorem: If two angles form a linear pair, then they are supplementary.
- Congruent Supplements Theorem: If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent.
- Right Angle Congruence Theorem: All right angles are congruent.
- Congruent Complements Theorem: If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent.
In Architecture, proofs are used to reason with the conclusions made about the buildings. It helps architects answer how safe the building it is and how sturdy the structure is. It is what confirmed that structures, such as the Parthenon, the Leaning Tower of Pisa, and the Hanging Gardens of Babylon, still stand today.