# Talk:Ideas in Geometry/Instructive examples

Megan Fitzgerald’s comment on Problem 9: Proving Parallel Bisectors in Quadrilaterals

I think that even though it looks like this group had to make a few assumptions, a good job was done on this problem. In this case actually, I think that you would have to make the assumptions. Given the nature of the problem, they seem like pretty safe ones. In order to find out about whether or not the lines are parallel they would have to use both the interior and exterior angles. Without making these assumptions there wouldn’t be a way to figure it out.

As Megan stated before, I think that something were assumed in order to get this problem to work out. I think maybe you could explore those assumptions that were made and just explain them. Or just state that you are assuming, as said above I think you need to assume that the lines are parallel. Overall I think you guys did a great job --Heather Brozowski (Hbrozowski 02:14, 9 October 2010 (UTC))

## Lesson One: Euclid's Axioms

[edit source]Lesson One: Euclid's Axioms

Euclid was known as the “Father of Geometry.” In his book, The Elements, Euclid begins by stating his assumptions to help determine the method of solving a problem These assumptions were known as the five axioms. An axiom is a statement that is accepted without proof. In order they are: 1. A line can be drawn from a point to any other point. 2. A finite line can be extended indefinitely. 3. A circle can be drawn, given a center and a radius. 4. All right angles are ninety degrees. 5. If a line intersects two other lines such that the sum of the interior angles on one side of the intersecting line is less than the sum of two right angles, then the lines meet on that side and not on the other side. (also known as the Parallel Postulate) To explain, axioms 1-3 establish lines and circles as the basic constructs of Euclidean geometry. The fourth axiom establishes a measure for angles and invariability of figures. The fifth axiom basically means that given a point and a line, there does not exist a line through that point parallel to the given line. The following picture shows a diagram for each axiom. File:5axioms.png