Summary 1, theory
Euclid's Elements
The first book of the Elements consists of definitions and postulates. Another word for postulate is axiom. An axiom is a proposition that can not be proved but is considered to be self-evident. By using the wordings of the postulates it is possible to make so called ruler and straightedge constructions.
The first postulate is: To draw a straight line from any point to any point
This should be read as: It is possible to draw a straight line from any point to any point
The following is a simplified version.
Definitions and postulates
You should be able to state following definitions and postulates.
Definition 1 Two lines in a plane that do not intersect are parallel.
Definition 2 Two triangles are congruent if the corresponding sides and the corresponding angles are pair wise the same. If the triangles ΔABC and ΔDEF are congruent you can write it like this ΔABC≅ΔDEF
Postulate 1 You can draw one and only one line through two points.
number one the
hit list of famous postulates
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Postulate 2 The Parallel Postulate Given a straight line and a point not on the line, you can draw one and only one line through the point that is parallel to the first line.
Postulate 3 The Method of Superposition You can move, rotate and flip any geometrical figure without changing the shape or the size. (This is not a postulate in the Elements but it is used in the books anyway. For a detailed discussion check this out.)
Theorems
The three first cases you did as GeoGebra-exercises are called the congruence theorems.
You should be able to prove theses theorems using the definitions and postulates above.
Theorem 1 The Side-Angle-Side Congruence Theorem, SAS
Theorem 2 Isosceles triangles If two sides of a triangle are equal, then the opposite angles are equal.
Theorem 3 The Side-Side-Side Congruence Theorem, SSS
Theorem 4 The Angle-Side-Angle Congruence Theorem, ASA
by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License
