Posulate and theoroem
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This document outlines several postulates of geometry: 1) Postulates are assumed to be true without proof and serve to explain undefined terms and prove other statements. 2) Seven postulates are described relating to lines, points, planes, and distances on a number line. 3) Five additional postulates are listed relating to line segments, circles, angles, and the intersection of lines cut by a transversal.
Posulate and theoroem
- 2. Postulates are statements that are assumed to be true without proof. Postulates serve two purposes - to explain undefined terms, and to serve as a starting point for proving other statements.
- 4. A) Unique Line Assumption: Through any two points, there is exactly one line. Note: This doesn't apply to nodes or dots. B) Dimension Assumption: Given a line in a plane, there exists a point in the plane not on that line. Given a plane in space, there exists a line or a point in space not on that plane. C) Number Line Assumption: Every line is a set of points that can be put into a one-to-one correspondence with real numbers, with any point on it corresponding to zero and any other point corresponding to one. Note: This doesn't apply to nodes or dots. This was once called the Ruler Postulate.
- 5. D) Distance Assumption: On a number line, there is a unique distance between two points. E) If two points lie on a plane, the line containing them also lies on the plane. F) Through three noncolinear points, there is exactly one plane. G) If two different planes have a point in common, then their intersection is a line.
- 7. A) Two points determine a line segment. B) A line segment can be extended indefinitely along a line. C) A circle can be drawn with a center and any radius. D) All right angles are congruent. Note: This part has been proven as a theorem. See below, proof. E) If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180 degrees, then the lines will intersect on that side of the transversal.