Basic Theorems of Geometry

One of the concepts in Geometry is to use simple assumptions (postulates and properties) and to build up a complicated system of mathematics by proving theorems. Once a theorem has been proved, it can be used to build even more complicated theorems.

Sometimes there are related ideas that are so close to the original theorem that they are not considered a separate theorem. These are called corollaries.

All the postulates, theorems, and corollaries are listed in your textbook beginning on page R1.


Postulates

Two points defines a line You can draw exactly one and only one straight line through any two points
Two lines that intersect If two lines intersect, then they intersect in exactly one and only one point
Two planes that intersect If two planes intersect, then they intersect in exactly one and only one straight line
Three points defines a plane Through any three noncollinear points (points that don't line up) you can draw exactly one plane
Ruler Postulate The distance between any two points on a line is the absolute value of the difference of the coordinates
Segment Addition Postulate If three points are collinear (all on the same line) and B is between A and C, then AB + BC = AC
Protractor Postulate Let OA and OB be opposite rays in a plane. Then all of the rays that can be drawn with the endpoint at O that can be drawn on one side of line AB will line up with a measure on the protractor.

OA is paired with 0 degrees and OB is paired with 180 degrees

The measure of an angle formed by any two rays on the same side of line AB is given by m∠ COD = m∠ AOC - m∠ AOD

Angle Addition Postulate If point B is in the interior of angle AOC, then

m∠ AOB + m∠ BOC = m∠ AOC

If angle AOC is a straight angle, then 

m∠ AOB + m∠ BOC = 180 degrees

Theorems

Vertical Angles Theorem Vertical angles are congruent.
Congruent Supplements Theorem If two angles are supplements of congruent angles (or the same angle) then the two angles are congruent.
Congruent Complements Theorem If two angles are complements of congruent angles (or the same angle) then the two angles are congruent.
Triangle Angle-Sum Theorem The sum of the measures of the angles in a triangle is 180 degrees
Exterior Angle Theorem The measure of each exterior angle in a triangle equal the sum of the measures of its two remote interior angles.
  Corollary: The measure of an exterior angle in a triangle is greater than the measure of either of its remote interior angles.
Polygon Interior Angle-Sum Theorem The sum of the measures of the interior angles of an n-gon is (n-2)180
Polygon Exterior Angle-Sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360 degrees
  Two lines that are parallel to a third line are parallel to each other.
  In a plane, two lines that are perpendicular to a third line are parallel to each other.

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Last Updated: 9/28/2011
K. A. Shearer 2006-2011

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