Download Axiom 1. Quantities which are equal to the same quantity are equal

GEOMETRY AXIOMS
(Updated 1/25/12)
Axiom 1. Quantities which are equal to the same quantity are equal
to each other. (If a = b and c = b, then a = c.)
Axiom 2. Any quantity may be substituted for its equal in a
mathematical expression. (If x + 1 = 7 and if x = 2y, then 2y + 1 = 7.)
Axiom 3. If equals be added to equals, the sums are equal.
Axiom 4. If equals be subtracted from equals, the remainders are
equal.
Axiom 5. If equals be multiplied by equals, the products are equal.
Axiom 6. If equals be divided by equals, the quotients are equal.
Axiom 7. The whole equals the sum of its parts.
Axiom 8. The whole is greater than any of its parts.
Axiom 9. If a and b are any two magnitudes of the same kind then
either a < b, or a > b or a = b.
Axiom 10. Only one straight line can be drawn through two points.
Axiom 11. The straight line segment is the shortest line that can be
drawn between two points.
Axiom 12. All right angles are equal.
Axiom 13. An angle has only one bisector.
Axiom 14. A segment has only one mid-point.
Theorem 0. If a straight line stands on a straight line, then it
makes two supplementary angles.
Postulate 1. It is possible to draw a straight line from any point to
any other point.
Postulate 2. It is possible to extend a line segment continuously in
a straight line.
Postulate 3. It is possible to describe a circle with any center and
any radius.
Postulate 4. All right angles are equal to one another.
Postulate 5. If a straight line falling on two straight lines make the
interior angles on the same side less than two right angles, the two
straight lines, if produced indefinitely, intersect on that side on
which are the angles less than the two right angles.
Chapter 3 Theorems
Theorem 3.3.1. If two lines are parallel, then the interior angles on
the same side of the transversal are supplementary.
Theorem 3.3.2. If two lines cut by a transversal are parallel, then
the corresponding angles are equal.
Theorem 3.3.3. If two lines cut by a transversal are parallel, then
the alternate interior angles are equal.
Alternate Interior Angles Postulate. If a transversal intersects two
lines so that the alternate interior angles are equal, then the lines
are parallel.
Theorem 3.7.1. If corresponding angles are equal, then the lines
are parallel.
Theorem 3.8.1. If two lines are cut by a transversal so that the
angles on the same side of the transversal are supplementary, then
the lines are parallel.
Theorem 5.4.1. In rectangle ABCD with diagonal AC,
∠1 = ∠4 and ∠2 = ∠3.
1
C
2
1
111
3
A
4
Polygon vertices. The total number of diagonals of a polygon is
equal to one half of the product of number of vertices and the
number of diagonals from each vertex.
Theorem 5.9.1. Angle sum theorem: The sum of the angle
measures in any triangle is 180°.
Corollary to 5.9.1. An exterior angle of a triangle is equal to the
sum of the two non-adjacent angles.