 POSTULATES AND THEOREMS

Postulates
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.

Euclid's Postulates

Two points determine a line segment.
A line segment can be extended indefinitely along a line.
A circle can be drawn with a center and any radius.
All right angles are congruent.
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.

Point-Line-Plane Postulates

Unique Line Assumption: Through any two points, there is exactly one line.

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.

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. This was once called the Ruler Postulate.

Distance Assumption: On a number line, there is a unique distance between two points.
If two points lie on a plane, the line containing them also lies on the plane.
Through three noncolinear points, there is exactly one plane.
If two different planes have a point in common, then their intersection is a line.

Theorems

Theorems are statements that can be deduced and proved from definitions, postulates, and previously proved theorems.
Line Intersection Theorem: Two different lines intersect in at most one point.

Betweenness Theorem:
If C is between A and B and on line AB, then AC + CB = AB.

Related Theorems:

Theorem: If A, B, and C are distinct points and AC + CB = AB, then C lies on line AB .

Pythagorean Theorem: a
2 + b2 = c2, if c is the hypotenuse.

General:
 Reflexive Property A quantity is congruent (equal) to itself. a = a Symmetric Property If a = b, then b = a. Transitive Property If a = b and b = c, then a = c. Addition Postulate If equal quantities are added to equal quantities, the sums are equal. Subtraction Postulate If equal quantities are subtracted from equal quantities, the differences are equal. Multiplication Postulate If equal quantities are multiplied by equal quantities, the products are equal. (also Doubles of equal quantities are equal.) Division Postulate If equal quantities are divided by equal nonzero quantities, the quotients are equal. (also Halves of equal quantities are equal.) Substitution Postulate A quantity may be substituted for its equal in any expression. Partition Postulate The whole is equal to the sum of its parts.Also: Betweeness of Points: AB + BC = ACAngle Addition Postulate: m Construction Two points determine a straight line. Construction From a given point on (or not on) a line, one and only one perpendicular can be drawn to the line.

Angles:

 Right Angles All right angles are congruent. Straight Angles All straight angles are congruent. Congruent Supplements Supplements of the same angle, or congruent angles, are congruent. Congruent Complements Complements of the same angle, or congruent angles, are congruent. Linear Pair If two angles form a linear pair, they are supplementary. Vertical Angles Vertical angles are congruent. Triangle Sum The sum of the interior angles of a triangle is 180º. Exterior Angle The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.The measure of an exterior angle of a triangle is greater than either non-adjacent interior angle. Base Angle Theorem(Isosceles Triangle) If two sides of a triangle are congruent, the angles opposite these sides are congruent. Base Angle Converse(Isosceles Triangle) If two angles of a triangle are congruent, the sides opposite these angles are congruent.

Triangles:

 Side-Side-Side (SSS) Congruence If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Side-Angle-Side (SAS) Congruence If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. Angle-Side-Angle (ASA) Congruence If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. Angle-Angle-Side (AAS) Congruence If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. Hypotenuse-Leg (HL) Congruence (right triangle) If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent. CPCTC Corresponding parts of congruent triangles are congruent. Angle-Angle (AA) Similarity If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. SSS for Similarity If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar. SAS for Similarity If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar. Side Proportionality If two triangles are similar, the corresponding sides are in proportion. Mid-segment Theorem(also called mid-line) The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. Sum of Two Sides The sum of the lengths of any two sides of a triangle must be greater than the third side Longest Side In a triangle, the longest side is across from the largest angle.In a triangle, the largest angle is across from the longest side. Altitude Rule The altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse. Leg Rule Each leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse.

Parallels:

 Corresponding Angles If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Corresponding Angles Converse If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel. Alternate Interior Angles If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Alternate Exterior Angles If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. Interiors on Same Side If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary. Alternate Interior AnglesConverse If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel. Alternate Exterior AnglesConverse If two lines are cut by a transversal and the alternate exterior angles are congruent, the lines are parallel. Interiors on Same Side Converse If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel. 