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 PropertyA quantity is congruent (equal) to itself. a = a
Symmetric PropertyIf a = b, then b = a.
Transitive PropertyIf 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 PostulateIf equal quantities are multiplied by equal quantities, the products are equal. (also Doubles of equal quantities are equal.)
Division PostulateIf equal quantities are divided by equal nonzero quantities, the quotients are equal. (also Halves of equal quantities are equal.)
Substitution PostulateA quantity may be substituted for its equal in any expression.
Partition PostulateThe whole is equal to the sum of its parts.
Also: Betweeness of Points: AB + BC = AC
Angle Addition Postulate: m
ConstructionTwo points determine a straight line.
ConstructionFrom a given point on (or not on) a line, one and only one perpendicular can be drawn to the line.

Angles:

Right AnglesAll right angles are congruent.
Straight AnglesAll straight angles are congruent.
Congruent SupplementsSupplements of the same angle, or congruent angles, are congruent.
Congruent ComplementsComplements of the same angle, or congruent angles, are congruent.
Linear PairIf two angles form a linear pair, they are supplementary.
Vertical AnglesVertical angles are congruent.
Triangle SumThe sum of the interior angles of a triangle is 180º.
Exterior AngleThe 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) CongruenceIf 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.
CPCTCCorresponding 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 SimilarityIf the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.
SAS for SimilarityIf 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 ProportionalityIf 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 SideIn a triangle, the longest side is across from the largest angle.
In a triangle, the largest angle is across from the longest side.
Altitude RuleThe altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse.
Leg RuleEach 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 AnglesIf 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 Angles
Converse
If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel.
Alternate Exterior Angles
Converse
If two lines are cut by a transversal and the alternate exterior angles are congruent, the lines are parallel.
Interiors on Same Side ConverseIf two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel.

Quadrilaterals:

Parallelograms





About Sides
* If a quadrilateral is a parallelogram, the opposite
sides are parallel.
* If a quadrilateral is a parallelogram, the opposite
sides are congruent.
About Angles* If a quadrilateral is a parallelogram, the opposite
angles are congruent.
* If a quadrilateral is a parallelogram, the
consecutive angles are supplementary.
About Diagonals* If a quadrilateral is a parallelogram, the diagonals
bisect each other.
* If a quadrilateral is a parallelogram, the diagonals
form two congruent triangles.
Parallelogram Converses







About Sides
* If both pairs of opposite sides of a quadrilateral
are parallel, the quadrilateral is a parallelogram.
* If both pairs of opposite sides of a quadrilateral
are congruent, the quadrilateral is a
parallelogram.
About Angles* If both pairs of opposite angles of a quadrilateral
are congruent, the quadrilateral is a
parallelogram.
* If the consecutive angles of a quadrilateral are
supplementary, the quadrilateral is a parallelogram.
About Diagonals


* If the diagonals of a quadrilateral bisect each
other, the quadrilateral is a
parallelogram.
* If the diagonals of a quadrilateral form two
congruent triangles, the quadrilateral is a
parallelogram.
ParallelogramIf one pair of sides of a quadrilateral is BOTH parallel and congruent, the quadrilateral is a parallelogram.
RectangleIf a parallelogram has one right angle it is a rectangle
A parallelogram is a rectangle if and only if its diagonals are congruent.
A rectangle is a parallelogram with four right angles.
RhombusA rhombus is a parallelogram with four congruent sides.
If a parallelogram has two consecutive sides congruent, it is a rhombus.
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
A parallelogram is a rhombus if and only if the diagonals are perpendicular.
SquareA square is a parallelogram with four congruent sides and four right angles.
A quadrilateral is a square if and only if it is a rhombus and a rectangle.
TrapezoidA trapezoid is a quadrilateral with exactly one pair of parallel sides.
Isosceles TrapezoidAn isosceles trapezoid is a trapezoid with congruent legs.
A trapezoid is isosceles if and only if the base angles are congruent
A trapezoid is isosceles if and only if the diagonals are congruent
If a trapezoid is isosceles, the opposite angles are supplementary.

Circles:

RadiusIn a circle, a radius perpendicular to a chord bisects the chord and the arc.
In a circle, a radius that bisects a chord is perpendicular to the chord.
In a circle, the perpendicular bisector of a chord passes through the center of the circle.
If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
Chords

In a circle, or congruent circles, congruent chords are equidistant from the center. (and converse)

In a circle, or congruent circles, congruent chords have congruent arcs. (and converse0
In a circle, parallel chords intercept congruent arcs
In the same circle, or congruent circles, congruent central angles have congruent chords (and converse)
TangentsTangent segments to a circle from the same external point are congruent
ArcsIn the same circle, or congruent circles, congruent central angles have congruent arcs. (and converse)
AnglesAn angle inscribed in a semi-circle is a right angle.

In a circle, inscribed angles that intercept the same arc are congruent.

The opposite angles in a cyclic quadrilateral are supplementary
In a circle, or congruent circles, congruent central angles have congruent arcs.