These are the typical geometry properties, postulates, theorems, corollaries and definitions commonly used in high school Geometry. The subjects are organized to correspond somewhat to each chapter in a typical textbook:

Many of these theorems are difficult to visualize. I hope to include a PDF formatted page that diagrams some of the more complicated theorems in the future. 

Elementary Geometry Proofs use one of five types of justifications below:

Properties: These are the rules you typically learned to use in algebra.
Definitions: Because we use the names of things in Geometry, definitions are required to properly use these terms in a proof. Proofs using terms like Perpendicular, Right, Congruent, Supplement will usually require the use of a definition in the proof since these are "invented" terms.
Postulates: These are "obvious" assumptions we make about Geometry. These cannot be formally proven, but are clearly true to most observers, because of the way the human mind understands how things work.
Theorems: These are rules that can be proven using postulates and other theorems. These are sometimes more complicated and can save a lot of time when they occur frequently in proofs.
Corollaries: These are fairly obvious Sub-Theorems (technically they are also Theorems) based on some important Theorem, but not important enough to be given its own name. When used, they refer to the Theorem they fall under. [ Corollary to the Theoreom
*Converse to Postulates and Theorems: Many Postulates and Theorems have converses that are also true. Converses are conditional statements where the "if" and "then" statements have been switched. Since many justifications have true converses, this saves having to give them separate names. [Converse to the Theorem] Properties and Definitions are always bi-conditional and do not use the term "Converse" when used in reverse.

Properties of Equality: If something is added, subtracted, multiplied, divided, squared, etc. to one side of an equation, it must be done to the other side. (There are also similar Properties of Inequality except that the direction of the inequality changes when multiplied/divided by a negative number) [If a=2 then a+1=3]
Reflexive Property: Any expression is always equal to itself. (There are also Reflexive Properties for Parallels, Congruence and Similarity) [a=a]
Symmetric Property: In an equation, the left and right sides are interchangeable. (There are also Symmetric Properties for Inequality, Parallels, Congruence and Similarity) [If a=b then b=a]
Transitive Property (Chain Rule): If the first expression is equal to the second and the second is equal to a third, then the first is equal to the third. This applies to longer chains with four, five or more expressions as well. (There are also Transitive Properties for Inequality, Parallels, Congruence and Similarity) [If a=b and b=c then a=c]
Substitution Property: If an expression or part of an expression is equal to another. The first can be replaced in every case by the second. Transitive Property is a special case of Substitution. [If a+b=c and b=2 then a+2=c] In Geometry proofs, this reason is given in cases when you are simplifying or combining like terms. [If 2a+3a=45 then 5a=45] 

Linear Postulate: For any two points, there is one and only one line that passes thru them.
Planar Postulate: For any three non-collinear points, there is one and only one plane that passes thru them.
Parallel Postulate: (In a plane) Given a line and a point not on the line, there is only one line that passes thru he point and is parallel to that line.
Perpendicular Postulate: (In a plane) Given a line and a point, there is only one line that passes thru the point and is perpendicular to that line.
Definition of Congruence: If two measurable geometric figures (usually segments and angles) are congruent, they have equal measure.
The Addition Postulates: The measures of two or more adjacent pieces of the same figure (segment, angle, arc, area, volume, etc.) add up to the measure of the whole figure.
Definition of Midpoint: A point that cuts a segment into two congruent/equal parts.
Definition of Bisector: A segment, ray or line that cuts a segment/angle into two congruent/equal parts.
Definition of Straight Angle: A straight line interpreted as an angle with the vertex located between the opposite rays.
Definition of Straight Angle Measure: A straight angle measures 180 degrees.
Definition of Right Angle: An angle that measures 90 degrees.
Right Angle Congruence Theorem: If two angles are right, they are congruent.
Converse to Right Angle Congruence Theorem: If one of two congruent angles is right then the other is right.
Definition of Supplementary Angles: Two angles whose sum is 180 degrees.
Definition of Complementary Angles: Two angles whose sum is 90 degrees.
Supplements Theorem ("Linear Pair Postulate"): If two adjacent angles form a straight angle, then they are supplementary. (Two such angles are called a Linear Pair.)
Vertical Angle Theorem: Vertical angles are congruent.
Congruent Supplements Theorem: If one of two congruent angles is supplementary to a third angle, then the second is supplementary to the third.
Converse to Congruent Supplements Theorem: If an angle is supplementary to two other angles, then those angles are congruent to each other.
Congruent Complements and Converse: (Same as for Congruent Supplements.) 

Linear Postulate: For any two points, there is one and only one line that passes thru them.
Planar Postulate: For any three non-collinear points, there is one and only one plane that passes thru them.
Parallel Postulate: (In a plane) Given a line and a point not on the line, there is only one line that passes thru he point and is parallel to that line.
Perpendicular Postulate: (In a plane) Given a line and a point, there is only one line that passes thru the point and is perpendicular to that line.
Definition of Congruence: If two measurable geometric figures (usually segments and angles) are congruent, they have equal measure.
The Addition Postulates: The measures of two or more adjacent pieces of the same figure (segment, angle, arc, area, volume, etc.) add up to the measure of the whole figure.
Definition of Midpoint: A point that cuts a segment into two congruent/equal parts.
Definition of Bisector: A segment, ray or line that cuts a segment/angle into two congruent/equal parts.
Definition of Straight Angle: A straight line interpreted as an angle with the vertex located between the opposite rays.
Definition of Straight Angle Measure: A straight angle measures 180 degrees.
Definition of Right Angle: An angle that measures 90 degrees.
Right Angle Congruence Theorem: If two angles are right, they are congruent.
Converse to Right Angle Congruence Theorem: If one of two congruent angles is right then the other is right.
Definition of Supplementary Angles: Two angles whose sum is 180 degrees.
Definition of Complementary Angles: Two angles whose sum is 90 degrees.
Supplements Theorem ("Linear Pair Postulate"): If two adjacent angles form a straight angle, then they are supplementary. (Two such angles are called a Linear Pair.)
Vertical Angle Theorem: Vertical angles are congruent.
Congruent Supplements Theorem: If one of two congruent angles is supplementary to a third angle, then the second is supplementary to the third.
Converse to Congruent Supplements Theorem: If an angle is supplementary to two other angles, then those angles are congruent to each other.
Congruent Complements and Converse: (Same as for Congruent Supplements.) 

Definition of Rise: The difference between the y-coordinates of two points.
Definition of Run: The difference between the x-coordinates of two points.
Definition of Slope: The ratio of rise to run.
Definitions of Intercept (x or y): The point where a line intersects the x or y-axis.
Distance Formula: Uses Pythagorean Theorem with the difference of the x-coordinates one leg, the difference of the y-coordinates the other leg and the distance the hypotenuse.
Property of Parallel Lines: Parallel lines have the same slope.
Property of Perpendicular Lines: Perpendicular lines have slopes that are opposite and reciprocal from each other (their product is -1). Lines with zero slope are perpendicular to lines with no (undefined) slope and vice versa.

Corresponding Angle Postulate: If two parallel lines are cut by a transversal, the corresponding angles are congruent.
Converse to Corresponding Angle Postulate: If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.
Alternate Interior and Alternate Exterior Angle Theorems and their Converses: (Same as for Corresponding Angle Postulate.)
Consecutive Interior Angle Theorem: If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.
Converse to Consecutive Interior Angle Theorem: If two lines are cut by a transversal and consecutive interior angles are supplementary, then the lines are parallel.
Transitive Parallels Postulate: If two lines are parallel to the same line, they are parallel to each other.
Definition of Perpendicular Lines: Two lines that form right angles.
Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other.
Converse to Perpendicular Transversal Theorem: If two lines are perpendicular to the same line, they are parallel to each other.

Definition of Isosceles Triangle: A triangle with at least two equal/congruent sides.
Definition of Equilateral Triangle: A triangle with all sides equal/congruent.
Triangle Sum Theorem: The sum of the interior angles of a triangle is 180 degrees.
Third Angle Theorem: If two sets of corresponding angles in two triangles are congruent, the third angles are congruent.
Right Triangle Corollary to the Triangle Sum Theorem: The acute angles of a right triangle are complementary.
Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
Definition of Congruent Triangles (Figures): Corresponding parts (segments and angles) of congruent triangles (figures) are congruent ("CPCTC").
*Congruence of Triangles and other Figures is Transitive: (See Transitive Property.)
SSS Triangle Congruence Postulate: If the corresponding sides of two triangles are congruent, the triangles are congruent.
SAS Triangle Congruence Postulate: If two sets of corresponding sides and the included angles are congruent, the triangles are congruent.
ASA Triangle Congruence Postulate: If two sets of corresponding angles and the included sides are congruent, the triangles are congruent.
AAS Triangle Congruence Theorem: If two sets of corresponding angles and a set of corresponding non-included sides are congruent, the triangles are congruent.
HL Right Triangle Congruence Postulate: If a set of corresponding legs and the hypotenuses of two right triangles are congruent, the triangles are congruent. (This is a special case for SSA triangles which is not true in all cases.)
Isosceles Triangle Theorem (Base Angle Theorem): If two sides of a triangle are congruent/equal, their opposite sides are congruent/equal.
Converse to Isosceles Triangle Theorem (Base Angle Theorem): If two angles of a triangle are congruent/equal, their opposite sides are congruent/equal.
Equilateral Triangle Theorem: If all three sides of a triangle are congruent/equal, all three angles are congruent/equal.
Converse to Equilateral Triangle Theorem: If all three angles of a triangle are congruent/equal, all three sides are congruent/equal.
Equilateral Triangle Corollary to the Triangle Sum Theorem: The measures of each angle of an equilateral triangle is 60 degrees. 

Quadrilateral Sum Theorem: The sum of the interior angles of a quadrilateral is 360 degrees.
Definition of Parallelogram: A quadrilateral with both sets of opposite sides parallel.
Parallelogram Theorem (Sides): Both sets of opposite sides are congruent.
Parallelogram Theorem (Angles): Both sets of opposite angles are congruent.
Parallelogram Theorem (Diagonals): Diagonals bisect each other.
Parallelogram Theorem (Consecutives): All pairs of consecutive angles are supplementary.
Converses to Parallelogram Theorems: If a quadrilateral satisfies the conditions of any of the previous theorems, it is a parallelogram.
Parallel and Congruent Theorem: If a quadrilateral has one set of opposite sides parallel and congruent, it is a parallelogram.
Definition of Rhombus: A parallelogram with all sides congruent.
Rhombus Theorem (Perpendiculars): The diagonals of a rhombus are perpendicular.
Rhombus Theorem (Angle Bisectors): The diagonals of a rhombus bisect the opposite angles.
Converse to Rhombus Theorems: If the figure is a parallelogram and satisfies the conditions of either of the above theorems, it is a rhombus.
Definition of Rectangle: A parallelogram with all four angles right.
Rectangle Theorem: The diagonals of a rectangle are congruent.
Converse to Rectangle Theorem: If the figure is a parallelogram and has congruent diagonals, it is a rectangle.
Definition of Square: A quadrilateral that is both a rhombus and a rectangle.
Definition of Trapezoid: A quadrilateral with one set of opposite sides (bases) parallel. (Other sides are called legs.)
Trapezoid Theorem: Consecutive angles are supplementary.
Converse to Trapezoid Theorem: If a quadrilateral has a set of consecutive angles supplementary, it is a trapezoid.
Definition of Isosceles Trapezoid: A trapezoid with two congruent legs.
Isosceles Trapezoid Theorem (Base Angles): The base angles of an isosceles trapezoid are congruent.
Isosceles Trapezoid Theorem (Diagonals): The diagonals of an isosceles trapezoid are congruent.
Converse to Isosceles Trapezoid Theorems: If a figure is a trapezoid and satisfies the conditions of either of the two theorems, it is an isosceles trapezoid.
Definition of Kite: A quadrilateral with two sets of adjacent sides congruent.
Kite Theorem (Perpendicular Diagonals): The diagonals of a kite are perpendicular.
Kite Theorem (Bisecting Diagonal): One diagonal bisects the other and the kite’s opposite angles.
Kite Theorem (Opposite Angles): The non-bisected angles of a kite are congruent.

Perpendicular Bisector Theorem: A point that lies on the perpendicular bisector of a segment is equidistant to the endpoints of that segment.
Converse to the Perpendicular Bisector Theorem: If a point is equidistant to the endpoints of a segment, it lies on the perpendicular bisector of that segment.
Angle Bisector Theorem: If a point lies on an angle bisector, it is equidistant to the sides of that angle.
Converse to the Angle Bisector Theorem: If a point is equidistant to the sides of an angle, it lies on the angle’s bisector.
Definition of Median: A segment drawn from a vertex of a triangle to the midpoint of the opposite side.
Definition of Altitude: A segment drawn from a vertex of a triangle and perpendicular to the opposite side.
Definition of Circumcenter: The point of concurrency between the three perpendicular bisectors of a triangle.
Circumcenter Theorem: The circumcenter is equidistant to the vertices of the triangle.
Definition of Incenter: The point of concurrency between the three angle bisectors of a traingle.
Incenter Theorem: The incenter is equidistant to the three sides of a triangle.
Definition of Centroid: The point of concurrency between the three medians of a triangle.
Centroid Theorem: The centroid divides each median into segments whose measures are in the ratio 1 to 2.
Definition of Orthocenter: The point of concurrency between the three altitudes of a triangle.
Definition of Midsegment: A segment drawn between the midpoints of two sides of a triangle.
Midsegment Theorem (Parallel): The midsegment is parallel to the third side of a triangle.
Midsegment Theorem (Length): The midsegment is half the measure of the third side. 

Properties of Inequality: Same as properties of equality except multiplication/division by negative numbers causes a change in the direction of the inequality.
Transitive Property of Inequality: Same as for equality as long as the direction of the inequalities in each link of the chain points the same direction. (This property applies even when there are some equalities in the chain.)
Triangle Inequality Postulate: The sum of the measures of any two sides of a triangle is greater than the measure of the third.
Exterior Angle Inequality Theorem: The measure of the exterior angle of a triangle is greater than the measure of either remote interior angle.
Triangle Comparison Theorem: The measures of the angles of a triangle have the same rank by size as their opposite sides. (Ranks also apply to parts equal in rank.)
Converse to Triangle Comparison Theorem: The measures of the sides of a triangle have the same rank by size as their opposite angles. (Ranks also apply to parts equal in rank.)
SSS Inequality Theorem (Hinge Theorem): If two sets of corresponding sides of triangles are congruent, the triangle with the greater third side will have the greater included angle also.
SAS Inequality Theorem (Converse to Hinge Theorem): If two sets of corresponding sides of triangles are congruent, the triangle with the greater included angle will have the greater third side also.

Definition of Reflection: A transformation where a figure is copied by "flipping" it over an axis/line.
Definition of Rotation: A transformation where a figure is copied by "turning" it about some point.
Definition of Translation: A transformation where a figure is copied by "sliding" it to a new location without changing its orientation.
Congruent Isometries Postulate: Figures (preimages) subject to one or more of the above transformations produces a congruent figure (image).
Angle of Reflection-Rotation Theorem: The angle of rotation between preimage and image is twice the measure of the angle of incidence between two axes of reflection that produce the same effect/image.

Proportion Property (Cross-Products): The cross-products in a proportion are equal.
Proportion Property (Means and Extremes): Switching the means, extremes or both in a proportion, produces a true proportion.
Proportion Property (Proportional Parts): If proportional parts are added or subtracted to the corresponding parts of a proportion, the result preserves that proportion.
Definition of Similarity: In similar figures, corresponding angles are congruent and corresponding sides/lengths are proportional.
*Similarity of Triangles and other Figures is Transitive: (See Transitive Property.)
AA Similarity Theorem: If two sets of corresponding angles in two triangles are congruent, the triangles are similar.
SSS Similarity Theorem: If all three sets of corresponding sides of two triangles are proportional, the triangles are similar.
SAS Similarity Theorem: If two sets of corresponding sides of two triangles are proportional and their included angles are congruent, the triangles are similar.
Parallel Proportionality Theorem: Corresponding parts of two transversals cut by parallel lines (or by the intersection of the two transversals) are proportional.
Angle Bisector Proportionality Theorem: If a triangle is cut by an angle bisector, the mirror image parts of the two triangles formed are proportional. 

Pythagorean Theorem: The sum of the squares of two legs of a right triangle is equal to the square of the hypotenuse.
Converse to the Pythagorean Theorem: If the sum of the squares of two sides of a triangle are equal to the square of the third side, the triangle is right.
Corollaries to the Pythagorean Theorem: If the square of the longest side is greater than the sum of the squares of the other two, the triangle is obtuse. If the longest side is less than, then the triangle is acute.
Definition of Geometric Mean: The square root of the product of two numbers. (Occurs in proportions when one cross-product is equal to a cross-product with two equal parts.)
Similar Right Triangle Theorem: When an altitude is drawn from the right angle of a triangle, three similar right triangles are formed.
Corollary to Similar Right Triangle Theorem: Either leg of the main right triangle or the altitude is the geometric mean of the two segments of the hypotenuse that touch that leg/altitude. (Sometimes called the Tripod Touch Technique)
Property of 45-45-90 Triangle: The ratio of the lengths of leg:hypotenuse is 1:%2.
Property of 30-60-90 Triangle: The ratio of the lengths of short leg:long leg:hypotenuse is 1:%3:2.

Definition of Sine: The ratio of the opposite leg to hypotenuse in relation to the angle of sine. (SOHCAHTOA)
Definition of Cosine: The ratio of the adjacent leg to hypotenuse in relation to the angle of cosine. (SOHCAHTOA)
Definition of Tangent: The ratio of the opposite to adjacent legs in relation to the angle of tangent. (SOHCAHTOA)
Law of Sines "Theorem": The sides of a triangle are proportional to the sines of the opposite angles.
Law of Cosines "Theorem": The square of a side of a triangle is equal to the sum of the squares of the other two sides subtracted by 2 times each of the other two sides times the cosine of the opposite angle.

Circles: