Theorems and Postulates for Geometry
This is a partial listing of the more popular theorems, postulates and properties needed when working with Euclidean proofs. You need to have a thorough understanding of these items.
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 = AC Angle Addition Postulate: m<ABC + m<CBD = m<ABD 
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 nonadjacent interior angles. The measure of an exterior angle of a triangle is greater than either nonadjacent 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:
SideSideSide (SSS) Congruence  If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. 
SideAngleSide (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. 
AngleSideAngle (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. 
AngleAngleSide (AAS) Congruence  If two angles and the nonincluded side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. 
HypotenuseLeg (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. 
AngleAngle (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. 
Midsegment Theorem (also called midline)  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 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 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. 
Quadrilaterals:
Parallelograms  About Sides 

About Angles 
 
About Diagonals 
 
Parallelogram Converses
 About Sides 

About Angles 
 
About Diagonals

 
Parallelogram  If one pair of sides of a quadrilateral is BOTH parallel and congruent, the quadrilateral is a parallelogram.  
Rectangle  If 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.  
Rhombus  A 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.  
Square  A 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.  
Trapezoid  A trapezoid is a quadrilateral with exactly one pair of parallel sides.  
Isosceles Trapezoid  An 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:
Radius  In 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)  
Tangents  Tangent segments to a circle from the same external point are congruent 
Arcs  In the same circle, or congruent circles, congruent central angles have congruent arcs. (and converse) 
Angles  An angle inscribed in a semicircle 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. 