
This is a partial listing of the more popular rules (theorems, postulates, and properties) that you will be using in your study of Geometry.
First a few words that refer to types of geometric "rules":
• A theorem is a statement (rule) that has been proven true using facts, operations and other rules that are known to be true. These are usually the "big" rules of geometry. A short theorem referring to a "lesser" rule is called a lemma.
• A corollary is a followup to an existing proven theorem. Corollaries are offshoots of a theorem that require little or no further proof.
• A postulate (or axiom) is a statement (rule) that is taken to be true without proof. Euclid derived many of the rules for geometry starting with a series of definitions and only five postulates.
• A property is a quality or characteristic belonging to something.
For example, the real numbers have the associative, commutative and distributive properties.

Your textbook (and your teacher) may want you to remember these "rules" with slightly different wording.
Be sure to follow the directions from your teacher. 
Real Number Properties:
Reflexive Property 
A quantity is 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. 
Division Postulate 
If equal quantities are divided by equal nonzero quantities, the quotients are equal. 
Substitution Postulate 
A quantity may be substituted for its equal in any expression. 
Segments:
Ruler Postulate 
Points on a line can be paired with the real numbers. 
Segment Addition Postulate

The whole is equal to the sum of its parts.
When B lies between A and C on a segment,
AB + BC = AC

Midpoint of Segment 
The midpoint of a segment is a point on the segment forming two congruent segments (equal segments).

Bisector of Segment 
The bisector of a segment is a line, a ray, or segment which cuts the given segment into two congruent segments (equal segments). 
Euclid's Postulate 1 
A straight line segment can be drawn joining any two points.

Euclid's Postulate 3 
Any straight line segment can be extended indefinitely in a straight line. 
Angles:
Angle Addition Postulate 
The whole is equal to the sum of its parts.
m∠ABD + m∠DBC = m∠ABC 


Right Angles
(Euclid's Postulate 4) 
All right angles are congruent (equal in measure).
(They all have a measure of 90º.)

Straight Angles 
All straight angles are congruent (equal in measure).
(They all have a measure of 180º.)

Vertical Angles 
Vertical angles are congruent (equal in measure).
m∠1 = m∠2
m∠3 = m∠4 


Triangle Sum 
The sum of the measures 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.

Base Angle Theorem
(Isosceles Triangle) 
If two sides of a triangle are congruent, the angles opposite these sides are congruent (equal in measure). 
Base Angle Converse
(Isosceles Triangle) 
If two angles of a triangle are congruent, the sides opposite these angles are congruent (equal in length). 
Angles forming a straight line 

Angles around a point 

Complementary Angles 
Two angles the sum of whose measures is 90º. 
Supplementary Angles 
Two angles the sum of whose measures is 180º. 
Triangles:
Pythagorean Theorem 
c^{2} = a^{2} + b^{2}
In a right triangle, the square of the hypotenuse equals the sum of the square of the lengths of the legs. 
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. 
Largest Angle 
In a triangle, the largest angle is across from the longest side 
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. 
Sides of Similar Δs 
Corresponding sides of similar triangles are in proportion. 
Parallels:
Construction 
Through a point not on a line, one and only one parallel to that line can be drawn. 
Construction 
From a given point on (or not on) a line, one and only one perpendicular can be drawn to the line. 
Corresponding Angles 
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 
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. 

Quadrilaterals:
Quadrilateral 
• a figure with exactly four sides
• the sum of the interior angles is 360º

Parallelograms 
About Sides 
• opposite sides are parallel
• opposite
sides are congruent 
About Angles 
• opposite angles are congruent
• consecutive angles are supplementary 
About Diagonals 
• diagonals bisect each other
• diagonals form two congruent triangles 
Rectangle 
• is a parallelogram
• has 4 right angles
• diagonals are congruent 
Rhombus 
• is a parallelogram
• has 4 congruent sides
• diagonals bisect the angles
• diagonals are perpendicular

Square 
• has all the properties of a parallelogram, a rectangle, and a rhombus 
Trapezoid 
• has at least one pair of parallel sides

Isosceles Trapezoid 
• has at least one pair of parallel sides
• legs congruent
• base angles congruent
•
diagonals are congruent
• opposite angles supplementary

Kite 
• figure with four sides
• two distinct pairs of adjacent sides congruent
• diagonals perpendicular
• one pair opposite angles congruent
• one diagonal creates 2 isosceles triangles
• one diagonal creates 2 congruent triangles
• one diagonal bisects the angles
• one diagonal bisects the other


Area (A), Volume (V), Surface Area (SA):
Rectangle 
A_{rectangle} = l × w = b • h
l= length; w = width; b = base; h = height 
Parallelogram 
A_{parallelogram} = b • h 
Triangle 
A_{Δ} = ½ • b• h 
Trapezoid 
A_{trapezoid} = ½ h (b_{1} + b_{2}) 
Regular Polygon 
A_{regular polygon} = ½ • a • p
a = apothem; p = perimeter 
Circle (circumference) 
C = 2πr = πd
r = radius; d = diameter

Circle (area) 
A_{circle} = πr^{2} 
Rectangular Solid 
SA formula assumes a "closed box" with all 6 sides. 
Cube
[special case of rectangular solid] 
SA formula assumes a "closed box" with all 6 sides. s = side 
Cylinder 
SA formula assumes a "closed container" with a top and a bottom. 
Cone 
SA formula assumes a "closed container", with a bottom. s = slant height 
Sphere 

Right Prism 
V_{right prism} = B • h; SA = 2B + p • h
B = area of the base; h = height; p = perimeter of base 
Pyramid
[assuming all of the faces (not the base) are the same] 
B = area of the base; h = height; p = perimeter of base; s = slant height


NOTE: The reposting of materials (in part or whole) from this site to the Internet
is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Use". 

