
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. 
Angles:
Adjacent Angles 
Two angles that share a common vertex, a common side, and no common interior points (don't overlap).
m∠ABD and m∠DBC are adjacent. m∠ABC and m∠DBC are not adjacent 


Linear Pair 
Two adjacent angles whose noncommon sides for a straight line. 
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 Interior 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.

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 
Congruent Triangles 
Triangles that are congruent if there corresponding angles are congruent and their corresponding sides are congruent. 
Shortcuts to verify congruent triangles 
SSS, ASA, AAS, SAS, HL(in right triangles) 
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:
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 
• is a four sided polygon
•
a figure with exactly four sides

Parallelograms 
• is a quadrilateral w/ both pairs of opposite sides parallel
• opposite
sides are equal in length 
Rectangle 
• is a parallelogram with 4 right angles
• two pairs of parallel sides
• opposite sides of equal length 
Rhombus 
• is a parallelogram with all 4 sides of equal length
• two pairs of parallel sides

Square 
• is a parallelogram with 4 sides of equal length and 4 right angles
• two pairs of parallel sides 
Trapezoid 
• quadrilateral with at least one pair of parallel sides

Isosceles Trapezoid 
• is a trapezoid with congruent base angles
• at least one pair of parallel sides
• legs congruent

Kite 
• is a quadrilateral with two sets of adjacent sides equal
• not a trapezoid and not a parallelogram


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}) or decompose 
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
(also called right rectangular prism) 
SA formula assumes a "closed box" with all 6 sides. 
Cube
[special case of rectangular solid with all edges equal) 
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
(rectangular or triangular) 
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


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