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Geometric Rules Quick Reference
[Junior Level]
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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 follow-up to an existing proven theorem. Corollaries are off-shoots 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.

star 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.

star 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.

star Angles:

Angle Addition Postulate 
The whole is equal to the sum of its parts.
m∠ABD + m∠DBC = m∠ABC
anglediagram
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
vert
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 non-adjacent 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
180angles
Angles around a point
anglesaround
Complementary Angles
Two angles the sum of whose measures is 90º.
Supplementary Angles
Two angles the sum of whose measures is 180º.

star Triangles:

Pythagorean Theorem
c2 = a2 + b2
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
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) Congruence
If 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.
CPCTC
Corresponding 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.
Sides of Similar Δs
Corresponding sides of similar triangles are in proportion.

star 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.

star 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
quadfamily

star Area (A), Volume (V), Surface Area (SA):

Rectangle
Arectangle = l × w = b • h
l= length; w = width; b = base; h = height
Parallelogram
Aparallelogram = b • h
Triangle
AΔ = ½ • b• h
Trapezoid
Atrapezoid = ½ h (b1 + b2)
Regular Polygon
Aregular polygon = ½ • a • p
a = apothem; p = perimeter
Circle (circumference)
C = 2πr = πd
r =
radius; d = diameter
Circle (area)
Acircle = πr2
Rectangular Solid
recsolidformula
SA formula assumes a "closed box" with all 6 sides.
Cube
[special case of rectangular solid]
cubeformula
SA formula assumes a "closed box" with all 6 sides. s = side
Cylinder
cylinderformula
SA formula assumes a "closed container" with a top and a bottom.
Cone
coneformual1
SA formula assumes a "closed container", with a bottom. s = slant height
Sphere
sphereformula
Right Prism
Vright 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]
pyrformula
B
= area of the base; h = height; p = perimeter of base; s = slant height

 



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