Polygons

A polygon is a plane shape with straight sides.

Is it a Polygon?

Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).

Polygon (straight sides)	Not a Polygon (has a curve)	Not a Polygon (open, not closed)

Polygon comes from Greek. Poly- means "many" and -gon means "angle".

Types of Polygons

Regular or Irregular

If all angles are equal and all sides are equal, then it is regular, otherwise it is irregular

Regular		Irregular

Concave or Convex

A convex polygon has no angles pointing inwards. More precisely, no internal angle can be more than 180°.
If any internal angle is greater than 180° then the polygon is concave. (Think: concave has a "cave" in it)

Convex		Concave

Simple or Complex

A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself! Many rules about polygons don't work when it is complex.

Simple Polygon (this one's a Pentagon)		Complex Polygon (also a Pentagon)

More Examples

Irregular Hexagon		Concave Octagon		Complex Polygon (a "star polygon", in this case a pentagram)

Play With Them! Try Interactive Polygons ... make them regular, concave or complex.

Names of Polygons

		If it is a Regular Polygon...
Name	Sides	Shape	Interior Angle
Triangle (or Trigon)	3		60°
Quadrilateral (or Tetragon)	4		90°
Pentagon	5		108°
Hexagon	6		120°
Heptagon (or Septagon)	7		128.571°
Octagon	8		135°
Nonagon (or Enneagon)	9		140°
Decagon	10		144°
Hendecagon (or Undecagon)	11		147.273°
Dodecagon	12		150°
Triskaidecagon	13		152.308°
Tetrakaidecagon	14		154.286°
Pentadecagon	15		156°
Hexakaidecagon	16		157.5°
Heptadecagon	17		158.824°
Octakaidecagon	18		160°
Enneadecagon	19		161.053°
Icosagon	20		162°
Triacontagon	30		168°
Tetracontagon	40		171°
Pentacontagon	50		172.8°
Hexacontagon	60		174°
Heptacontagon	70		174.857°
Octacontagon	80		175.5°
Enneacontagon	90		176°
Hectagon	100		176.4°
Chiliagon	1,000		179.64°
Myriagon	10,000		179.964°
Megagon	1,000,000		~180°
Googolgon	10100		~180°
n-gon	n		(n-2) × 180° / n

Types of Triangles

Triangles can be classified by various properties relating to their angles and sides. The most common classifications are described on this page.

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Classifications of Triangles

Right Triangles
 

A right triangle has one 90° and a variety of often-studied properties including: Right Triangle Calculator Proof of Pythagorean Theorem Pythagorean Triplets Sine, Cosine, Tangent Pictures of Right Triangles 7 24 25 Right Triangle Images 3 4 5 Right Triangles 5 12 13 Right Triangles

The Equilateral triangle shown on the left has three equal sides and three equal angles.
Each angle is 60°

The Isosceles triangle shown on the left has two equal sides and two equal angles.

The Scalene Triangle has no congruent sides. In other words, each side must have a different length..

The Acute Triangle has three acute angles (an acute angle measures less than 90°)

The  Obtuse Triangle has an obtuse angle (an obtuse angle has more than 90°). In the picture on the left, the shaded angle is the obtuse angle that distinguishes this triangle

Since the total degrees in any triangle is 180°, an obtuse triangle can only have one angle that measures more than 90°.

Points, lines and planes

The word ‘geometry’ derives from the Greek words for ‘earth’ (geo) and ‘to measure’ (metron).

A construction is a geometric drawing for which only a compass and a straightedge may be used.

a compass: for making circles; for transferring the distance between two existing points

ruler/straightedge: used for drawing a line between two existing points; no distance markings will be used; no measurements are allowed

A conjecture is an educated guess.
Using specific observations and examples to arrive at a conjecture is called inductive reasoning.
For example, you might make the conjecture that for all real numbers x and y, the distance between them is given by the formula y−x.
A counterexample is a specific example that shows that a conjecture is not always true.
For example, here is a counterexample to the previous conjecture:
Let y=5 and x=7. Then, y−x=5−7=−2, which is not the distance between them.
Deductive reasoning uses logic, and statements that are already accepted to be true, to reach conclusions.
The methods of mathematical proof are based on deductive reasoning.
Point, line, and plane are three undefined terms to get us started in the study of geometry—we will just agree on their meaning.
A point represents an exact location. It is represented with a dot. Capital letters, like P, are frequently used to denote points.

DEFINITIONS space; geometric figure

Space is the set of all points.

A geometric figure is a subset of space.

That is, a geometric figure is any collection of points.
Of course, there are certain important geometric figures (like triangles and circles) that will be studied throughout the course.
A line has length only; it has no width or thickness; it extends forever in both directions.
A line will be denoted using a lowercase script letter, like ℓ.
If A and B are two distinct points, then they determine a unique line which will be denoted by AB←→ or BA←→.


A plane is a flat surface that extends infinitely in all directions; it has length and width only; it has no thickness.
A plane will be denoted using an uppercase script letter, like P.
If A, B, and C are three distinct noncollinear points (see below), then they determine a unique plane which will be denoted by ABC.


Note: In the following definitions, the prefix ‘co’ means ‘same’.

DEFINITIONS collinear/noncollinear/coplanar/noncoplanar points

collinear points: three or more points lying on the same line


noncollinear points: points not lying on the same line
coplanar points: points lying in the same plane
noncoplanar points: points not lying in the same plane

Coplanar can also refer to other geometric figures.
For example, two lines are coplanar if and only if they lie in the same plane.

Rational and real numbers

Numbers are classified according to type. The first type of number is the first type you ever learned about: the counting, or "natural" numbers:

1, 2, 3, 4, 5, 6, ...

The next type is the "whole" numbers, which are the natural numbers together with zero:

0, 1, 2, 3, 4, 5, 6, ...

Then come the "integers", which are zero, the natural numbers, and the negatives of the naturals:

..., –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, ...

The next type is the "rational", or fractional, numbers, which are technically regarded as ratios (divisions) of integers. In other words, a fraction is formed by dividing one integer by another integer.
Note that each new type of number contained the previous type within it. The wholes are just the naturals with zero thrown in. The integers are just the wholes with the negatives thrown in. And the fractions are just the integers with all their divisions thrown in. (Remember that you can turn any integer into a fraction by putting it over the number 1. For example, the integer 4 is also the fraction ⁴/₁.) Since you learned these number types in the same order as their hierarchy, it's easy to remember their order.
Once you're learned about fractions, there is another major classification of numbers: the ones that can't be written as fractions. Remember that fractions (also known as rational numbers) can be written as terminating (ending) or repeating decimals (such as 0.5, 0.76, or 0.333333....). On the other hand, all those numbers that can be written as non-repeating, non-terminating decimals are non-rational, so they are called the "irrationals". Examples would be sqrt(2) ("the square root of two") or the number pi ("3.14159...", from geometry). The rationals and the irrationals are two totally separate number types; there is no overlap.
Putting these two major classifications, the rationals and the irrationals, together in one set gives you the "real" numbers. Unless you have dealt with complex numbers (the numbers with an "i" in them, such as 4 – 3i), then every number you have ever seen has been a "real" number. "But why", you ask, "are they called 'real' numbers? Are there 'pretend' numbers?" Well, yes, actually there are, though they're actually called "imaginary" numbers; they are what is used to make the complex numbers, and is what the "i" stands for.
The commonest question I hear regarding number types is something along the lines of "Is a real number irrational, or is an irrational number real, or neither... or both?" Unless you know about complexes, everything you've ever done has used real numbers. Unless the number has an "i" in it, it's a real.
Here are some typical number-type questions (assuming that you haven't yet learned about imaginaries and complexes):   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

• True or False: An integer is a rational number.

Since any integer can be formatted as a fraction by putting it over 1, then this is true.

• True or False: A rational is an integer.

Not necessarily; ⁴/₁ is an integer, but ²/₃ is not! So this is false.

• True or False: A number is either a rational or an irrational, but not both.

True!  In decimal form, a number is either non-terminating and non-repeating (so it's an irrational) or not (so it's a rational); there is no overlap between these two number types!

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Classify according to number type; some numbers may be of more than one type.

• 0.45

This is a terminating decimal, so it can be written as a fraction: ⁴⁵/₁₀₀ = ⁹/₂₀. Since this fraction does not reduce to a whole number, then it's not an integer or a natural. And everything is a real, so the answer is: rational, real

• 3.14159265358979323846264338327950288419716939937510...

You probably recognize this as being pi, though this may be more decimal places than you customarily use. The point, however, is that the decimal does not repeat, so pi is an irrational. And everything (that you know about so far) is a real, so the answer is: irrational, real

• 3.14159

Don't let this fool you! Yes, you often use something like this as an approximation of pi, but it isn't pi! This is a rounded decimal approximation, and, since this approximation terminates, this is actually a rational, unlike pi which is irrational! The answer is: rational, real

• 10

Obviously, this is a counting number. That means it is also a whole number and an integer. Depending on the text and teacher (there is some inconsistency), this may also be counted as a rational, which technically-speaking it is. And of course it's also a real. The answer is: natural, whole, integer, rational (possibly), real

• ⁵/₃

This is a fraction, so it's a rational. It's also a real, so the answer is: rational, real

• 1 ²/₃

This can also be written as ⁵/₃, which is the same as the previous problem. The answer is: rational, real

• –sqrt(81)

Your first impulse may be to say that this is irrational, because it's a square root, but notice that this square root simplifies: –sqrt(81) = –9, which is just an integer. The answer is: integer, rational, real

• ^{– 9}/₃

This is a fraction, but notice that it reduces to –3, so this may also count as an integer. The answer is: integer (possibly), rational, real

Except for the section where you have to classify numbers according to type, you really won't need to be terribly familiar with this hierarchy. It's more important to know what the terms mean when you hear them. For instance, if your teacher talks about "integers", you should know that the term refers to the counting numbers, their negatives, and zero.

Adding and substracting integers

How to Add and Subtract
Positive and Negative Numbers

Numbers Can be Positive or Negative:

Negative Numbers (−)	Positive Numbers (+)
(This is the Number Line, read about Using The Number Line)

"−" is the negative sign.	"+" is the positive sign

No Sign Means Positive

If a number has no sign it usually means that it is a positive number.

Example: 5 is really +5

Adding Positive Numbers

Adding positive numbers is just simple addition.

Example: 2 + 3 = 5

is really saying

"Positive 2 plus Positive 3 equals Positive 5"

You could write it as (+2) + (+3) = (+5)

Subtracting Positive Numbers

Subtracting positive numbers is just simple subtraction.

Example: 6 − 3 = 3

is really saying

"Positive 6 minus Positive 3 equals Positive 3"

You could write it as (+6) − (+3) = (+3)

Balloons and Weights

This basket has balloons and weights tied to it: The balloons pull up (positive) And the weights drag down (negative)

Here is what adding and subtracting positive numbers looks like:

You can add balloons (you are adding positive value) the basket gets pulled upwards (positive)

You can take away balloons (you are subtracting positive value) the basket gets pulled downwards (negative)

Now let's see what adding and subtracting negative numbers looks like:

You can add weights (you are adding negative values) the basket gets pulled downwards (negative)

And you can take away weights (you are subtracting negative values) the basket gets pulled upwards (positive)

That last one was interesting ... subtracting a negative made the basket go up

Subtracting a Negative is the same as Adding

Example: What is 6 − (−3) ?

6−(−3) = 6 + 3 = 9

Example: What is 14 − (−4) ?

14−(−4) = 14 + 4 = 18

We also found that taking away balloons (subtracting positives) or adding weights (adding negatives) both made the basket go down.

And Positive and Negative Together ...

Subtracting a Positive or Adding a Negative is Subtraction

Example: What is 6 − (+3) ?

6−(+3) = 6 − 3 = 3

Example: What is 5 + (−7) ?

5+(−7) = 5 − 7 = −2

The Rules:

It can all be put into two rules:

	Rule		Example
	Two like signs become a positive sign	+(+)	3+(+2) = 3 + 2 = 5
		−(−)	6−(−3) = 6 + 3 = 9
	Two unlike signs become a negative sign	+(−)	7+(−2) = 7 − 2 = 5
		−(+)	8−(+2) = 8 − 2 = 6

They are "like signs" when they are like each other (in other words: the same).

So, all you have to remember is:

Two like signs become a positive sign

Two unlike signs become a negative sign

Example: What is 5+(−2) ?

+(−) are unlike signs (they are not the same), so they become a negative sign.

5+(−2) = 5 − 2 = 3

Example: What is 25−(−4) ?

−(−) are like signs, so they become a positive sign.

25−(−4) = 25+4 = 29

Example: What is −6+(+3) ?

+(+) are like signs, so they become a positive sign.

−6+(+3) = −6 + 3 = -3

Start at −6 on the number line, move forward 3, and you end up at −3