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October 7, 2009

Geometry Notes — Polygons

Filed under: Geometry, Geometry Notes — Tags: — bowman @ 4:55 pm

Polygon – a plane figure that is formed by three or more segments such that
1)  each segment intersects exactly two other segments at an endpoint    and
2)  no two segments with a common endpoint are collinear.

Polygon – From Greek: polys “many” + gonia “angle”
Polygons are a number of coplanar line segments, each connected end to end to form a closed shape.

Polygons are one of the most all-encompassing shapes in geometry. From the simple triangle up through squares, rectangles, trapezoids, to dodecagons and beyond.

This is polygon ABCDE. or polygon AEDCB. or polygon BCDEA. or polygon BAEDC. or yawn. A polygon is named by picking one of the vertices and proceeding either clockwise or counterclockwise. No rearranging letters.

Points A, B, C, D, and E are this polygon’s vertices.

A, ∠B, ∠C, ∠D, and ∠E are this polygon’s angles.

AB, BC, CD, DE, and AE are this polygon’s sides.

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Diagonal – a segment that connects two non-consecutive vertices

AC and AD are diagonals of this polygon starting at vertex A.

A diagonal of a polygon is a line segment joining two vertices. From any given vertex, there is no diagonal to the vertex on either side of it, since that would lay on top of a side. Also, there is obviously no diagonal from a vertex back to itself.

The formula n(n – 3)/2 where n is the number of sides in the polygon, gives the number of distinct diagonals a polygon can have.

A pentagon will have 5(5 – 3)/2 = 5 distinct diagonals

A triangle will have 3(3 – 3)/2 = 0 diagonals

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Convex polygon – a polygon with all diagonals in the interior of the polygon

If a polygon is not convex, it is called concave or non-convex

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A polygon is classified or named according to the number of its sides

3 sides        triangle

4 sides        quadrilateral


5 sides        pentagon


6 sides        hexagon


7 sides        heptagon


8 sides        octagon


9 sides        nonagon


10 sides      decagon

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12 sides      dodecagon

42 sides      tetracontakaidigon

672 sides    hexahectaheptacontakaidigon

There are some that wish to name every possible polygon, but there seems little point in doing so. Beyond about 10 sides, most people call them an “n-gon”. For example a 15-gon has 15 sides.
n sides        n – gon

25 sides      25 – gon

The above examples were first a simple convex polygon and second a regular polygon. Notice how as another side was added, the polygons became more circular.

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Equilateral polygon – a polygon with all its sides all congruent.

Equiangular polygon – a polygon with all its angles all congruent.

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Triangles are the only polygons that are automatic equiangular if they are equilateral and vice versa.

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Regular polygon – a polygon that is both equilateral and equiangular

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Regular polygons tend to look like someone tried to make a circle out of some straight lines. In fact, if you have a polygon with very many sides, it looks a lot like a circle from a distance. The sides and vertices are evenly spread around a central point, and regular polygons are convex – all the vertices point ‘outwards’.

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…. Regular 20-gon – The more sides, a regular polygon looks more and more like a circle.

…..Regular 40-gon – Or is this a circle? No, its a regular 40-gon.

September 30, 2009

Geometry Notes — Triangles (art)

Filed under: Geometry, Geometry Notes — Tags: — bowman @ 8:29 pm

I just love geometry art. Outstanding.

September 15, 2009

Geometry Vocabulary Study Guide — Chapter 2 Test Part 2

Filed under: Geometry, Geometry Assignment, Geometry Notes — Tags: — bowman @ 10:18 am

geo chapter 2.2 study guide

August 28, 2009

Geometry Notes — Conditionals and Other Statements

Filed under: Geometry, Geometry Notes — Tags: — bowman @ 7:02 am

A mathematical sentence is one in which a fact or complete idea is expressed.  Because a mathematical sentence states a fact, many of them can be judged to be true or false.

Conditional – a statement that clearly states a hypothesis and a conclusion in an if-then form

Conditionals are statements that say if one thing happens, another will follow. Conditionals are symbolically written as pq. The ⟹ stands for “implies.” The p stands for the hypothesis, the given or the problem. It is the first part of the conditional. The q stands for the conclusion, the prove, or the answer. It is the “then” part of the if-then statement.

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The definition of a right angle is… an angle that measures exactly 90.

Here is that definition written as a conditional: If an angle is a right angle, then it measures exactly 90.

hypothesis (p):  an angle is a right angle
conclusion (q):  it measures exactly 90

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Here is the definition of perpendicular lines written as a conditional: If lines are perpendicular, then they form a right angle.

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And here is something silly: If an animal is a whale, then it is a mammal.

If a mathematical sentence is incorrect you must provide a counterexample, which is a single example used to prove a statement false.

An if-then statement or conditional statement is a statement formed when one thing implies another, but not necessarily the other way around.

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Converse – a statement formed by interchanging a conditional’s hypothesis and conclusion. Symbolically this is written as qp. q implies p.

Here is the definition of a right angle written as a converse statement: If an angle measures exactly 90, then it is a right angle.

hypothesis:  an angle measures exactly 90
conclusion:  it is a right angle

The definition of perpendicular lines written as a converse statement: If lines form a right angle, then they are perpendicular.

And then the silly: If an animal is a mammal, then it is a whale.

The two boring math definition are still true statements when written as converse statements. The silly statement is incorrect. To show that, I would present a single counterexample such as a pony — a pretty pony. I could have said geometry student. A pony and a geometry student are both fine examples of creatures that are mammals but are not whales.

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Biconditional – a statement formed by combining a true conditional with a true converse in an if and only if form. Symbolically this is written as pq.

The definition of a right angle written as a biconditional: An angle is a right angle if and only if it measures 90.

The definition of perpendicular lines written as a biconditional: Lines are perpendicular iff they form a right angle.

No biconditional can be written about the mammal and whale statement because the statement’s converse was not true.

All geometry definitions can be written as biconditionals.

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A point is a midpoint iff it divides a segment into two congruent segment.

Remember, I like definition that clearly state what the item is followed why that item is important.

A midpoint is first and foremost a point… with a responsibility.

A angle bisector is a ray first… then it does something.

An obtuse angle is first an angle… blah blah blah.

The hypothesis of the conditional and the first part of the biconditional is what the item is first and foremost. The conclusion of the conditional and the second part of the biconditional is the result, the responsibility, the action, the thing that makes the item super special to us all.

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Inverse – a statement formed by negating the original conditional’s hypothesis and conclusion. Symbolically, the inverse is written as ~p ⟹ ~q. The ~ symbol mean “not”. ~p ⟹ ~q is read “not p implies not q.”

Here is the definition of a right angle written as an inverse statement: If an angle is not a right angle, then it does not measure 90.

The definition of perpendicular lines written as an inverse statement: If lines are not perpendicular, then they do not form a right angle.

And again the silly: If an animal is not a whale, then it is not a mammal.

The boring geometry definitions are still true. The silly whale/mammal one is not. A salamander is not a whale and it is not a mammal.

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Contrapositive – a statement formed by interchanging a conditional’s hypothesis and conclusion and by negating both. Symbolically a contrapositive is written as ~q ⟹ ~p. not q implies not p.

Here is the definition of a right angle written as a contrapositive statement: If an angle does not measure 90, then it is not a right angle.

The definition of perpendicular lines written as a contrapositive statement: If lines do not form a right angle, then they are not perpendicular.

And the silly one: If an animal is not a mammal, then it is not a whale.

If you’re thinking about the truth of these statements, you find that they are all true.

The truth of mathematical statement appear in pairs. Conditional, Converse, Inverse, Contrapositive — 4 statements. In pairs, either

  • all are true
  • 2 are true, 2 are false
  • or, all are false

The “verses” go together when it comes to truth. Conditionals and contrapositives also go together when it comes to truth.

Geometry Notes / Assignment — History — Euclid

Filed under: Geometry, Geometry Assignment, Geometry Notes, History of Math — Tags: — bowman @ 7:01 am

The word geometry comes from the Greek word geometrein (geo meaning earth, and metrein meaning to measure). Geometry was originally the science of measuring the land.

Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid’s text Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history. His method consists of assuming a small set of postulate, and then proving many theorems from those postulate. Although many of Euclid’s results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fit together into a comprehensive deductive and logical system.

Euclid was a disciple of the Platonic school, founded by Plato in 387 B.C. in Athens. Neither the year nor place of Euclid birth have been established, nor the circumstances of his death, although he is known to have lived and worked in Alexandria for much of his life. Around 300 B.C. he produced the definitive treatment of geometry in his 13-volume Elements. In compiling his masterpiece Euclid built on the experiences and achievements of his predecessors — Pythagoras, Archytas, Eudoxus, and Theaetetus.

The book synthesized earlier knowledge about geometry, and was used for centuries in western Europe as a geometry textbook. The text began with definitions, postulates, and common opinions, then proceeded to obtain results by rigorous geometric proof. Euclid used the method of proof by exhaustion.

The Elements begins with definitions and five postulates. The first three postulates are postulates of construction. For example the first postulate states that it is possible to draw a straight line between any two points. These postulates also implicitly assume the existence of points, lines and circles and then the existence of other geometric objects are deduced from the fact that these exist. There are other assumptions in the postulates which are not explicit. For example it is assumed that there is a unique line joining any two points. Similarly postulates two and three, on producing straight lines and drawing circles assume the uniqueness of the objects of whose construction is being postulated. The fourth and fifth postulates are of a different nature. Postulate four states that all right angles are equal. This may seem obvious but it actually assumes that space in homogeneous — meaning that a figure will be independent of the position in space in which it is placed. The famous fifth, or parallel, postulate states that one and only one line can be drawn through a point parallel to a given line. Euclid’s decision to make this a postulate led to Euclidean geometry. It was not until the 19th century that this postulate was dropped and non-euclidean geometries were studied.

The Elements also included five “common notions”: 1) Things that equal the same thing also equal one another. 2) If equals are added to equals, then the wholes are equal. 3) If equals are subtracted from equals, then the remainders are equal. 4) Things that coincide with one another equal one another. 5) The whole is greater than the part.

The Elements is divided into 13 books.

Books one to six deal with plane geometry. In particular books one and two set out basic properties of triangles, parallels, parallelograms, rectangles and squares.

Book three studies properties of the circle.

Book four deals with problems about circles and is thought largely to set out work of the followers of Pythagoras.

Book five lays out the work of Eudoxus on proportion applied to commensurable and incommensurable magnitudes.

Book six looks at applications of the results of book five to plane geometry.

Books seven to nine deal with number theory. In particular book seven is a self-contained introduction to number theory and contains the Euclidean algorithm for finding the greatest common divisor of two numbers.

Book eight looks at numbers in geometrical progression.

Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus. Euclid changed the proofs of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus.

Books eleven to thirteen deal with three-dimensional geometry. In book eleven the basic definitions needed for the three books together are given. The theorems then follow a fairly similar pattern to the two-dimensional analogues previously given in books one and four.

The main results of book twelve are that circles are to one another as the squares of their diameters and that spheres are to each other as the cubes of their diameters. These results are certainly due to Eudoxus. Euclid proves these theorems using the “method of exhaustion” as invented by Eudoxus.

The Elements ends with book thirteen which discusses the properties of the five regular polyhedra and gives a proof that there are precisely five. This book appears to be based largely on an earlier treatise by Theaetetus.

Little is known about Euclid other than his writings. He is considered the Father of Geometry. Euclid may not have been a first class mathematician, meaning that he did indeed build on the findings of other math greats, but the long lasting nature of The Elements makes him the leading mathematics teacher of ancient times or perhaps of all time.  Outstanding.

August 19, 2009

Geometry Notes — Congruent Angles

Filed under: Geometry, Geometry Notes — Tags: — bowman @ 7:00 pm

Congruent angles – angles with the same measure

Since mBIG = mCOW , then ∠BIG ≅ ∠COW .

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Angle bisector – a ray that divides an angle into two congruent angles

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Adjacent angles – two angles that share a common vertex and side but no common interior points

∠1 and ∠2 are adjacent angles. They have the same vertex. They share a common side. No interior point in ∠1 is in ∠2, no interior point in ∠2 is in ∠1.

These angle pairs are not adjacent.  Why?

∠3 and ∠4 are not adjacent. They do not have the same vertex.

∠5 and ∠6 are not adjacent. They do not share a common side. However, they are both adjacent to the angle between them, but they are not adjacent with each other.

∠7 and ∠8 are not adjacent. This is an odd drawing, ∠8 is the entire drawing, and ∠7 is the angle on the right. All the points are would be in the interior of ∠7 are also in the interior of ∠8, this violates the definition of adjacent angles. ∠7 is adjacent to the angle on the left. The measure of ∠7 and the measure of this angle on the left combine to equal the measure of ∠8 — angle addition at it finest.

SPN is a straight angle.  mSPN = 180.  What do you know about the measures of ∠1 and ∠2? The sum of the measures of these angles is 180. More about this special number later.

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Perpendicular lines – lines that intersect to form a right angle

Since ce, then ∠1 is a right angle.

perpendicular art

August 18, 2009

Geometry Notes — Angles

Filed under: Geometry, Geometry Notes — Tags: — bowman @ 6:15 pm

A ray is the part of the line which consists of the given point and the set of all points on one side of the end point.

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Angle – a figure formed by two rays with the same endpoint

Two rays that have the same endpoint form an angle. That endpoint is called the vertex, and the rays are called the sides of the angle. In geometry, an angle is measured in degrees from 0° to 180°. The number of degrees indicates the size of the angle.

Point A is the initial point of both rays. Point A is the angle’s vertex.

There are three different ways to name the same angle

  • By the letter of the vertex
  • By the number in its interior
  • By the letters of three points that form it. The center letter is always the letter of the vertex.

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Angles are classified according to their measures.

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Acute angle – an angle with measure between 0° and 90°

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Right angle – an angle with measure exactly 90°

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Obtuse angle – an angle with measure between 90° and 180°

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Straight angle – an angle with measure exactly 180°

This is a line as well. This is the only angle whose measure you can assume.

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A reflex angle is an angle whose measure is more than 180º

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Angle Addition Postulate -

In other words, if an angle is divided into parts, then the measure of the angle equals the sum of the measures of the non-overlapping parts (angles).

August 16, 2009

Geometry Notes — Segments

Filed under: Geometry, Geometry Notes — Tags: — bowman @ 7:03 pm

A line segment is a part of a line that is bounded by two distinct endpoints, and contains every point on the line between its endpoints


The bar over the two letters indicates it is a line segment, rather than a line, which goes on forever in both directions.

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Length of a segment – a positive real number that is the distance between a segment’s two endpoints.

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Ruler Postulate: The points on a line can be matched, one-to-one, with the set of real numbers. The real number that corresponds to a point is the coordinate of the point. The distance, AB , between two points, A and B ,  on a line is equal to the absolute value of the difference between the coordinates of A and B .

An example of the Ruler Postulate is the Exit Numbers on the Interstate. Kingston is Exit 352. Harriman is Exit 347. Crossville is Exit 317. Using these number, you can determine the distances from one town to another, as long as you know the exit number. Kingston to Crossville = 352 – 317 = 35 miles.

Its important to remember that distance is always positive. Even if you walk backwards or drive the wrong direction, you still went a positive distance.

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Segment Addition Postulate – If B is between A and C , then  AB + BC = AC .
A , B , and C are collinear with B somewhere between A and C , therefore the length AB plus the length BC equals the entire length AC .

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Congruent segments – segments with the same length

Since the lengths are equal, then the segments are congruent.

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Midpoint – the point that divides a segment into two congruent segments

Only a line segment can have a midpoint. A line cannot since it goes on indefinitely in both directions, and so has no midpoint. A ray cannot because it has only one end, and so no midpoint.

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Segment bisector – a line, segment, ray, or plane that intersects a segment at its midpoint

Rarely do segments bisect each other.  You must know which segment is bisecting and which segment is being bisected. In general ‘to bisect’ something means to cut it into two equal parts. The ‘bisector’ is the thing doing the cutting.
With a segment bisector, you are cutting a line segment into two equal lengths with another object – the bisector. This bisector can be a line, a ray, another segment, or a plane.

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081709_0017_1.png……..081709_0017_2.png

081709_0017_3.png……..081709_0017_4.png

August 13, 2009

Geometry Notes — Segments, Rays, Parallel Lines and Planes

Filed under: Geometry, Geometry Notes — Tags: — bowman @ 9:15 pm

A line segment is a part of a line that is bounded by two distinct endpoints, and contains every point on the line between its endpoints.


In the figure above, the line segment would be called AB because it links the two points A and B. The bar over the two letters indicates it is a line segment, rather than a line, which goes on forever in both directions.
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A ray is the part of the line which consists of the given point and the set of all points on one side of the end point.


In the figure, the ray would be called AB because starts at point A and passes through B on it’s way to infinity. The single arrow over the second letter indicates it is a ray, and the arrow direction indicates that A is the point where the ray starts.


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Opposite rays are two rays that both start from a common point and go off in exactly opposite directions. Because of this the two rays form a single straight line through the common endpoint T.


When the two rays are opposite, the points B, T, and R are collinear.
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Intersection – the set of points two figures have in common

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If two lines intersects, their intersection is a single point.

If two lines do not intersect, then they are either parallel or skew.
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Parallel lines – coplanar lines that do not intersect

Skew lines – non-coplanar lines that do not intersect

Parallel planes – planes that do not intersect.


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I just love geometric art.

August 11, 2009

Geometry Notes — Points, Lines, and Planes

Filed under: Geometry, Geometry Notes — Tags: — bowman @ 8:17 pm

The 3 “undefined terms in geometry” are the building blocks for the rest of the subject. They are like the foundation of a house – with a strong foundation, a course in mathematics is built.

The 3 terms are point, line, and plane.

Points are the basis of all geometry. Points are zero-dimensional. That means that they have no height, length, or width.

A point is like a star in the night sky. Unlike stars, geometric points have no size.

A point shows location.

A point is represented by a dot.

Points are named using capital letters. Points have no size, but we draw them as a “dot” on our paper so that we know where they are.

A point shows an exact position or location in space. It is important to understand that a point is not a thing, but a place.

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A line is a one-dimensional figure. A line has length, but no width or height.

A line is made up of an infinite number of points. Points in the same line are called collinear.

A line is like the edge of a ruler that never ends.

Imagine taking a ruler and drawing a line – now imagine if that line kept going straight forever.

The line you have is thick enough for you to see, but you need to imagine that your line is so thin that you can’t see it – it has no thickness at all.

That is a geometric line.

We draw arrows at the end of the line. This tells us that our line extends forever.

A line is identified by naming two points on the line or by writing a lowercase letter of choice after the line.

The notation, for example, AB (written with a line symbol <—> over the letters), is read as “line AB” and refers to the line that has points A and B, but there are infinite number of points on a line.

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Planes are two-dimensional. A plane has length and width, but no height, and extends infinitely on all sides.

Planes are thought of as flat surfaces, like a table top.

A plane is like a flat piece of land (like a football field) that extends forever.

Imagine that you can pick that football field up, and put it anywhere in the air that you like. You can even turn it side ways, or diagonally.

A plane is a flat edge (like a piece of paper) that has no thickness and extends forever.

Just like lines, planes too can be named in two different ways.

The first way is to name it with a capital scripted (cursive) letter.

Planes can also be named by naming any three non-collinear points that are within the plane.

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Space is the set of all points. It is made up of an infinite number of planes.

Collinear points are points that are all on the same line

“between” is a very simple preposition that is important in geometry. For a point to be between two other points, all three points must be collinear.

Coplanar points are points that are all on the same plane

Coplanar lines are lines that are all on the same plane

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Intersection – The set of points two figures have in common.

  • when 2 lines intersect, the intersection is a single point
  • when a line intersects a plane, the intersection is point
  • when 2 planes intersect, the intersection is line

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Postulate 1: Through any two points there is exactly one line.

Postulate 2: If two lines intersect, then they intersect in exactly one point.

Postulate 3: If two planes intersect, then they intersect in exactly one line.

Postulate 4: Through any three non-collinear points there is exactly one plane.

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