• any points that are on the same line
  • the line does not have to be drawn in--just has to be possible
• 2 points are ALWAYS collinear--POSTULATE**

• any points that are all one one plane
• 3 non-collinear points will define a unique plane--POSTULATE**
  • 3 collinear points are all coplanar but are on several different planes
  • 4 points may or may not be coplanar
• POSTULATE (a.k.a. AXIOM)
  • simple, obviously true statements that we accept with out any proof
  • THEOREMS--will discuss in later chapters, have to be proven before we accept them
• segment
  • a piece of a line
  • 2 endpoints
  • many (too many to count) points between the endpoints
  • 1 dimensional
  • CAN be measured
  • NAME--2 capital letters with the symbol over the top
    • the letters MUST be the ENDPOINTS
  • 2 capital letter WITHOUT the symbol means 'the measurement of the segment' and refers to a number not the segment itself
• ray
  • a piece of a line
  • one endpoint
  • many (too many to count) points on the ray
  • NAME--2 capital letters with the symbol over the top
    • the FIRST letter must be the endpoint
    • the second letter can be any other point that is on the ray
  • 1 dimensional
  • cannot be measured
  • OPPOSITE RAYS
    • have to share the same endpoint
    • have to be collinear
• intersection
  • the shared point(s) when 2 or more objects cross each other
    • the intersection is NOT the combined objects
    • only the points that are shared by all the objects in question
  • the verb 'intersect' means that the 2 objects share at least one common point
  • the noun 'intersection' means 'the locations of all the common points shared by the 2 object'

section 2

• you do not have to line up the END of the ruler like you were taught in elementary school
• you can use the ruler from the middle and do a subtraction
• they taught you to use the end of the ruler for CONVIENCE--not out of necessity
  • distance is a number and therefore, you do not use the symbol over the 2 capital letters

• difficult word to define without using the word 'between'
• between is only used when the points are COLLINEAR
• the point that is between the other 2 'is the one that cannot be the endpoint of the segment containing all 3 points'

• the 3 points MUST be collinear
• If B is between A and C, then AB + BC = AC
• the absolute value symbol simply means that measurements are never negative
  • if you always put the larger number first, it is not an issue, you will always get a positive answer

• 2 things that are the same size and the same shape
• seems a lot like the word 'equal'
  • equal refers to things that are 'the same size' but does not say anything about the shape of the thing
  • equal is used to refer to numbers only
• congruent refers to objects or pictures of objects
  • for a while you will find 'congruent' uncomfortable but will get used to it over time
  • most students will find themselves using the wrong symbol for the first few weeks--keep trying

section 3

• midpoint
  • the exact middle point of the millions and millions of points on a segment
    • it sounds silly but the midpoint is a POINT
  • the midpoint defines 2 congruent segments
    • denoted in a picture by tick marks near the middle of each of the segments
    • if one segment is marked with one tick mark and the other is marked with 2 tick marks, then the midpoint is NOT the point indicated
  • the measures of these segments are equal
• bisector
  • a point, line, segment, ray, or plane that intersects a segment at the segments midpoint
    • a bisector is the thing that is cutting the segment into 2 equal pieces
    • the midpoint is where the cutting is taking place
  • a noun (it is a thing)
• to bisect
  • a verb that means something is being cut in half
    • you will hear all of these words--you have to train yourself to hear how the word is being used
    • as with everything, it gets easier with practice
• midpoint formula
  • either endpoint of the segment can be used as point 1, either can be considered to be the second point
  • must pay attention to the sign (positive or negative) on the coordinates and calculate carefully
  • the coordinates of the midpoint should always fall between the coordinates of the endpoint
    • sometimes students forget to divide by 2 and the coordinates they calculate will not fall in the right range
    • sometimes students subtract instead of adding the initial coordinates, this should also be noticeable if you check your answer to see if it is reasonable
    • checking your work is an important step in any class
      • in English, you check for spelling, grammar, etc.
      • in math, you check to see if the answer makes any sense in the context of the problem
      • take the time to check things--it makes you look smarter
• distance formula
  • based on the Pythagorean Theorem for a right triangle
    • a² + b² = c²
  • the square root symbol may not be on everyone's calculator
    • you can leave the square root symbol around the number in an answer
    • we will deal more with square roots later--for now just leave it
  • technically a square root problem has 2 answers
    • one positive, one negative
    • we use only the positive one since the answer represents a measurement of a segment
      • measures of segments are always positive
      • so this is a reasonable answer
• formulas (in general)
  • a lot of students are resistant to formulas
    • sorry--they are a necessary evil
    • sometimes you will find that you do not really need the formula
    • if you UNDERSTAND the formula and where it fits with the picture of what you are doing, sometimes you can do the calculation without the formula
      • for example--the midpoint
        1. think about (or draw) the segment on graph paper
        2. where does the midpoint need to be (I guess you need to know what a midpoint is first)
        3. think about what would be a reasonable answer
        4. try to figure out how to make the coordinates of the endpoints calculate something close
        5. adding the 2 x's and dividing by 2, adding the 2 y's and dividing by 2
        6. see it worked!!
  • a lot of students worry about memorizing formulas
    • if you ask, I will tell you which ones I expect you to know and which ones I will give you on a test
    • some test (like the MME or ACT) the people who wrote the test will give you a formula sheet
      • in these cases, you can get a copy of these sheets ahead of time so that you know what you will be expected to have memorized and what is unnecessary
    • try not to stress too much over this kind of thing

section 4

• angle
  • sides of an angle are ALWAYS rays
    • I will (and you will too eventually) get lazy and draw them looking more like segments--they are still rays
  • vertex is the common endpoint
    • plural is 'vertices'
    • if the 2 rays in question do not have a common endpoint, then it is NOT and angle
  • naming the angle should be simple (and it is) but it sounds complicated--it does not take long to get used to it
    • the angle symbol is always the same
      • DO NOT try to make it match the size of the angle in the picture
      • DO NOT try to make it point the direction of the angle in the picture
    • use the angle symbol followed by 3 capital letters
      • the first and last letters can be ANY non-endpoint, one from each of the rays
      • the middle letter must be the common endpoint
        • there are (at least) 2 different 3 letter names for any angle
    • use the angle symbol followed by only 1 capital letter
      • the letter must be the vertex of the angle
      • CAN ONLY USE THIS METHOD WHEN there is only one angle with that particular vertex
    • use the angle symbol followed by a number
      • this is used to simplify the naming process
      • can only be used in certain circumstances--details in class
    • when you see a lower case letter 'm' in front of the angle symbol, it means that you are talking about the measurement of the angle rather than the angle itself
• 4 kinds of angles
  • measuring an angle has nothing to do with the lengths of the rays (since they are infinite)
  • measuring an angle has to do with the amount of space separating the rays
    • THERE IS ONLY ONE MEASUREMENT FOR EACH ANGLE--not one measurement for each ray
    • MEASURING IS DONE WITH A PROTRACTOR--not a ruler
  • acute angle--any angle measuring more than 0^o and less than 90^o
  • right angle--any angle measuring exactly 90^o
    • symbol for right angle--a little box drawn into the picture at the corner of the angle
      • even if the angle does not 'look like a right angle' if the symbol is there, it IS a right angle
  • obtuse angle--any angle that measures more than 90^o and less than 180^o
  • straight angle--any angle that measures exactly 180^o
    • these angles look like lines on purpose
  • angles more than 180^o
    • these angles exist but are not usually addressed until you get to trig class
    • we will talk about these angles briefly when we deal with circles--particularly when we deal with major arcs
  • angles less than 0^o
    • again these angles are usually left to deal with in a trig class
  • the degree symbol--should be included after any number that represents an angle measurement
    • if you see a number written (usually inside an angle) near the vertex and it has a degree symbol--then the number is the measurement of the angle
    • if you see a similar number written in the same place but it does not have the degree symbol--then it is probably being used as the name of the angle
    • if you are not careful, when you write the degree symbol, it may end up looking like the number 0
• Protractor Postulate
  • very similar to the Ruler Postulate
• Angle Addition Postulate
  • very similar to the Segment Addition Postulate
    • please note that the angles must have the same vertex
    • please note that the angles cannot overlap
• congruent angles
  • congruent still means that the things are the same size--in this case, the same degree measure
  • congruent still means that the things are the same shape--in this case, they are both angles
• angle bisector
  • bisector still means that something is cutting something in half
    • an angle can be divided in half by a line, a ray, a segment, or a plane
    • the 2 pieces are congruent angles
      • the picture is usually marked with small curved arcs near the vertex
      • using the same number of arcs indicates that the angles are congruent
      • using different numbers of arcs indicates that the angles are different sizes--does NOT indicate which is larger
  • angles DO NOT have midpoints
    • think about this--it would be very difficult to get people to agree on where the middle point is since the rays go on forever and cannot be measured

  constructions

  • constructions--technical drawings
    • different from 'sketches' or 'drawings' which do not have to be accurate
  • tools--only 2 tools allowed
    • compass--to draw ALL curves and make ALL measurements
    • straight edge--not a ruler, used only to draw straight
  • methods
    • constructions are technical and perfect
      • there is a correct procedure for each construction
      • you need to know what to do and in what order
      • you will have to keep up while I am demonstrating in class
    • you should take detailed notes the procedure for each construction and practice each at home
    • you cannot get good at these without some practice
      • doing one or 2 in class is not going to be enough to get good at it
      • your hands and your brain need to work together--practice, practice, practice
  • some simple constructions
    1. dividing a circle into 6 congruent pieces
    2. copying a segment
       1. doubling the length of a segment
       2. tripling (etc.) the length of a segment
    3. midpoint of a segment (a.k.a. bisecting a segment)
    4. copying an angle
       1. doubling an angle
    5. bisecting an angle
• ASSIGNMENT: practice your constructions--they will be on the test!!

section 5

• angle relationships
  • complementary--2 angles whose measures ADD UP TO 90^o
    • the angles do not have to be next to each other, but they can be
    • to find an angle's complement, subtract its measure from 90^o
    • some angles DO NOT have a complement
      • right angles
      • obtuse angles
      • straight angles
  • supplementary--2 angles whose measures ADD UP TO 180^o
    • the angles do not have to be next to each other, but they can be
    • to find an angle's supplement, subtract its measure from 180^o
      • only straight angles do not have a supplement
  • adjacent
    • the 2 angles must share the same vertex
    • the 2 angles must share a common ray
    • the 2 angles must NOT overlap
  • non-adjacent
    • any 2 angles that are not adjacent
      • there are lots of angles that intersect or share some portion of themselves but are not adjacent
      • try not to be confused--check the details of the definition and compare it to the picture in question
  • vertical angles--formed when 2 lines intersect
    • you always get 2 pair of vertical angles when 2 lines cross each other
    • usually there is one pair that is obtuse and one pair that is acute
  • linear pair of angles--any 2 angles that are BOTH supplementary and adjacent
    • 4 pair of linear pairs are created when 2 lines intersect
    • 1 linear pair is created from a line intersecting a ray at the ray's endpoint

section 6

• polygons--must have all 3 of these characteristics
  1. closed
  2. straights sides
  3. only one interior region
• sides--always segments
  • segments are named with 2 capital letters with the symbol over them
  • sides can be measured
  • the segments must all intersect other segments at their endpoints
  • equilateral--a polygon that has all its sides the same measure
• angles--there are always the same number of sides as there are angles in a polygon
  • the angles are formed by the intersection of the sides to create a vertex
  • equiangular--a polygon that has all its angles the same size
  • regular--a polygon that is both equilateral AND equiangular
    • it is possible for a polygon to be equilateral and not equiangular
    • it is possible for a polygon to be equiangular and not equilateral
    • it is possible for a polygon to be both
• convex v. concave--all polygons must be one or the other--never both
  • if even one vertex of the entire polygon 'pokes into' the polygon, the whole thing is called concave
  • we only address the concept of convex or concave when we are dealing with polygons
    • so you do not have to decide what kind of polygon something is when it is not a polygon to begin with
• naming polygons--polygons are often classified by the number of sides
  • you should know the following names by heart
    • 3 sides--triangle
    • 4 sides--quadrilateral
    • 5 sides--pentagon
    • 6 sides--hexagon
    • 7 sides--heptagon
    • 8 sides--octagon
    • 9 sides--nonagon
    • 10 sides--decagon
    • 12 sides--dodecagon
    • 13 sides--13-gon
      • yes, this is a real name--I did not make this up
    • 73 sides--73-gon
    • n sides--n-gon
• diagonal--all polygons except triangles have diagonals
  • a diagonal does not have to 'slant' across the polygon--like you usually think of with a square or a rectangle
  • technical definition of diagonal--any segment that connect non-adjacent vertices of a polygon
    • quadrilaterals only have 2 diagonals
    • pentagons have 5 diagonals
    • hexagons have 9 of them
      • try drawing and counting at the same time, you will find them

section 7

• area--you have done this since elementary school
• area is the number of squares (all uniform sized) that can be fit inside a closed 2 dimensional space
  • the units on area are always 'squared'
  • of simple polygons
    • square
    • rectangle
    • triangle
  • of circles
    • exact answer v. approximate answer
      • if you round pi off to be 3.14 (or one of the other approximations for pi) you are not going to get an exact answer, you get an approximate answer
      • if you leave the pi symbol in your answer, the problem is easier to calculate (often accomplished without a calculator or scrap paper) and is the exactly correct answer
• perimeter--the distance around all the sides of a polygon
  • of polygons--you simply add all the side measures
    • there are formulas for some of the more common polygons--but they really all say the same thing
    • the units on perimeter are NEVER 'squared'
    • you are simply wrapping a tape measure all the way around the outside of the polygon and reading off the measurement
  • of circles--since these are not polygons--they do not have perimeters
    • circumference--the distance around the circle
      • basically, circumference is just the perimeter of a circle, but you will never see it this way in a book
      • again, the units are NEVER 'squared'
      • you can leave the pi symbol in your answers to make them more correct and easier to calculate