Math

• Mathematical system


  Divided into four parts: undefined terms, defined terms, postulates, and theorems
• Undefined terms


  Terms that cannot be precisely defined in modern mathematics, but are accepted by description
• Defined terms


  Terms that have a formal definition, and can be used to define even more terms
• Postulates


  Statements accepted as true without proof
• Theorems


  Statements that can be proven
• Undefined terms in geometry

  • Point
  • Line
  • Plane
• Defined terms


  • Collinear points
  • Coplanar points
  • Subsets of a line
• Postulates under point, line, and plane

  • 6 postulates
• Theorems for point, line, and plane

  • 3 theorems
• Relationship between undefined terms, defined terms, postulates, and theorems


  From undefined terms, we can have defined terms, which can then lead to postulates, and then theorems can be proven from the postulates
• Representations of a line


  • Edge of a ruler/notebook/paper
  • Rope
  • Pencil (excluding the tip)
• Representations of a plane

  • Board
  • Carpet
  • Notebook/notepad/paper
• A point has only location, but no dimension, length, thickness, or area
• A point is named using a capital letter
• A line is a straight, continuous arrangement of infinitely many points, with infinite length in two directions
• Naming a line


  Using a single lowercase script letter, or by any two points on the line
• Two points determine a line
• A plane is a flat surface that extends infinitely along its length and width, with no thickness
• Line


  A line is determined by at least two points and extends infinitely in both directions
• Line segment


  A part of a line consisting of two endpoints and all the points in between
• Ray


  A line with one endpoint that extends infinitely in one direction
• Opposite rays


  Rays with a common endpoint but extending in opposite directions
• If ray BA and ray BC are opposite rays, then point B is between points A and C
• Elements in the diagram

  • Points: A, W, X, Y, Z
  • Lines: A, B, C, D
• Another name for line B is line XY or line YZ
• Another name for line A is line WZ or line Z
• Plane


  A flat surface that extends infinitely along its length and width
• The plane formed by the four lines is plane L
• Ways to name the plane

  • Plane WXY
  • Plane XYZ
  • Plane WZY
• The intersection of line A and line AD is point A
• Two-column proof


  A formal proof consisting of a list of statements and the reasons why those statements are true, using deductive reasoning and accepted information
• Methods of mathematical proof

  • Deductive reasoning
  • Using given information
  • Using definitions
  • Using postulates
  • Using logical equivalences
  • Using previously proven statements
• Postulates/theorems to prove triangle congruence

  • SAS (Side-Angle-Side)
  • ASA (Angle-Side-Angle)
  • SSS (Side-Side-Side)
  • AAS (Angle-Angle-Side)
• Proving triangle congruence using postulates

  1. Identify given information

  2. Mark the figure

  3. Apply the appropriate postulate/theorem

  4. Conclude the triangles are congruent
• Vertical angles are congruent
• Midpoint divides a segment into two congruent parts
• Triangle congruence postulates and theorems

  SAS (side-angle-side), SSS (side-side-side), ASA (angle-side-angle), AAS (angle-angle-side)
• Right triangle congruence theorems

  HL (hypotenuse-leg), LA (leg-angle), LLL (leg-leg-leg)
• Angle bisector


  A line, segment or ray that divides an angle into two equal parts
• Constructing an angle bisector


  1. Draw a point on one side of the angle

  2. Draw an arc from the vertex to the other side of the angle

  3. Draw a line from the vertex to the point on the other side