MATHEMATICAL SYSTEM G8 Q3

• System

  • a significant role in building an organization or company.
  • Serves as an essential building block that support ones organizations
• A typical mathematical system has the following four parts:

  • undefined terms

  • defined terms

  • axioms and postulates

  • theorems
• Undefined Terms

  • a term that cant be defined or terms that dont need a formal definition
• Defined Terms

  • terms that have a formal definition and can be defined using other geometrical forms
• 3 Undefined Terms:

  • point
  • line
  • plane
• A point is a location in space. It has no size or shape.
• A line is made up of points. It has no thickness or width and extends infinitely in two directions.
• A plane is a flat surface made up of points. It has no depth and extends infinitely in all directions.
• Space is the set of all points (objects in space have length, width, and height)
• Collinear points is the points all in one line.
• Coplanar points are points all in one place.
• Name the following figures:

  1. Line

  2. Segment

  3. Ray

  4. Point

  5. Angle

  6. Plane

  7. Congruent

  
• Determine the segment in the following figure:

  Answer: Segment AB

  
• AC + CB = AB

  
• A ray has one endpoint and extends to another direction up to infinity.
• An angle is formed by two rays with the same endpoint.

  
• The common endpoint of the two rays in an angle is called a vertex. Determine the vertex in the following figure:

  Answer: Point G

  
• Angle Bisector

  • a ray or segment that divides an angle into two equal angles.

  
• Congruent Angles

  • angles that have the same measure. The angles can be separate or they can be together, as long as they are congruent.
• Adjacent Angles

  • two angles in a plane that have a common vertex and common side.
• Determine the type of angle:

  Congruent Angles

  
• Postulates

  • known as basic assumptions. Assumptions are statements accepted without proof.

  •Some will seem very basic.
• P1: Ruler Postulate

  Remember: The distance is always positive!

  Eg: |23-8|= |15|= 15

  |8-23|= |-15| = 15

  
• P2: Segment Addition Postulate

  • If B is between A and C, then AB+BC= AC

  
• P3: Protractor Postulate The measure of AOB is equal to the absolute value of the position of Ray OA on the protractor and the position of OB on the protractor. Refer to the figure below

  
• P4: Angle Addition Postulate

  The sum of two angle measures that are joined by a common ray will be equal to the measure of the angle they form.

  Refer to the figure below:

  ABD+DBC=ABC

  
• Supplementary Angles

  •two angles that add up to 180°.
• Determine the type of angle shown in the figure below:

  Supplementary Angles

  
• Linear Pair

  •a pair of adjacent formed with two intersecting lines. They are also supplementary.
• Linear Pair Theorem

  • if two angles form a linear pair, then they are supplementary.
• Vertical Angles

  • two non adjacent angles formed by two intersecting lines.

  If AOB and COD are vertical angles, then AOC and BOD are also vertical angles.

  
• Verticsl Angles Theorem

  •vertical angles are congruent.