Mathematical Systems

Mathematical Systems The four parts of a mathematical system are: Vocabulary 1. “Undefined” terms Terms such as point, line, and plane are classified as undefined because they do not fit into any set or category that has been previously determined. 2. Definitions ( defined terms ) The four characteristics of a good definition are: 1. It names the term being defined; 2. It places the term into a set or category; 3. It distinguishes itself from other terms in that category ( without providing unnecessary facts) 4. It is reversible.** ** If a triangle is isosceles, then it has two congruent sides. If a triangle has two congruent sides, then it is isosceles. DEFINITIONS: Distance Midpoint The distance of AB is AB. The distance between two points is the length of a straight line segment that links them. A point on a line segment that divides it M is the midpoint of AB if M is into two equal parts. The halfway point on AB and of a line segment AM = MB The midpoint M bisects the line segment, AB Principles 3. Postulates ( axioms, conjectures ) Postulate – a statement that is assumed to be true. 4. Theorems Theorem – a statement that follows logically from previous definitions and principles; a statement that can be proved to be true. Corollary – a theorem that follows from another theorem as a “by-product”; a theorem that is easily proved as the consequence of another theorem. Lemma – a theorem that is introduced and proved so that a later theorem can be proved (“helping theorem”). Symbols and Definitions in Geometry Symbols save time and space when writing. Here are the most common geometrical symbols: Symbol Meaning Example In Words ABC has 3 equal sides Triangle ABC has three equal sides Angle m ABC is 45° The measure of the angle formed by ABC is 45 degrees. Perpendicular AB CD The line AB is perpendicular to line CD Parallel EF GH The line EF is parallel to line GH Degrees 360° makes a full circle Triangle is 90° A right angle is 90 degrees AB The line between A and B Right Angle (90°) Line Segment "AB" Line "AB" The infinite line that includes A and B Ray "AB" The line that starts at A, goes through B and continues on Congruent (same shape and size) ABC DEF Triangle ABC is congruent to triangle DEF Similar (same shape, different size) DEF MNO Triangle DEF is similar to triangle MNO Therefore a=b b=a Parallel MN || CD The corresponding congruent sides are marked with small straight line segments called hash marks. The corresponding congruent angles are marked with arcs. Congruent Angles and Congruent Line Segments CAB  FDE ABC  DEF ACB  DFE AC  DF AB  DE BC  EF a equals b, therefore b equals a MN is parallel to CD CAB is congruent toFDE ABC is congruent toDEF ACB is congruent toDFE AC is congruent to DF AB is congruent to DE BC is congruent to EF  Postulate 1: Through any two points, there is exactly one line.  Postulate 2: The measure of any line segment is a unique positive number. The measure (or length) of AB is a positive number, AB. Postulate 3: If X is a point of AB and A-X-B (X is between A and B), then AX + XB = AB Postulate 4: If two lines intersect, then they intersect in exactly one point Postulate 5: Through any three noncollinear points, there is exactly one plane. Postulate 6: If two planes intersect, then their intersection is a line.+ Postulate 7: If two points lie in a plane, then the line joining them lies in that plane. Theorem 1.1: The midpoint of a line segment is unique.       Example 1: State the postulate or theorem you would use to justify the statement made about each figure.