Y. Davis Geometry Notes Chapter 2

Inductive Reasoning Looking at examples for a pattern to determine what you think will happen next.

Conjecture A statement based on inductive reasoning.

Counterexample An example that proves a conjecture is false

Statement A sentence that is either true or false. Statements are represented by letters such as p and q p: A rectangle is a quadrilateral

Truth Value Of a statement is either true (T) or false (F) p: A rectangle is a quadrilateral Truth Value: T

Negation Has an opposite meaning as well as an opposite truth value. ~p: A rectangle is not a quadrilateral Truth Value: F

Compound statements Two or more statements joined by using the words and or or.

Conjunction A compound statement that uses the word and to join two statements (True if only both statements are true) p: A rectangle is a quadrilateral. Truth Value: T q: A rectangle is convex. Truth Value : T A rectangle is a quadrilateral and convex. Truth Value: T

Disjunction A compound statement that use the word or. (is true as long as one statement is true) p: Malik studies geometry. q: Malik studies chemistry. Malik studies geometry or chemistry.

Truth Tables Can be used to organize the truth values of statements. Truth tables can be used to determine the truth value of negations and compound statements

Venn Diagram Diagrams that show all possible logical relations between sets.

Conditional statement A statement that can be written in if-then form Hypothesis—statement that follows the word if. Conclusion—statement that follows the word then. p→q read: If p, then q

Related conditional statements Converse– formed by switching the hypothesis and conclusion. q→p “If q, then p.” Inverse—formed by negating the hypothesis and conclusion. ~p→~q “If not p, then not q.” Contrapositive—formed by negating and switching the hypothesis and conclusion. ~q→ ~p “If not q, then not p.”

Logically equivalent Two statements with the same truth value. Examples of logically equivalent statements: A conditional and its contrapositive. A converse and inverse of a conditional statement.

Biconditional statement The conjunction of a conditional statement and its converse. Written in “if and only if” form. p↔q “p if and only if q”

Deductive Reasoning Uses facts, rules, definitions, or properties to reach logical conclusions from given statements.

Valid Logically correct

Ways to prove validity Law of Detachment – if a conditional statement is true, and the hypothesis is true, then the conclusion must be true. Venn Diagrams Law of Syllogism—If the conclusion of one conditional statement is the hypothesis of another and both conditional statements are true, then the original hypothesis leads to the final conclusion.

Postulate (Axiom) A statement that is true without proof.

Postulate 2.1 Through any two points there is exactly one line.

Postulate 2.2 Through any 3 non-collinear points there is exactly one plane.

Postulate 2.3 A line contains at least two points.

Postulate 2.4 A plane contains at least three non-collinear points

Postulate 2.5 If two point lie in a plane, then the entire line containing those points lies in that plane.

Postulate 2.6 If two lines intersect, then their intersection is exactly one point.

Postulate 2.7 If two planes intersect, then their intersection is a line.

Proof A logical argument in which each statement you make is supported by a statement that is accepted as true.

Theorem A statement that has been proven to be true.

Paragraph (informal) Proof Involves writing a paragraph to explain why a conjecture for a given situation is true.

Theorem 2.1: Midpoint Theorem

Algebraic Properties of equality for all real numbers a, b and c Addition Property– If a=b, then a+c=b+c. Subtraction Property – If a=b, then a-c=b-c. Multiplication Property – If a=b, then ac=bc. Division Property – If a=b & c≠0, then a/c=b/c. Reflexive Property–a=a Symmetric Property– If a=b, then b=a. Transitive Property– If a=b, & b=c, then a=c. Substitution Property– If a=b, then a can replace b in any expression or equation. Distributive Property – a(b+c)=ab+ac.

Algebraic Proof Consist of algebraic statements, with the algebraic properties as a justification (reason) for each statement.

Two-column (formal) Proof Includes statements and reasons organized in two columns

Postulate 2.8: Ruler Postulate The points on any line or line segment can be put into one-to-one correspondence with real numbers. (You can measure segments with a ruler.)

Postulate 2.9: Segment Addition Postulate If A, B and C are collinear, then point B is between A and C if and only if AB+BC=AB. (The sum of the parts of a segment is equal to the whole segment.)

Theorem 2.2: Properties of Segment Congruence Reflexive– Symmetric– Transitive–

Postulate 2.10: Protractor Postulate Given any angle the measure can be put in a one-to-one correspondence with the real numbers between 0 and 180. (You can measure angles with a protractor.)

Postulate 2.11: Angle Addition Postulate (The sum of the parts of an angle is equal to the whole angle.)

Theorem 2.3: Supplement Theorem If 2 angles form a linear pair, then they are supplementary angles.

Theorem 2.4: Complement Theorem If the non-common sides of 2 adjacent angles form a right angle, then the angles are complementary angles.

Theorem 2.5: Properties of Angle Congruence Reflexive– Symmetric– Transitive–

Theorem 2.6: Congruent Supplements Theorem Angles supplementary to the same angle or to congruent angles are congruent.

Theorem 2.7 Congruent Complements Theorem Angles complementary to the same angle or to congruent angles are congruent.

Theorem 2.8: Vertical Angles Theorem If two angles are vertical angles, then they are congruent.

Theorem 2.9 Perpendicular lines intersect to form four right angles.

Theorem 2.10 All right angles are congruent.

Theorem 2.11 Perpendicular lines form congruent adjacent angles.

Theorem 2.12 If two angles are congruent and supplementary, then each angle is a right angle.

Theorem 2.13 If two congruent angles form a linear pair, then they are right angles.