[go: up one dir, main page]
Everand Logo

Welcome to Everand!

  • Discover millions of ebooks, audiobooks, and so much more with a free trial

    From $11.99/month after trial. Cancel anytime.

    Geometry I Essentials
    Geometry I Essentials
    Geometry I Essentials
    Ebook141 pages36 minutes

    Geometry I Essentials

    By The Editors of REA

    ()

    Read preview

    About this ebook

    REA’s Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Geometry I includes methods of proof, points, lines, planes, angles, congruent angles and line segments, triangles, parallelism, quadrilaterals, geometric inequalities, and geometric proportions and similarity.
    LanguageEnglish
    PublisherResearch & Education Association
    Release dateJan 1, 2013
    ISBN9780738672700
    Geometry I Essentials

    Read more from The Editors Of Rea

    Related to Geometry I Essentials

    Related ebooks

    Mathematics For You

    View More
    View More

    Related podcast episodes

    Related categories

    Reviews for Geometry I Essentials

    0 ratings

    0 ratings0 reviews

    What did you think?

    Tap to rate

    Review must be at least 10 words

      Book preview

      Geometry I Essentials - The Editors of REA

      more!

      CHAPTER 1

      Method of Proof

      1.1 Logic

      Definition 1

      A statement is a sentence that is either true or false, but not both.

      Definition 2

      If a and b are statements, then a statement of the form "a and b" is called the conjunction of a and b, denoted by a ∧ b.

      Definition 3

      The disjunction of two statements a and b is shown by the compound statement "a or b," denoted by a ∨ b.

      Defnition 4

      The negation of a statement q is the statement "not q," denoted by ~ q.

      Definition 5

      The compound statement "if a, then b," denoted by a → b, is called a condition statement or an implication.

      "If a is called the hypothesis or premise of the implication; then b" is called the conclusion of the implication.

      Further, statement a is called the antecedent of the implication, and statement b is called the consequent of the implication.

      Definition 6

      The converse of a → b is b → a.

      Definition 7

      The contrapositive of a → b is ~ b → ~ a.

      Definition 8

      The inverse of a → b is ~ a → ~ b.

      Definition 9

      The statement of the form "p if and only if q," denoted by p ↔ q, is called a biconditional statement.

      Definition 10

      An argument is valid if the truth of the premises means that the conclusion must also be true.

      Definition 11

      Intuition is the process of making generalizations based on insight.

      Basic Principles, Laws, and Theorems

      Any statement is either true or false (The Law of the Excluded Middle).

      A statement cannot be both true and false (The Law of Contradiction).

      The converse of a true statement is not necessarily true.

      The converse of a definition is always true.

      For a theorem to be true, it must be true for all cases.

      A statement is false if one false instance of the statement exists.

      The inverse of a true statement is not necessarily true.

      The contrapositive of a true statement is true and the contrapositive of a false statement is false.

      If the converse of a true statement is true, then the inverse is true. Likewise, if the converse is false, the inverse is false.

      Statements that are either both true or false are said to be logically equivalent.

      If a given statement and its converse are both true, then the conditions in the hypothesis of the statement are both necessary and sufficient for the conclusion of the statement.

      If a given statement is true but its converse is false, then the conditions are sufficient but not necessary for the conclusion of the statement.

      If a given statement and its converse are both false, then the conditions are neither sufficient nor necessary for the statement’s conclusion.

      1.2 Deductive Reasoning

      An arrangement of statements that would allow you to deduce the third one from the preceding two is called a syllogism. A syllogism has three parts:

      The first part is a general statement concerning a whole group. This is called the major premise.

      The second part is a specific statement which indicates that a certain individual is a member of that group. This is called the minor premise.

      The last part of a syllogism is a statement to the effect that the general statement which applies to the group also applies to the individual. This third statement of a syllogism is called a deduction.

      EXAMPLE A: Properly Deduced Argument

      Major Premise: All birds have feathers.

      Minor Premise: An eagle is a bird.

      Enjoying the preview?
      Page 1 of 1