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Questions and Answers

What is a conjecture?

  • A specific example
  • A logical statement
  • An unproven statement based on observations (correct)
  • A proven statement

    • What is a counterexample?

    A specific case for which the conjecture is false.

    Define a conditional statement.

    A logical statement that has two parts: a hypothesis and conclusion.

    What does the if-then form consist of?

    <p>The 'if' part contains the hypothesis and the 'then' part contains the conclusion.</p> Signup and view all the answers

    What is negation in logic?

    <p>The opposite of the original statement.</p> Signup and view all the answers

    What happens in the converse of a statement?

    <p>The hypothesis and conclusion are exchanged.</p> Signup and view all the answers

    Define the inverse of a conditional statement.

    <p>Negate both the hypothesis and the conclusion.</p> Signup and view all the answers

    What is the contrapositive of a statement?

    <p>First write the converse and then negate both the hypothesis and the conclusion.</p> Signup and view all the answers

    What are equivalent statements?

    <p>When two statements are both true or both false.</p> Signup and view all the answers

    What is a biconditional statement?

    <p>A statement that contains the phrase 'if and only if'.</p> Signup and view all the answers

    What does deductive reasoning use to form arguments?

    <p>Uses facts, definitions, accepted properties, and the laws of logic.</p> Signup and view all the answers

    State the law of detachment.

    <p>If the hypothesis of a true conditional statement is true, then the conclusion is also true.</p> Signup and view all the answers

    Explain the law of syllogism.

    <p>If hypothesis p leads to conclusion q, and hypothesis q leads to conclusion r, then hypothesis p leads to conclusion r.</p> Signup and view all the answers

    What defines a line perpendicular to a plane?

    <p>If and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point.</p> Signup and view all the answers

    What is a proof?

    <p>A logical argument that shows a statement is true.</p> Signup and view all the answers

    Describe a two-column proof.

    <p>Has numbered statements and corresponding reasons that show an argument in a logical order.</p> Signup and view all the answers

    What is a theorem?

    <p>A statement that can be proven.</p> Signup and view all the answers

    State the Linear Pair Postulate (Postulate 12).

    <p>If two angles form a linear pair, then they are supplementary.</p> Signup and view all the answers

    What does the Vertical Angles Congruence Theorem (Theorem 2.6) state?

    <p>Vertical angles are congruent.</p> Signup and view all the answers

    State the Congruent Complements Theorem (Theorem 2.5).

    <p>If two angles are complementary to the same angle, then they are congruent.</p> Signup and view all the answers

    What is stated in the Congruent Supplements Theorem (Theorem 2.4)?

    <p>If two angles are supplementary to the same angle, then they are congruent.</p> Signup and view all the answers

    What does the Right Angles Congruence Theorem (Theorem 2.3) indicate?

    <p>All right angles are congruent.</p> Signup and view all the answers

    State the Congruence of Angles (Theorem 2.2).

    <p>Angle congruence is reflexive, symmetric, and transitive.</p> Signup and view all the answers

    What do you know about the Congruence of Segments (Theorem 2.1)?

    <p>Segment congruence is reflexive, symmetric, and transitive.</p> Signup and view all the answers

    What is the Reflexive Property of Equality?

    <p>A = A.</p> Signup and view all the answers

    What does the Symmetric Property of Equality state?

    <p>If A = B, then B = A.</p> Signup and view all the answers

    Define the Transitive Property of Equality.

    <p>If A = B and B = C, then A = C.</p> Signup and view all the answers

    What does the Distributive Property state?

    <p>A(b + c) = ab + ac.</p> Signup and view all the answers

    What is the Addition Property of Equality?

    <p>If a = b, then a + c = b + c.</p> Signup and view all the answers

    State the Subtraction Property of Equality.

    <p>If a = b, then a - c = b - c.</p> Signup and view all the answers

    What does the Multiplication Property of Equality state?

    <p>If a = b, then ac = bc.</p> Signup and view all the answers

    What is the Division Property of Equality?

    <p>If a = b and c is not equal to 0, then a/c = b/c.</p> Signup and view all the answers

    Explain the Substitution Property of Equality.

    <p>If a = b, then a can be substituted for b in any equation or expression.</p> Signup and view all the answers

    What should you know about using inductive and deductive reasoning?

    <p>You use inductive reasoning to make conjectures, and deductive reasoning to show whether the conjecture is true or false, and you should also know how to write proofs.</p> Signup and view all the answers

    What skills do you need to be able to do in geometry?

    <p>Use inductive reasoning, analyze conditional statements, apply deductive reasoning, use postulates and diagrams, and prove statements about segments and angles.</p> Signup and view all the answers

    Study Notes

    Vocabulary Terms

    • Conjecture: An unproven statement derived from observations.
    • Counterexample: A particular case that disproves a conjecture.
    • Conditional Statement: A logical structure combining a hypothesis and a conclusion.
    • If-then Form: The format in which the hypothesis is introduced with "if" and the conclusion follows with "then".
    • Negation: The statement that expresses the opposite of the original statement.
    • Converse: A statement formed by reversing the hypothesis and conclusion of a conditional statement.
    • Inverse: A statement formed by negating both the hypothesis and conclusion of a conditional statement.
    • Contrapositive: Created by writing the converse of a conditional statement and negating both parts.
    • Equivalent Statements: Statements that hold the same truth value, both true or both false.
    • Biconditional Statement: A logical statement connecting two conditions with "if and only if".

    Reasoning and Proof

    • Deductive Reasoning: Involves reasoning from established facts and logical rules to arrive at conclusions.
    • Law of Detachment: If the hypothesis of a true conditional statement is true, then the conclusion must also be true.
    • Law of Syllogism: Allows one to conclude a direct connection between two conditional statements, forming a new conditional statement.

    Geometry Concepts

    • Line Perpendicular to a Plane: Exists if a line intersects a plane at a single point and is perpendicular to all lines through that point that lie in the plane.
    • Proof: A structured logical argument demonstrating the truth of a statement.
    • Two-column Proof: A method of proof that lists statements and their corresponding reasons side by side for clarity.

    Theorems and Postulates

    • Theorem: A statement that has been proven based on previously established statements and principles.
    • Linear Pair Postulate: States that when two angles form a linear pair, they are supplementary.
    • Vertical Angles Congruence Theorem: Claims that vertical angles are congruent.
    • Congruent Complements Theorem: If two angles are each complementary to the same angle, they are congruent.
    • Congruent Supplements Theorem: If two angles are supplementary to the same angle, they are congruent.
    • Right Angles Congruence Theorem: Asserts that all right angles are congruent.
    • Congruence of Angles Theorem: Describes the properties of angle congruence (reflexive, symmetric, transitive).
    • Congruence of Segments Theorem: Outlines the properties of segment congruence (reflexive, symmetric, transitive).

    Properties of Equality

    • Reflexive Property: Every quantity is equal to itself (A=A).
    • Symmetric Property: If one quantity equals a second, then the second quantity equals the first (If A=B, then B=A).
    • Transitive Property: If one quantity equals a second, and that second quantity equals a third, then the first quantity equals the third (If A=B and B=C, then A=C).
    • Distributive Property: Describes how to multiply a number by a sum (A(b+c)= ab+ac).
    • Addition Property: States that if two quantities are equal, adding the same amount maintains the equality.
    • Subtraction Property: States that subtracting the same amount from equal quantities keeps them equal.
    • Multiplication Property: States that multiplying equal quantities by the same number keeps them equal.
    • Division Property: States that dividing equal quantities by the same non-zero number keeps them equal.
    • Substitution Property: Allows one to replace a quantity with equal quantity in any expression.

    Important Skills

    • Distinguish between inductive reasoning (used to form conjectures) and deductive reasoning (used to validate conjectures).
    • Understand assumptions in mathematical statements and the limitations on what can be assumed.
    • Skillfully articulate and construct proofs.
    • Apply postulates and utilize diagrams effectively in problem-solving.
    • Utilize algebraic properties for reasoning.
    • Prove relationships concerning segments and angles, especially angle pair relationships.

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