Reasoning and Proof flashcards

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What is inductive reasoning?Patterns and Conjectures: Using observed cases to propose a general conjecture.
Does a pattern prove a universal claim?Patterns and Conjectures: No; it only suggests a conjecture.
What is a conjecture?Patterns and Conjectures: A proposed statement believed to be true.
What should follow a conjecture?Patterns and Conjectures: Testing and, where possible, a deductive proof.
Can several examples establish an “all” claim?Patterns and Conjectures: Not by themselves.
What is a counterexample?Patterns and Conjectures: A valid case that makes a universal claim false.
How many counterexamples disprove “all”?Patterns and Conjectures: One.
What must a counterexample satisfy?Patterns and Conjectures: The claim’s stated conditions.
Does a counterexample prove a different universal claim?Patterns and Conjectures: No; it disproves the original claim.
Counterexample to “all primes are odd”?Patterns and Conjectures: The prime number 2.
What is deductive reasoning?Patterns and Conjectures: Applying accepted facts and valid logic to derive a conclusion.
What must be true for a sound conclusion?Patterns and Conjectures: The reasoning is valid and the premises are true.
Is a drawing alone a proof?Patterns and Conjectures: No.
Can a theorem justify a proof step?Patterns and Conjectures: Yes, when its conditions hold.
How does deduction differ from pattern guessing?Patterns and Conjectures: It uses logical rules to guarantee a conclusion from premises.
Form of a conditional?Conditional Statements: If p, then q.
What is the hypothesis?Conditional Statements: The condition after “if”.
What is the conclusion?Conditional Statements: The result after “then”.
When is p → q false?Conditional Statements: When p is true and q is false.
Does p → q assert its converse?Conditional Statements: No.
Converse of p → q?Conditional Statements: q → p.
Inverse of p → q?Conditional Statements: ¬p → ¬q.
Contrapositive of p → q?Conditional Statements: ¬q → ¬p.
Which related statement is equivalent to p → q?Conditional Statements: Its contrapositive.
Does a true conditional guarantee a true converse?Conditional Statements: No.
Meaning of “if and only if”?Conditional Statements: Both directions of the conditional hold.
Symbol for a biconditional?Conditional Statements: ↔.
When is p ↔ q true?Conditional Statements: When p and q have the same truth value.
What must be checked before writing a biconditional?Conditional Statements: The statement and its converse.
What does a definition often give?Conditional Statements: Necessary and sufficient conditions.
Reflexive property of equality?Properties of Equality and Congruence: a = a.
Symmetric property of equality?Properties of Equality and Congruence: If a=b, then b=a.
Transitive property of equality?Properties of Equality and Congruence: If a=b and b=c, then a=c.
Substitution property?Properties of Equality and Congruence: Replace an expression by an equal expression.
Does equality reverse direction?Properties of Equality and Congruence: Yes, by symmetry.
Addition property of equality?Properties of Equality and Congruence: Adding the same amount to both sides preserves equality.
Subtraction property of equality?Properties of Equality and Congruence: Subtracting the same amount from both sides preserves equality.
Multiplication property of equality?Properties of Equality and Congruence: Multiplying both sides by the same amount preserves equality.
When can both sides be divided by a value?Properties of Equality and Congruence: When that value is nonzero.
Why must operations be done on both sides?Properties of Equality and Congruence: To preserve equality.
Reflexive congruence?Properties of Equality and Congruence: A geometric object is congruent to itself.
Symmetric congruence?Properties of Equality and Congruence: If AB ≅ CD, then CD ≅ AB.
Transitive congruence?Properties of Equality and Congruence: If AB ≅ CD and CD ≅ EF, then AB ≅ EF.
Congruent segments have what?Properties of Equality and Congruence: Equal lengths.
Congruent angles have what?Properties of Equality and Congruence: Equal angle measures.
First step for 3x+5=20?Algebraic Proofs: Subtract 5 from both sides.
After 3x=15, what next?Algebraic Proofs: Divide both sides by 3.
Why may you subtract from both sides?Algebraic Proofs: Subtraction property of equality.
What value of x solves 3x+5=20?Algebraic Proofs: 5.
What does checking a solution mean?Algebraic Proofs: Substitute it into the original equation.
Distributive property?Algebraic Proofs: a(b+c)=ab+ac.
Expand 2(x+3).Algebraic Proofs: 2x+6.
Expand 3(x−4).Algebraic Proofs: 3x−12.
What are like terms?Algebraic Proofs: Terms with the same variable factors and powers.
Can 2x and 3 be combined as like terms?Algebraic Proofs: No.
What does a proof statement need?Algebraic Proofs: A valid reason.
Can a proof assume what it must prove?Algebraic Proofs: No.
Why is division by zero invalid?Algebraic Proofs: It is undefined.
What can justify a proof step?Algebraic Proofs: A given, definition, property, postulate, or theorem.
What is the final statement of a proof?Algebraic Proofs: The desired conclusion.
What theorem gives equal opposite angles at intersecting lines?Geometric Proofs: Vertical Angles Theorem.
How much do linear-pair angles sum to?Geometric Proofs: 180°.
What allows two interior angle parts to sum to a whole?Geometric Proofs: Angle Addition Postulate.
Are adjacent angles always supplementary?Geometric Proofs: No; only with additional conditions such as a linear pair.
What must be known to use a linear pair?Geometric Proofs: Adjacent angles with opposite noncommon rays.
What does a midpoint establish?Geometric Proofs: Two equal subsegment lengths.
Segment addition when B is between A and C?Geometric Proofs: AB+BC=AC.
What does a segment bisector do?Geometric Proofs: Intersects a segment at its midpoint.
Does equal length alone prove M is the midpoint of AB?Geometric Proofs: No; M must also lie on AB.
If AB=BC and B is between A and C, what is B?Geometric Proofs: The midpoint of AC.
What are givens?Geometric Proofs: Facts supplied at the start.
What is the goal of a proof?Geometric Proofs: The conclusion to establish.
Is apparent equality in a sketch a valid given?Geometric Proofs: No.
How should proof steps be ordered?Geometric Proofs: Each follows from givens or earlier justified steps.
What should be checked before applying a theorem?Geometric Proofs: Its stated conditions.