Comprehensive Accelerated Geometry Study Guide: Conditional Statements, Logic, and Proofs

Foundations of Logical Statements and Compound Operations

• Statement Definition: In formal mathematical logic, a statement is a declarative sentence that is definitively either true ($\text{T}$) or false ($\text{F}$), but not both simultaneously. Lowercase letters such as $p$, $q$, $r$, and $s$ are used to represent individual statements.
• Truth Value: The truth value of a statement denotes its truth or falsity. It is denoted by $\text{T}$ if the statement is true and $\text{F}$ if the statement is false.
• Negation ($\sim$):
  • The negation of a statement has the opposite meaning as well as the opposite truth value of the original statement.
  • Symbolic notation: The negation of statement $p$ is written as $\sim p$, read as "not $p$".
  • If $p$ is true, then $\sim p$ is false; conversely, if $p$ is false, then $\sim p$ is true.
  • Example: For the statement $r$: "The temperature is $25^\circ\text{F}$", its negation $\sim r$ is "The temperature is not $25^\circ\text{F}$".
• Compound Statements: A compound statement is formed by combining two or more individual statements using logical connectors.
• Conjunction ($\land$):
  • A conjunction is a compound statement formed by joining two or more statements with the connector word "and".
  • Symbolic notation: $p \land q$, read as "$p$ and $q$".
  • Truth condition: A conjunction is true if and only if all of its component statements are true. If at least one component statement is false, the entire conjunction is false.
• Disjunction ($\lor$):
  • A disjunction is a compound statement formed by joining two or more statements with the connector word "or".
  • Symbolic notation: $p \lor q$, read as "$p$ or $q$".
  • Truth condition: A disjunction is true as long as at least one of its component statements is true. It is false only when both component statements are false.
• Evaluations of Compound Statements:
  • Given the component statements:
    • $p$: Mrs. Cain teaches math. (Truth value: $\text{T}$)
    • $q$: Mr. Acone teaches video production. (Truth value: $\text{F}$)
  • Statement $p \land q$: "Mrs. Cain teaches math and Mr. Acone teaches video production." Because $q$ is false, the conjunction $p \land q$ has a truth value of $\text{FALSE}$.
  • Statement $p \lor q$: "Mrs. Cain teaches math or Mr. Acone teaches video production." Because $p$ is true, the disjunction $p \lor q$ has a truth value of $\text{TRUE}$.
  • Statement $\sim p \land q$: "Mrs. Cain does not teach math and Mr. Acone teaches video production." Because $\sim p$ is false and $q$ is false, the statement has a truth value of $\text{FALSE}$.
  • Statement $p \lor \sim q$: "Mrs. Cain teaches math or Mr. Acone does not teach video production." Because $p$ is true and $\sim q$ is true, the statement has a truth value of $\text{TRUE}$.

Conditional Statements

• Definition: A conditional statement is an "if-then" statement that establishes a mathematical relationship connecting a hypothesis to a conclusion.
• Structural Components:
  • Hypothesis ($p$): The phrase immediately following the word "if". It represents the premise or condition.
  • Conclusion ($q$): The phrase immediately following the word "then". It represents the result or effect produced when the hypothesis is met.
  • Symbolic notation: $p \rightarrow q$, read as "If $p$, then $q$" or "$p$ implies $q$".
• Identifying Hypotheses and Conclusions across Syntactic Variations:
  • Standard conditional form: "If $3x + 4 = -5$, then $x = -3$".
    • Hypothesis ($h$): $3x + 4 = -5$
    • Conclusion ($c$): $x = -3$
  • Standard conditional form: "If you take a class in television broadcasting, then you will film a sporting event".
    • Hypothesis ($h$): You take a class in television broadcasting.
    • Conclusion ($c$): You will film a sporting event.
  • Standard conditional form: "If it snows tomorrow, then school will be cancelled".
    • Hypothesis: It snows tomorrow.
    • Conclusion: School will be cancelled.
  • Inverted syntax ("$q$ if $p$"): In natural language, the conclusion may appear before the hypothesis. The clause attached to "if" always remains the hypothesis regardless of position.
    • Sentence: "School will be cancelled if it snows tomorrow".
      • Hypothesis: It snows tomorrow.
      • Conclusion: School will be cancelled.
    • Sentence: "You can register to vote if you are at least 18 years old".
      • If-then conditional form: "If you are at least 18 years old, then you can register to vote".
• Translating General Declarative Statements into Conditional (If-Then) Form:
  • Historical aphorism: "Those who do not remember the past are condemned to repeat it" (George Santayana).
    • If-then form: "If you do not remember the past, then you are condemned to repeat it".
  • Geometric property: "Adjacent angles share a common vertex and a common side".
    • If-then form: "If two angles are adjacent, then they share a common vertex and a common side".
  • Geometric definition: "A square must have four congruent sides".
    • If-then form: "If a shape is a square, then it must have four congruent sides".
  • Geometric classification: "A triangle with all congruent angles is an equilateral triangle".
    • If-then form: "If a triangle has all congruent angles, then it is an equilateral triangle".
  • Institutional policy: "Older campers who attend Woodland Falls Camp are expected to work".
    • Hypothesis ($h$): Campers are older campers who attend Woodland Falls Camp.
    • Conclusion ($c$): Campers are expected to work.
    • If-then form: "If campers are older campers who attend Woodland Falls Camp, then they are expected to work".
• Truth Value of a Conditional Statement ($p \rightarrow q$):
  • A conditional statement is defined as false in only one specific scenario: when the hypothesis ($p$) is true and the conclusion ($q$) is false.
  • If the hypothesis ($p$) is false, the conditional statement is automatically considered true (vacuously true), irrespective of whether the conclusion ($q$) is true or false.
  • Standard Truth Table for $p \rightarrow q$:

        | $p$ | $q$ | $p \rightarrow q$ |         | :---: | :---: | :---: |         | $\text{T}$ | $\text{T}$ | $\text{T}$ |         | $\text{T}$ | $\text{F}$ | $\text{F}$ |         | $\text{F}$ | $\text{T}$ | $\text{T}$ |         | $\text{F}$ | $\text{F}$ | $\text{T}$ |

*   **The Broken Promise Scenario:** Sally promises: "If I catch the early train home, then I'll take you swimming."
    *   Case 1 ($p = \text{T}, q = \text{T}$): Sally catches the early train home and takes you swimming. She kept her promise; the statement is $\text{true}$.
    *   Case 2 ($p = \text{T}, q = \text{F}$): Sally catches the early train home but does not take you swimming. She broke her promise; the statement is $\text{false}$.
    *   Case 3 ($p = \text{F}, q = \text{T}$): Sally does not catch the early train home, yet she still takes you swimming. She has not broken her promise; the statement is $\text{true}$.
    *   Case 4 ($p = \text{F}, q = \text{F}$): Sally does not catch the early train home and does not take you swimming. The condition was not met, so she broke no promise; the statement is $\text{true}$.
*   **Evaluating Isolated Truth Values:**
    *   "If $a$ and $b$ are negative, then $a + b$ is also negative": Truth value is $\text{true}$, because the sum of two negative numbers is always negative.
    *   "If the moon has purple spots, then it is June": Truth value is $\text{true}$. Because the moon never has purple spots, the hypothesis is permanently false; a conditional with a false hypothesis is true by default until a counterexample can show the hypothesis true and conclusion false simultaneously.

Related Conditional Statements and Logical Equivalence

• Definitions of Related Conditionals:
  • Conditional: The initial statement relating hypothesis to conclusion ($p \rightarrow q$, "If $p$, then $q$").
  • Converse: Formed by exchanging (reversing) the hypothesis and conclusion ($q \rightarrow p$, "If $q$, then $p$").
  • Inverse: Formed by negating both the hypothesis and the conclusion of the original conditional ($\sim p \rightarrow \sim q$, "If not $p$, then not $q$").
  • Contrapositive: Formed by both negating and reversing the hypothesis and conclusion ($\sim q \rightarrow \sim p$, "If not $q$, then not $p$").
• Logical Equivalence:
  • Two statements are logically equivalent if and only if they possess identical truth values under all possible truth conditions.
  • A conditional statement ($p \rightarrow q$) and its contrapositive ($\sim q \rightarrow \sim p$) are always logically equivalent.
  • A converse statement ($q \rightarrow p$) and its inverse ($\sim p \rightarrow \sim q$) are always logically equivalent.
  • A conditional statement is not necessarily logically equivalent to its converse or its inverse.
• Comprehensive Related Conditional Examples:
  • Instrument Example:
    • Conditional ($p \rightarrow q$): "If you play the trumpet, then you play a brass instrument". (Truth value: $\text{True}$; a trumpet is classified as a brass instrument).
    • Converse ($q \rightarrow p$): "If you play a brass instrument, then you play the trumpet". (Truth value: $\text{False}$; counterexample: you could play the trombone, tuba, or French horn).
    • Inverse ($\sim p \rightarrow \sim q$): "If you do not play the trumpet, then you do not play a brass instrument". (Truth value: $\text{False}$; counterexample: a musician can play the tuba without playing the trumpet).
    • Contrapositive ($\sim q \rightarrow \sim p$): "If you do not play a brass instrument, then you do not play the trumpet". (Truth value: $\text{True}$; because playing the trumpet requires playing a brass instrument, not playing a brass instrument precludes playing the trumpet).
  • Temperature Example:
    • Let $r$: "The temperature is $25^\circ\text{F}$", and $s$: "It is below freezing".
    • Conditional ($r \rightarrow s$): "If the temperature is $25^\circ\text{F}$, then it is below freezing". (Truth value: $\text{True}$, since freezing is $32^\circ\text{F}$ and $25^\circ\text{F} < 32^\circ\text{F}$).
    • Negation ($\sim r$): "The temperature is not $25^\circ\text{F}$".
    • Inverse ($\sim r \rightarrow \sim s$): "If the temperature is not $25^\circ\text{F}$, then it is not below freezing". (Truth value: $\text{False}$; counterexample: the temperature can be $30^\circ\text{F}$, which is not $25^\circ\text{F}$ but still below freezing).
    • Contrapositive ($\sim s \rightarrow \sim r$): "If it is not below freezing, then it is not $25^\circ\text{F}$". (Truth value: $\text{True}$, matching the conditional).
    • Converse ($s \rightarrow r$): "If it is below freezing, then it is $25^\circ\text{F}$". (Truth value: $\text{False}$; counterexample: it could be $20^\circ\text{F}$ or $30^\circ\text{F}$, both of which are below freezing but not equal to $25^\circ\text{F}$).
  • Segment Addition Example:
    • Conditional: "If $M$ is between points $K$ and $L$, then $KM + ML = KL$". (Truth value: $\text{True}$, by the Segment Addition Postulate).
    • Hypothesis: $M$ is between points $K$ and $L$.
    • Conclusion: $KM + ML = KL$.
    • Converse: "If $KM + ML = KL$, then $M$ is between points $K$ and $L$".
    • Inverse: "If $M$ is not between points $K$ and $L$, then $KM + ML \neq KL$".
• Contrapositive: "If $KM + ML \neq KL$, then $M$ is not between points $K$ and $L$". (Truth value: $\text{True}$, logically equivalent to the original conditional statement).
  • Broadcast Example:
    • Statement: "There will be a faculty guest on Eagle News if it is Friday".
    • If-then form ($p \rightarrow q$): "If it is Friday, then there will be a faculty guest on Eagle News". (Truth value: $\text{True}$, given premise).
    • Converse ($q \rightarrow p$): "If there is a faculty guest on Eagle News, then it is Friday". (Truth value: $\text{True}$/conditional-dependent).
    • Inverse ($\sim p \rightarrow \sim q$): "If it is not Friday, then there will not be a faculty guest on Eagle News".
    • Contrapositive ($\sim q \rightarrow \sim p$): "If there is not a faculty guest on Eagle News, then it is not Friday".
    • Biconditional: "There will be a faculty guest on Eagle News if and only if it is Friday". (Can be written if both conditional and converse hold true).
  • Idiom Example:
    • Statement: "Every good boy deserves fudge".
    • If-then form: "If you are a good boy, then you deserve fudge". (Truth value: $\text{True}$, assumed premise).
    • Converse: "If you deserve fudge, then you are a good boy". (Truth value: $\text{False}$; someone could deserve fudge for another reason without being a good boy).
    • Inverse: "If you are not a good boy, then you do not deserve fudge". (Truth value: $\text{False}$, matching the converse).
    • Contrapositive: "If you do not deserve fudge, then you are not a good boy". (Truth value: $\text{True}$, matching the conditional).
    • Biconditional: A true biconditional cannot be formed because the converse is false.
• Error Analysis in Constructing Inverses:
  • Given conditional: "If it is sunny, then I use sunscreen".
  • Alma's incorrect formulation of inverse: "If it is not sunny, then I use sunscreen".
  • Error identified: Alma failed to negate both parts of the statement. She only negated the hypothesis ($\sim p$) while leaving the conclusion ($q$) untouched, erroneously writing $\sim p \rightarrow q$.
  • Correct inverse: "If it is not sunny, then I do not use sunscreen" ($\sim p \rightarrow \sim q$).

Biconditional Statements

• Definition: A biconditional statement is the compound statement formed by joining a conditional statement and its converse using the logical connective "and".
  • Symbolic representation: $p \leftrightarrow q$, which is logically equivalent to $(p \rightarrow q) \land (q \rightarrow p)$.
  • Verbal syntax: Read as "$p$ if and only if $q$", frequently abbreviated as "$p$ iff $q$".
• Truth Criteria:
  • A biconditional statement is true if and only if both the conditional ($p \rightarrow q$) and its converse ($q \rightarrow p$) are true.
  • If either the conditional or the converse is false, a true biconditional statement cannot be written.
  • A counterexample to a biconditional statement occurs whenever the hypothesis is true while the conclusion is false, or whenever the hypothesis is false while the conclusion is true.
• Role in Mathematical Foundations:
  • Postulates and Theorems: Typically formulated as conditional statements ($p \rightarrow q$). Example: The Angle Addition Postulate states, "If $D$ is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$".
  • Definitions: All geometric definitions are biconditional statements ($p \leftrightarrow q$) because they are inherently reversible.
• Formulating Biconditionals from Conditionals:
  • Conditional: "If two lines intersect at right angles, then they are perpendicular". (Converse is also true).
  • Biconditional statement: "Two lines intersect at right angles if and only if they are perpendicular". (Truth value: $\text{True}$, $p \leftrightarrow q$).
• Deconstructing Biconditionals into Implied Conditionals ($p \rightarrow q$ and $q \rightarrow p$):
  • Midpoint Definition: "A point is a midpoint if and only if it splits a segment into two congruent parts".
    • Conditional ($p \rightarrow q$): "If a point is a midpoint, then it splits a segment into two congruent parts".
    • Converse ($q \rightarrow p$): "If a point splits a segment into two congruent parts, then the point is a midpoint".
  • Equilateral Triangle: "A triangle is equilateral if and only if it has three congruent sides".
    • Conditional ($p \rightarrow q$): "If a triangle is equilateral, then it has three congruent sides".
    • Converse ($q \rightarrow p$): "If a triangle has three congruent sides, then it is equilateral".
  • Product Sign Rule: "The product of two numbers is negative if and only if the numbers have opposite signs".
    • Conditional ($p \rightarrow q$): "If the product of two numbers is negative, then the numbers have opposite signs".
    • Converse ($q \rightarrow p$): "If two numbers have opposite signs, then the product of the two numbers is negative".
  • Complementary Angles: "Two angles are complementary if and only if the sum of their measures is 90 degrees".
    • Conditional ($p \rightarrow q$): "If two angles are complementary, then the sum of their measures is $90^\circ$".
    • Converse ($q \rightarrow p$): "If the sum of the measures of two angles is $90^\circ$, then the two angles are complementary".
  • Right Angles: "An angle is a right angle if and only if its measure is 90 degrees".
    • Conditional ($p \rightarrow q$): "If an angle is a right angle, then its measure is $90^\circ$".
    • Converse ($q \rightarrow p$): "If the measure of an angle is $90^\circ$, then it is a right angle".
  • Supplementary Angles: "Two angles are supplementary if and only if the sum of their measures is 180 degrees".
    • Conditional ($p \rightarrow q$): "If two angles are supplementary, then the sum of their measures is $180^\circ$".
    • Converse ($q \rightarrow p$): "If the sum of the measures of two angles is $180^\circ$, then the angles are supplementary".
  • Perpendicular Lines: "Two lines are perpendicular if and only if their intersection forms 4 right angles".
    • Conditional ($p \rightarrow q$): "If two lines are perpendicular, then their intersection forms 4 right angles".
    • Converse ($q \rightarrow p$): "If the intersection of two lines forms 4 right angles, then the lines are perpendicular".
• Feasibility Testing for Biconditionals:
  • Case A: Congruent Angles and Measures
    • Given $p$: Two angles are congruent; $q$: Two angles have equal measures.
    • Conditional ($p \rightarrow q$): "If two angles are congruent, then those two angles have equal measures". (Truth value: $\text{True}$
    • Converse ($q \rightarrow p$): "If two angles have equal measures, then they are congruent". (Truth value: $\text{True}$
    • Biconditional ($p \leftrightarrow q$): "Two angles are congruent if and only if the two angles have equal measures". (Valid and $\text{True}$
  • Case B: Congruent Angles and Adjacent Angles
    • Given $p$: Two angles are congruent; $q$: Two angles are adjacent.
    • Conditional ($p \rightarrow q$): "If two angles are congruent, then two angles are adjacent". (Truth value: $\text{False}$
    • Converse ($q \rightarrow p$): "If two angles are adjacent, then two angles are congruent". (Truth value: $\text{False}$
    • Biconditional ($p \leftrightarrow q$): Cannot be written as a valid biconditional because there are many types of congruent angles that are non-adjacent (e.g., vertical angles) and adjacent angles that are non-congruent; at least one statement is false.

Truth Tables and Compound Logical Expressions

• Row Calculation Principle: For a truth table containing $n$ distinct, independent statement variables, the total number of required truth value rows is given by the formula $2^n$.
  • For $n = 1$ statement: $2^1 = 2$ rows.
  • For $n = 2$ statements ($p, q$): $2^2 = 4$ rows.
  • For $n = 3$ statements ($p, q, r$): $2^3 = 8$ rows.
  • For $n = 4$ statements ($p, q, r, s$): $2^4 = 16$ rows.
  • For $n = 5$ statements: $2^5 = 32$ rows.
• Detailed Truth Table Constructions:
  • Truth Table for $(p \lor \sim p) \rightarrow (q \lor r)$:
    • Because $p \lor \sim p$ is a tautology, it evaluates to $\text{T}$ for all rows.
    • The implication is false only when the conclusion $q \lor r$ is false ($\text{F}$).

        | $p$ | $q$ | $r$ | $\sim p$ | $p \lor \sim p$ | $q \lor r$ | $(p \lor \sim p) \rightarrow (q \lor r)$ |         | :---: | :---: | :---: | :---: | :---: | :---: | :---: |         | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{F}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ |         | $\text{T}$ | $\text{T}$ | $\text{F}$ | $\text{F}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ |         | $\text{T}$ | $\text{F}$ | $\text{T}$ | $\text{F}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ |         | $\text{T}$ | $\text{F}$ | $\text{F}$ | $\text{F}$ | $\text{T}$ | $\text{F}$ | $\text{F}$ |         | $\text{F}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ |         | $\text{F}$ | $\text{T}$ | $\text{F}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ |         | $\text{F}$ | $\text{F}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ |         | $\text{F}$ | $\text{F}$ | $\text{F}$ | $\text{T}$ | $\text{T}$ | $\text{F}$ | $\text{F}$ |

*   *Truth Table for $(p \lor q) \lor (r \land \sim q)$:*
    *   Evaluates an 8-row compound disjunction.

        | $p$ | $q$ | $r$ | $p \lor q$ | $\sim q$ | $r \land \sim q$ | $(p \lor q) \lor (r \land \sim q)$ |         | :---: | :---: | :---: | :---: | :---: | :---: | :---: |         | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{F}$ | $\text{F}$ | $\text{T}$ |         | $\text{T}$ | $\text{T}$ | $\text{F}$ | $\text{T}$ | $\text{F}$ | $\text{F}$ | $\text{T}$ |         | $\text{T}$ | $\text{F}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ |         | $\text{T}$ | $\text{F}$ | $\text{F}$ | $\text{T}$ | $\text{T}$ | $\text{F}$ | $\text{T}$ |         | $\text{F}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ | $\text{F}$ | $\text{F}$ | $\text{T}$ |         | $\text{F}$ | $\text{T}$ | $\text{F}$ | $\text{T}$ | $\text{F}$ | $\text{F}$ | $\text{T}$ |         | $\text{F}$ | $\text{F}$ | $\text{T}$ | $\text{F}$ | $\text{T}$ | $\text{T}$ | $\text{T}$ |         | $\text{F}$ | $\text{F}$ | $\text{F}$ | $\text{F}$ | $\text{T}$ | $\text{F}$ | $\text{F}$ |

*   *Evaluation and Probability Analysis for $(p \land \sim q) \rightarrow (\sim r \lor s)$:*
    *   This compound expression utilizes 4 distinct variables ($p, q, r, s$), generating $2^4 = 16$ total rows.
    *   The conditional statement is false if and only if the hypothesis is true ($p \land \sim q = \text{T}$) and the conclusion is false ($\sim r \lor s = \text{F}$).
        *   Hypothesis condition ($p \land \sim q = \text{T}$): Requires $p = \text{T}$ and $q = \text{F}$.
        *   Conclusion condition ($\sim r \lor s = \text{F}$): Requires $\sim r = \text{F}$ (so $r = \text{T}$) and $s = \text{F}$.
        *   Therefore, the compound statement evaluates to $\text{False}$ in exactly one variable combination across all 16 rows: $(p, q, r, s) = (\text{T}, \text{F}, \text{T}, \text{F})$.
        *   In all other 15 rows, the conditional evaluates to $\text{True}$.
    *   Probability calculation: The probability that the conditional $(p \land \sim q) \rightarrow (\sim r \lor s)$ is true is: $\frac{15}{16}$.

Inductive Reasoning, Sequences, and Counterexamples

• Inductive Reasoning Definition: Inductive reasoning is the process of observing patterns, specific examples, or data trends to arrive at a general conclusion or rule.
• Conjectures:
  • A conjecture is an unproven conclusion, generalization, or educated guess reached using inductive reasoning.
  • Logical validity: A conjecture is not considered logically valid or formally proven, because observing a pattern hold true for several cases does not guarantee it holds true for all possible cases.
• Sequence Analysis and Term Prediction:
  • Sequence: $4, 8, 16, 32, \dots$
    • Pattern: Geometric sequence where each term is multiplied by 2 (doubling pattern).
    • Next predicted term: $32 \times 2 = 64$.
  • Sequence: $4, 8, 12, 16, \dots$
    • Pattern: Arithmetic sequence with a common difference of $+4$.
    • Next predicted term: $16 + 4 = 20$.
  • Sequence: $4, 8, 14, 22, \dots$
    • Pattern: Differences between consecutive terms increase by 2:
      • $8 - 4 = +4$
      • $14 - 8 = +6$
      • $22 - 14 = +8$
      • Next difference: $+10$
    • Next predicted term: $22 + 10 = 32$.
• Counterexamples:
  • Definition: A counterexample is a specific case, numerical value, or diagram that demonstrates that a conjecture or conditional statement is false.
  • To disprove a universal conjecture or conditional statement, exactly one valid counterexample is required.
• Rigorous Analysis of Conjectures and Counterexamples:
  • Conjecture: "If $M$ is between $A$ and $B$, then $AM \cong MB$".
    • Counterexample: Betweenness does not necessitate being the midpoint. Let segment $AB$ have length 6. Position point $M$ such that $AM = 2$ and $MB = 4$. Point $M$ lies between $A$ and $B$, but $AM \neq MB$.
• Conjecture: "If a number is divisible by 2, then it is divisible by 4".
  • Counterexample: The integer 30 (as well as 6, 10, 14, 18, 22). $30 / 2 = 15$, but $30 / 4 = 7.5$, which is not an integer.
• Conjecture: "If a student has Mr. Acone as a teacher, then the student takes Accelerated Geometry".
  • Counterexample: Mr. Acone also teaches standard math courses and video production. A student enrolled in Mr. Acone's video production class has Mr. Acone as a teacher but does not take Accelerated Geometry.
• Conjecture: "If $\angle ABC$ and $\angle DBC$ share ray $BC$, then $\angle ABC$ and $\angle DBC$ are adjacent".
  • Counterexample: By definition, adjacent angles cannot share interior points. If point $D$ lies in the interior of $\angle ABC$, the angles share the side $BC$, but their interiors overlap, meaning they are not adjacent.
• Conjecture: "If three points are coplanar, then they uniquely define a plane".
  • Counterexample: Three collinear points (points lying on the same straight line) are coplanar, but infinitely many distinct planes can intersect along that common line. Three non-collinear points are required to uniquely determine a single plane.
• Conjecture: "If a positive rational number is raised to the power of two, then it is greater than the number before it was raised to the power of two" (i.e., $x^2 > x$ for all positive rational numbers $x$).
  • Counterexample: Consider proper fractions strictly between 0 and 1, such as $x = \frac{1}{2}$. Raising to the power of two gives $(\frac{1}{2})^2 = \frac{1}{4}$. Since $\frac{1}{4} < \frac{1}{2}$, the squared value is less than the original number. (Additionally, for $x = 1$, $1^2 = 1$, which is equal, not strictly greater).
• Conjecture: "If $x$ and $y$ are whole numbers, then their difference, $x - y$, is a whole number".
  • Counterexample: Let $x = 2$ and $y = 5$ (both whole numbers). Their difference is $2 - 5 = -3$. Negative three is an integer, but it is not a whole number (the whole numbers are $0, 1, 2, 3, \dots$).
• Conjecture: "If water is heated, then it boils".
  • Counterexample: Heating water from $20^\circ\text{C}$ to $50^\circ\text{C}$ involves applying heat, but the water will not boil until it reaches its boiling threshold ($100^\circ\text{C}$ under standard atmospheric pressure).
• Conjecture: "If two triangles have equivalent angle measures, then they are congruent".
  • Counterexample: Two equilateral triangles with side lengths 2 and 5 both have angle measures of $60^\circ, 60^\circ, 60^\circ$. While their corresponding angles are equivalent, their corresponding side lengths are not equal; thus, they are similar triangles, not congruent triangles.
• Conjecture: "If it is raining, then it is spring".
  • Counterexample: It rains frequently during summer, autumn, and winter.
• Conjecture: "If your favorite color is blue, then you are a good speller".
  • Counterexample: Color preference has no causal or logical link to orthographic spelling competence.

Deductive Reasoning and Laws of Logic

• Deductive Reasoning Definition: Deductive reasoning is the logical process of using given facts, geometric definitions, accepted postulates, mathematical properties, and proven theorems to draw logically valid, indisputable conclusions.
• Comparison of Reasoning Methods:
  • Inductive Reasoning: Based on repeated observations and patterns; yields conjectures; never guarantees logical certainty; conclusions are not considered logically valid proofs.
  • Deductive Reasoning: Based on foundational logic, postulates, and theorems; yields rigorous conclusions; guarantees truth provided the original premises are true; logically valid.
• Utility of Symbolic Representation: Converting natural language sentences into abstract logical symbols (such as $p$, $q$, and $r$) removes linguistic ambiguity and reveals the underlying structure. This allows direct inspection to determine whether the Law of Detachment or the Law of Syllogism can be properly applied.
• Law of Detachment (Modus Ponens):
  • Formal Rule: If a conditional statement $p \rightarrow q$ is true, and its hypothesis $p$ is verified to be true, then the conclusion $q$ must also be true.
  • Structure:
    • Premise 1: $p \rightarrow q$
    • Premise 2: $p$
    • Valid Conclusion: $q$
  • Valid Application 1:
    • Given: If it rains this afternoon, then the soccer game will be cancelled ($p \rightarrow q$). It is raining this afternoon ($p$).
    • Valid Conclusion: The soccer game will be cancelled ($q$).
  • Valid Application 2:
    • Given: If you score an $85\%$ or greater on your test, then you will earn an A for your final grade ($p \rightarrow q$). You scored $89\%$ on your test ($p$, since $89\% \ge 85\%$).
    • Valid Conclusion: You earned an A for your final grade ($q$).
  • Valid Application 3:
    • Given: If you have a temperature above $100.4^\circ\text{F}$, then you have a fever ($p \rightarrow q$). Mario has a temperature of $101.2^\circ\text{F}$ ($p$, since $101.2^\circ\text{F} > 100.4^\circ\text{F}$).
    • Valid Conclusion: Mario has a fever ($q$), by the Law of Detachment.
  • Fallacy Analysis (Affirming the Conclusion / Converse Error):
    • Scenario A:
      • Given: If the Patriots play a late game on Sunday night, then your teacher will be tired on Monday ($p \rightarrow q$). Your teacher is tired on Monday ($q$).
      • Proposed Conclusion: The Patriots played a late game on Sunday night ($p$).
      • Validity: Not logically valid. Knowing that the conclusion ($q$) is true does not guarantee that the hypothesis ($p$) occurred; the teacher could be tired from insomnia, grading papers, or caring for a child.
    • Scenario B (Dakota's Error):
      • Given: "If my favorite team wins more than 55 games, they win the championship." Dakota observes: "My team won the championship, so they won more than 55 games."
      • Error Identified: Dakota assumed the hypothesis was true merely because the conclusion was true. The statement "My team won the championship" is the conclusion ($q$) of the conditional, not the hypothesis ($p$). The Law of Detachment cannot be applied in reverse.
• Law of Syllogism (Chain Rule):
  • Formal Rule: Given two true conditional statements such that the conclusion of the first conditional is the hypothesis of the second conditional, a third true conditional statement can be directly formed connecting the hypothesis of the first to the conclusion of the second.
  • Structure:
    • Premise 1: $p \rightarrow q$
    • Premise 2: $q \rightarrow r$
    • Valid Conclusion: $p \rightarrow r$
  • Valid Application 1:
    • Premise 1: If it is October, then it is autumn ($p \rightarrow q$).
    • Premise 2: If it is autumn, then you can get pumpkin-flavored anything ($q \rightarrow r$).
    • Valid Conclusion: If it is October, then you can get pumpkin-flavored anything ($p \rightarrow r$).
  • Valid Application 2:
    • Premise 1: If point $B$ is between $A$ and $C$, then $\vec{BA}$ and $\vec{BC}$ are opposite rays ($p \rightarrow q$).
    • Premise 2: If $\vec{BA}$ and $\vec{BC}$ are opposite rays, then $AB + BC = AC$ ($q \rightarrow r$).
    • Valid Conclusion: If $B$ is between $A$ and $C$, then $AB + BC = AC$ ($p \rightarrow r$).
  • Valid Application 3:
    • Premise 1: If you like sweet treats, then you will like ice cream ($p \rightarrow q$).
    • Premise 2: If you like ice cream, then you will like Crescent Ridge ($q \rightarrow r$).
    • Valid Conclusion: If you like sweet treats, then you will like Crescent Ridge ($p \rightarrow r$).
  • Valid Application 4:
    • Premise 1: If you eat too much, you get a stomach ache ($p \rightarrow q$).
    • Premise 2: If you get a stomach ache, you want to rest ($q \rightarrow r$).
    • Valid Conclusion: If you eat too much, you want to rest ($p \rightarrow r$).
  • Valid Application 5:
    • Premise 1: If two numbers are odd, the sum of the numbers is even ($p \rightarrow q$).
    • Premise 2: If a number is even, then the number is divisible by 2 ($q \rightarrow r$).
    • Valid Conclusion: If two numbers are odd, the sum is divisible by 2 ($p \rightarrow r$).
  • Valid Application 6 (Using Contrapositive Restructuring):
    • Premise 1: If the Eagles score the first goal, they will win the game ($p \rightarrow q$).
    • Premise 2: If they will not win the championship, then they will not win the game ($\sim r \rightarrow \sim q$).
    • Restructuring: Replace Premise 2 with its logically equivalent contrapositive: $q \rightarrow r$ ("If they win the game, they will win the championship").
    • Chain: $p \rightarrow q$ and $q \rightarrow r$.
    • Valid Conclusion: If the Eagles score the first goal, then they will win the championship ($p \rightarrow r$).
  • Non-Examples / Invalid Chain Structures:
    • Observation: Bus 7 was late to school on each of the past three Tuesdays. Proposed deductive conclusion: Bus 7 will be late next Tuesday. Validity: No logically valid conclusion can be deduced; this is purely an inductive observation.
    • Premise 1: If you plant these seeds, then corn will grow ($p \rightarrow r$).
    • Premise 2: If you water these seeds, then corn will grow ($q \rightarrow r$).
    • Validity: No logically valid conclusion. The conclusions match, but neither conclusion serves as a hypothesis for the other statement; the Law of Syllogism does not apply.
• Geometric Applications of Combined Laws of Logic:
  • Algebraic Segment Betweenness: Given that points $A$, $B$, and $C$ are collinear with $B$ between $A$ and $C$, the Segment Addition Postulate establishes that $AB + BC = AC$. If $AB = 3x$ and $BC = 5x - 3$, then substituting these expressions via detachment yields:         $AC = 3x + (5x - 3) = 8x - 3$
  • Interior Angle Deduction: Suppose point $D$ is in the interior of $\angle ABC$. By the Angle Addition Postulate, $m\angle ABD + m\angle DBC = m\angle ABC$. Applying algebraic properties of equality through detachment allows isolating the unknown measure:         $m\angle DBC = m\angle ABC - m\angle ABD$

Geometric Angle Pair Definitions and Relationships

• Adjacent Angles: Two coplanar angles that share a common vertex and a common side, but have no common interior points.
• Linear Pair: A pair of adjacent angles whose non-common sides are opposite rays. The angles in a linear pair are supplementary, and their measures sum to $180^\circ$.
• Vertical Angles: Two non-adjacent angles formed by two intersecting straight lines. Vertical angles are congruent ($\angle 1 \cong \angle 2$).
• Supplementary Angles: Two angles whose measures have a sum of $180^\circ$. The angles do not need to be adjacent (e.g., $m\angle 1 = 80^\circ$ and $m\angle 2 = 100^\circ$, so $80^\circ + 100^\circ = 180^\circ$).
• Complementary Angles: Two angles whose measures have a sum of $90^\circ$. The angles do not need to be adjacent (e.g., $m\angle 1 = 45^\circ$ and $m\angle 2 = 45^\circ$, or $30^\circ$ and $60^\circ$).
• Congruent Angles: Two angles that possess exactly equal degree measures ($m\angle 1 = m\angle 2$; e.g., $60^\circ = 60^\circ$).
• Perpendicular Lines: Two lines that intersect at right angles ($90^\circ$), forming 4 congruent right angles at their point of intersection.