Geometric Logic, Conditional Statements, and Deductive Reasoning

Conditional Statements

• Foundational Definitions:

  • Conditional Statement: An if-then statement consisting of two clauses.
  • Hypothesis: The phrase immediately following the word "if".
  • Conclusion: The phrase immediately following the word "then".

• Identifying Hypothesis and Conclusion:

  • Example 1: "If today is the first day of fall, then the month is September."
  • Hypothesis: Today is the first day of fall.
  • Conclusion: The month is September.
  • Example 2: "If $y - 3 = 5$, then $y = 8$."
  • Hypothesis: $y - 3 = 5$
  • Conclusion: $y = 8$
  • Example 3: "If you want to pass the test, then you should study."
  • Hypothesis: You want to pass the test.
  • Conclusion: You should study.
  • Example 4: "If two lines are parallel, then the lines never intersect."
  • Hypothesis: Two lines are parallel.
  • Conclusion: The lines never intersect.

• Symbolic Notation for Conditionals:

  • Written as $p \Rightarrow q$, which is read aloud as "if $p$ then $q$".
  • $p$ represents the hypothesis.
  • $q$ represents the conclusion.
  • Statement 1: Let $p$ = "the weather is warm" and $q$ = "we should go swimming".
  • $p \Rightarrow q$: If the weather is warm, then we should go swimming.
  • Statement 2: Let $p$ = "two segments have equal measures" and $q$ = "they are congruent".
  • $p \Rightarrow q$: If two segments have equal measures, then they are congruent.
  • Statement 3: Let $p$ = "yesterday was Wednesday" and $q$ = "today is Thursday".
  • $p \Rightarrow q$: If yesterday was Wednesday, then today is Thursday.

• Writing Given Sentences as Conditional Statements:

  • Sentence 1: "A rectangle is a polygon that has four right angles."
  • Conditional form: If a polygon is a rectangle, then it has four right angles.
  • Sentence 2: "An acute angle measures less than $90^{\circ}$."
  • Conditional form: If an angle is acute, then it measures less than $90^{\circ}$.
  • Sentence 3: "A tiger is an animal that is a mammal."
  • Conditional form: If an animal is a tiger, then it is a mammal.
  • Sentence 4: "Whole numbers that have $2$ as a factor are even."
  • Conditional form: If a whole number has $2$ as a factor, then it is even.

• Truth Value of Conditional Statements:

  • A conditional statement can have a truth value of either true or false.
  • True: Every single time the hypothesis is true, the conclusion is also guaranteed to be true.
  • False: To prove a conditional statement false, exactly one counterexample must be provided.

• Finding Counterexamples:

  • Statement 1: "If it is February, then there are only $28$ days this month."
  • Counterexample: In a leap year (such as the year $2024$), February contains $29$ days.
  • Statement 2: "If an integer is odd and less than $10$, then it is prime."
  • Counterexample: The integer $9$ is odd and less than $10$, but it is composite ($3 \times 3 = 9$).
  • Statement 3: "If the name of a state contains the word New, then the state borders an ocean."
  • Counterexample: New Mexico borders other states and Mexico, but does not border an ocean.
  • Statement 4: "If $x^2 = 25$, then $x = 5$."
  • Counterexample: Let $x = -5$. Since $(-5)^2 = 25$, the hypothesis holds while the conclusion $x = 5$ is false.

• Venn Diagrams and Conditional Statements:

  • A conditional statement "If $p$, then $q$" is modeled visually by nesting the set of items satisfying hypothesis $p$ entirely inside the set of items satisfying conclusion $q$.

• Example 1: "If it is a beagle, then it is a dog."
  • Inner circle: Beagles.
  • Outer circle: Dogs.
• Example 2: "If an angle has a measure of $40^{\circ}$, then it’s acute."
  • Inner circle: Angles with a measure of $40^{\circ}$.
  • Outer circle: Acute angles.
• Example 3: "If you live in Missouri, then you live in the Midwest."
  • Inner circle: People who live in Missouri.
  • Outer circle: People who live in the Midwest.
• Example 4: "$14$ is even."
  • Conditional formulation: If a number is $14$, then it is even.
  • Inner circle: The number $14$.
  • Outer circle: Even numbers.

Converses

• Definition and Mechanics of a Converse:

  • Converse: A statement formed by switching the hypothesis and the conclusion of a conditional statement.
  • Illustrative Example:
  • Conditional: If you draw a NINE, then draw three bonus cards.
  • Converse: If you draw $3$ bonus cards, then you drew a nine.

• Writing Converses:

  • Statement 1: "If you draw a four, then discard two cards."
  • Converse: If you discard two cards, then you draw a four.
  • Statement 2: "If two lines intersect to form right angles, then they are perpendicular."
  • Converse: If two lines are perpendicular, then they intersect to form right angles.
  • Statement 3: "If two lines are not parallel and do not intersect, then they are skew."
  • Converse: If two lines are skew, then they are not parallel and do not intersect.
  • Statement 4: "If $M$ is the midpoint of $\overline{AB}$, then $\overline{AM} \cong \overline{MB}$."
  • Converse: If $\overline{AM} \cong \overline{MB}$, then $M$ is the midpoint of $\overline{AB}$.

• Symbolic Notation for Converses:

  • Written as $q \Rightarrow p$, which is read aloud as "if $q$ then $p$".
  • Statement 1: Let $p$ = "a number is odd" and $q$ = "it’s not divisible by $2$".
  • Converse ($q \Rightarrow p$): If a number is not divisible by $2$, then it is odd.
  • Statement 2: Let $p$ = "two angles have a sum of $90^{\circ}$" and $q$ = "they are complementary".
  • Converse ($q \Rightarrow p$): If two angles are complementary, then they have a sum of $90^{\circ}$.
  • Statement 3: Let $p$ = "it is September $28$" and $q$ = "it is fall".
  • Converse ($q \Rightarrow p$): If it is fall, then it is September $28$.

• Truth Values of a Conditional and Its Converse:

  • A conditional statement and its converse do not necessarily have the same truth value.
  • Mammal/Dog Demonstration:
  • Conditional: If it is a dog, then it is a mammal. (Truth value: True)
  • Converse: If it is a mammal, then it is a dog. (Truth value: False)
  • Counterexample: A cat (or any other mammal that is not a dog).

• Evaluating Truth Values of Converses:

  • Statement 1: "If you are in Monett, then you are in Missouri."
  • Truth value of conditional: True.
  • Converse: If you are in Missouri, then you are in Monett.
  • Truth value of converse: False.
  • Counterexample: Being in St. Louis, Kansas City, or Springfield, Missouri.
  • Statement 2: "If a point is in the first quadrant, then its coordinates are positive."
  • Truth value of conditional: True.
  • Converse: If a point’s coordinates are positive, then it is in the first quadrant.
  • Truth value of converse: True.
  • Statement 3: "If it is raining, then there is precipitation."
  • Truth value of conditional: True.
  • Converse: If there is precipitation, then it is raining.
  • Truth value of converse: False.
  • Counterexample: Snow, sleet, or hail.
  • Statement 4: "If two angles have a measure of $90^{\circ}$, then the angles are congruent."
  • Truth value of conditional: True.
  • Converse: If two angles are congruent, then they have a measure of $90^{\circ}$.
  • Truth value of converse: False.
  • Counterexample: Two congruent angles that measure $45^{\circ}$.

Biconditional Statements

• Definition of a Biconditional:

  • Biconditional Statement: The statement formed by joining a conditional and its converse.
  • Formed by connecting the two parts using the phrase if and only if (often abbreviated as iff).
  • A biconditional statement is true if and only if both the original conditional statement and its converse are true.

• Forming Biconditional Statements:

  • Example 1:
  • Conditional: If two angles have the same measure, then the angles are congruent. (True)
  • Converse: If two angles are congruent, then the angles have the same measure. (True)
  • Biconditional: Two angles have the same measure if and only if the angles are congruent.
  • Example 2:
  • Conditional: If three points are collinear, then they lie on the same line. (True)
  • Converse: If three points lie on the same line, then they are collinear. (True)
  • Biconditional: Three points are collinear if and only if they lie on the same line.
  • Example 3:
  • Conditional: If a figure has eight sides, then it is an octagon. (True)
  • Converse: If a figure is an octagon, then it has eight sides. (True)
  • Biconditional: A figure has eight sides if and only if it is an octagon.

• Symbolic Notation for Biconditionals:

  • Written as $p \Leftrightarrow q$, which is read aloud as "$p$ if and only if $q$".
  • Statement 1: Let $p$ = "it is July $4\text{th}$" and $q$ = "it is Independence Day".
  • Biconditional ($p \Leftrightarrow q$): It is July $4\text{th}$ if and only if it is Independence Day.
  • Statement 2: Let $p$ = "points are coplanar" and $q$ = "they lie on the same plane".
  • Biconditional ($p \Leftrightarrow q$): Points are coplanar if and only if they lie on the same plane.
  • Statement 3: Let $p$ = "a figure is a pentagon" and $q$ = "it has $5$ sides".
  • Biconditional ($p \Leftrightarrow q$): A figure is a pentagon if and only if it has $5$ sides.

• Decomposing a Biconditional into Its Component Statements:

  • Biconditional 1: "A number is even if and only if it’s divisible by $2$."
  • Conditional: If a number is even, then it is divisible by $2$.
  • Converse: If a number is divisible by $2$, then it is even.
  • Biconditional 2: "A number is prime if and only if it has only two distinct factors, $1$ and itself."
  • Conditional: If a number is prime, then it has only two distinct factors, $1$ and itself.
  • Converse: If a number has only two distinct factors, $1$ and itself, then it is prime.
  • Biconditional 3: "Lines are skew if and only if they do not intersect and are noncoplanar."
  • Conditional: If lines are skew, then they do not intersect and are noncoplanar.
  • Converse: If lines do not intersect and are noncoplanar, then they are skew.

• Determining Whether a Biconditional Exists:

  • Statement 1: "If two angles share a side, then they are adjacent."
  • Converse: If two angles are adjacent, then they share a side.
  • Evaluation: The original conditional is false. Two overlapping angles can share a side without being adjacent if their interiors overlap.
  • Conclusion: A true biconditional does not exist.
  • Statement 2: "If it is October, then it is fall."
  • Converse: If it is fall, then it is October.
  • Counterexample: November is in fall, but it is not October.
  • Conclusion: A true biconditional does not exist.
  • Statement 3: "If a shape is a rectangle, then it has $4$ sides."
  • Converse: If a shape has $4$ sides, then it is a rectangle.
  • Counterexample: A trapezoid has $4$ sides, but it is not a rectangle.
  • Conclusion: A true biconditional does not exist.

Mathematical Definitions

• Characteristics of a Good Definition:

  • A good definition is a statement that classifies or identifies an object unambiguously.
  • Clearly Understood Terms: Words used must be commonly understood or previously defined.
  • Precise Language: Avoids vague qualifiers such as "large", "sort of", and "almost".
  • Reversibility: A good definition can always be stated as a true biconditional.

• Definition Discovery Activity: Polyglobs:

• Classification Analysis:

  • Left-hand reference examples demonstrate that Polyglobs possess exactly $3$ dark dots (eyes) arranged in a triangular cluster and exactly $4$ leg-like appendages.
  • Non-polyglobs violate these constraints by having only $2$ eyes, or having $5$ appendages, or having $2$ appendages.

• Identifying Marine Figures:

  • Figure A has $2$ eyes and $4$ tentacles; therefore, it is not a polyglob.
  • Figure B has $3$ eyes and $4$ tentacles; therefore, it is a polyglob.
  • Figure C has $3$ eyes and $2$ tentacles; therefore, it is not a polyglob.

• Definition: A polyglob is a creature with exactly $3$ eyes and exactly $4$ tentacles.

  • Expressing Definitions as Biconditionals:

• Base Definition: Perpendicular lines are two lines that intersect to form right angles.

  • Conditional: If two lines are perpendicular, then they intersect to form right angles. (True)
  • Converse: If two lines intersect to form right angles, then they are perpendicular. (True)
  • Biconditional: Two lines are perpendicular if and only if they intersect to form right angles.

• Definition 1: A right angle is an angle whose measure is $90^{\circ}$.

  • Conditional: If an angle is a right angle, then its measure is $90^{\circ}$.
  • Converse: If an angle has a measure of $90^{\circ}$, then it is a right angle.
  • Biconditional: An angle is a right angle if and only if its measure is $90^{\circ}$.

• Definition 2: A rectangle is a four-sided figure with at least one right angle.

  • Conditional: If a figure is a rectangle, then it is a four-sided figure with at least one right angle.
  • Converse: If a four-sided figure has at least one right angle, then it is a rectangle.
  • Biconditional: A figure is a rectangle if and only if it is a four-sided figure with at least one right angle.

• Definition 3: A triangle is a polygon with exactly three sides.

  • Conditional: If a polygon is a triangle, then it has exactly three sides.

  • Converse: If a polygon has exactly three sides, then it is a triangle.

  • Biconditional: A figure is a triangle if and only if it is a polygon with exactly three sides.

  • Evaluating Candidate Definitions:

• Candidate 1: "An apple is a fruit that contains seeds."

  • Evaluation: Not a good definition. It is not reversible because many other fruits contain seeds (e.g., oranges, watermelons).

• Candidate 2: "An angle bisector is a ray that divides an angle into two congruent angles."

  • Evaluation: Good definition. It is precise, utilizes clear terms, and is completely reversible.

• Candidate 3: "Opposite rays share a common endpoint."

  • Evaluation: Not a good definition. It is not reversible; any two rays sharing an endpoint can form an acute or obtuse angle without being opposite rays (opposite rays must form a straight line).

• Candidate 4: "A square is a figure with four right angles."

  • Evaluation: Not a good definition. It is not reversible because a non-square rectangle also has four right angles but is not a square.

• Candidate 5: "A penny is a coin worth one cent."

  • Evaluation: Good definition. It uniquely and reversibly identifies the coin in the currency system.

Inverse Statements

• Definition of Negation:

  • Negation: A statement that has the opposite truth value of the original statement.
  • Example: For the statement "The triangle is acute", its negation is "The triangle is not acute".

• Forming Negations:

  • Statement 1: "Angle $ABC$ is obtuse."
  • Negation: Angle $ABC$ is not obtuse.
  • Statement 2: "Line $m$ and line $n$ are not perpendicular."
  • Negation: Line $m$ and line $n$ are perpendicular.
  • Statement 3: "$m\angle XYZ > 70^{\circ}$"
  • Negation: $m\angle XYZ \le 70^{\circ}$
  • Statement 4: "Today is not Tuesday."
  • Negation: Today is Tuesday.

• Symbolic Notation for Negation:

  • Represented as $\sim p$, which is read aloud as "not $p$".
  • Statement 1: Let $p$ = "two angles are congruent".
  • $\sim p$: Two angles are not congruent.
  • Statement 2: Let $p$ = "the figure is a triangle".
  • $\sim p$: The figure is not a triangle.
  • Statement 3: Let $p$ = "you are not $16$ years old".
  • $\sim p$: You are $16$ years old.

• Definition of an Inverse:

  • Inverse: A statement that negates both the hypothesis and the conclusion of a conditional statement.
  • Illustrative Example:
  • Conditional: If a figure is a square, then it’s a rectangle.
  • Inverse: If a figure is not a square, then it’s not a rectangle.

• Writing Inverse Statements:

  • Statement 1: "If an angle is a straight angle, then it measures $180^{\circ}$."
  • Inverse: If an angle is not a straight angle, then it does not measure $180^{\circ}$.
  • Statement 2: "If a figure is a hexagon, then it has $6$ sides."
  • Inverse: If a figure is not a hexagon, then it does not have $6$ sides.
  • Statement 3: "If three points are on the same plane, then they are coplanar."
  • Inverse: If three points are not on the same plane, then they are not coplanar.
  • Statement 4: "If a point is in the center of a segment, then it is the midpoint."
  • Inverse: If a point is not in the center of a segment, then it is not the midpoint.

• Symbolic Notation for Inverse Statements:

  • Written as $\sim p \Rightarrow \sim q$, which is read aloud as "if not $p$ then not $q$".
  • Statement 1: Let $p$ = "it is $12:00\,\text{pm}$" and $q$ = "it is noon".
  • $\sim p \Rightarrow \sim q$: If it is not $12:00\,\text{pm}$, then it is not noon.
  • Statement 2: Let $p$ = "rays are opposite" and $q$ = "they share an endpoint and form a line".
  • $\sim p \Rightarrow \sim q$: If rays are not opposite, then they do not share an endpoint and form a line.
  • Statement 3: Let $p$ = "two angles are complementary" and $q$ = "the sum of the angles is $90^{\circ}$".
  • $\sim p \Rightarrow \sim q$: If two angles are not complementary, then the sum of the angles is not $90^{\circ}$.

• Determining the Truth Value of an Inverse:

  • Statement 1: "If you live in El Paso, then you live in Texas."
  • Inverse: If you do not live in El Paso, then you do not live in Texas.
  • Truth value: False.
  • Counterexample: Living in Dallas, Houston, or Austin, Texas.
  • Statement 2: "If a figure is a square, then all of its angles are right."
  • Inverse: If a figure is not a square, then not all of its angles are right.
  • Truth value: False.
  • Counterexample: A non-square rectangle is not a square, but all of its angles are right angles.
  • Statement 3: "If today is Thursday, then yesterday was Wednesday."
  • Inverse: If today is not Thursday, then yesterday was not Wednesday.
  • Truth value: True.
  • Statement 4: "If you’re in Geometry, then you have a test next week."
  • Inverse: If you are not in Geometry, then you do not have a test next week.
  • Truth value: False.
  • Counterexample: A student enrolled in Biology or Algebra who has a test next week.

Contrapositive Statements

• Definition of a Contrapositive:

  • Contrapositive: A statement formed by switching the hypothesis and conclusion of a conditional statement AND negating both of them.
  • Illustrative Example:
  • Conditional: If a figure is a square, then it is a rectangle.
  • Contrapositive: If the figure is not a rectangle, then the figure is not a square.

• Writing Contrapositive Statements:

  • Statement 1: "If an animal is a robin, then it is a bird."
  • Contrapositive: If an animal is not a bird, then it is not a robin.
  • Statement 2: "If two lines are skew, then they do not lie in the same plane."
  • Contrapositive: If two lines lie in the same plane, then they are not skew.
  • Statement 3: "If triangle $ABC$ is obtuse, then it has one angle greater than $90^{\circ}$."
  • Contrapositive: If triangle $ABC$ does not have one angle greater than $90^{\circ}$, then it is not obtuse.
  • Statement 4: "If a figure is a square, then it has $4$ congruent sides and $4$ right angles."
  • Contrapositive: If a figure does not have $4$ congruent sides and $4$ right angles, then it is not a square.

• Symbolic Notation for Contrapositives:

  • Written as $\sim q \Rightarrow \sim p$, which is read aloud as "if not $q$ then not $p$".
  • Statement 1: Let $p$ = "two planes intersect" and $q$ = "they meet in exactly one line".
  • $\sim q \Rightarrow \sim p$: If two planes do not meet in exactly one line, then they do not intersect.
  • Statement 2: Let $p$ = "a number is divisible by $8$" and $q$ = "it is divisible by $2$".
  • $\sim q \Rightarrow \sim p$: If a number is not divisible by $2$, then it is not divisible by $8$.

• Equivalent Statements and Truth Value Relationships:

  • Equivalent Statements: Statements that always share the exact same truth value.
  • A conditional statement and its converse or inverse may have different truth values.
  • The contrapositive of a conditional statement always has the same truth value as the original conditional statement.

Summary of Conditional Statement Forms

• Core Structural Summary:

  • Conditional: $p \Rightarrow q$
  • Example: If an angle is less than $90$ degrees, then it’s acute.
  • Converse: $q \Rightarrow p$
  • Example: If an angle is acute, then it’s less than $90$ degrees.
  • Biconditional: $p \Leftrightarrow q$
  • Example: An angle is less than $90$ degrees iff it’s acute.
  • Inverse: $\sim p \Rightarrow \sim q$
  • Example: If an angle is not less than $90$ degrees, then it’s not acute.
  • Contrapositive: $\sim q \Rightarrow \sim p$
  • Example: If an angle is not acute, then it’s not less than $90$ degrees.

• Comprehensive Form and Truth Value Analysis:

  • Base Statement: "The sum of two positive numbers is positive."
  • Conditional ($p \Rightarrow q$): If two numbers are positive, then their sum is positive.
  • Truth value: True.
  • Converse ($q \Rightarrow p$): If the sum of two numbers is positive, then the two numbers are positive.
  • Truth value: False.
  • Counterexample: Let the numbers be $10$ and $-2$. The sum is $10 + (-2) = 8$, which is positive, but $-2$ is not positive.
  • Biconditional ($p \Leftrightarrow q$): The sum of two numbers is positive if and only if the two numbers are positive.
  • Truth value: False (since the converse is false).
  • Inverse ($\sim p \Rightarrow \sim q$): If two numbers are not positive, then their sum is not positive.
  • Truth value: False.
  • Counterexample: Let the numbers be $10$ and $-2$. Not both numbers are positive, yet their sum is $8$, which is positive.
  • Contrapositive ($\sim q \Rightarrow \sim p$): If the sum of two numbers is not positive, then the two numbers are not positive.
  • Truth value: True (logically equivalent to the original true conditional).

Deductive Reasoning

• Foundations and Practical Applications:

  • Deductive Reasoning: The process of reasoning logically from given statements, facts, and definitions to arrive at a valid conclusion.
  • Professional applications include:
  • Physicians diagnosing illnesses from symptoms.
  • Carpenters calculating required building supplies from architectural plans.
  • Mechanics systematically diagnosing a car engine that fails to start.

• The Law of Detachment:

  • Definition: If a conditional statement is true and its hypothesis is true, then its conclusion must also be true.

• Introductory Scenario:

  • Conditional: If the football team wins on Friday night, then practice is canceled on Monday.

  • Given: The football team won by $7$ points Friday night.

  • Conclusion: Practice is canceled on Monday.

  • Applications of the Law of Detachment:

• Problem 1:

  • Statement: If a student gets an A on a final exam, then the student will pass the course.
  • Given: The student gets a $95\%$ on the final exam.
  • Conclusion: The student will pass the course.

• Problem 2:

  • Statement: If $M$ is the midpoint of a segment, then it divides the segment into two congruent segments.
  • Given: $M$ is the midpoint of $\overline{AB}$.
  • Conclusion: $M$ divides $\overline{AB}$ into two congruent segments ($\overline{AM} \cong \overline{MB}$).

• Problem 3:

  • Statement: If an angle is less than $90^{\circ}$, then it is acute.
  • Given: $\angle A$ is $80^{\circ}$.
  • Conclusion: $\angle A$ is acute.

• Problem 4:

  • Statement: If a figure is a triangle, then it has $2$ sides.

  • Given: Figure $ABC$ is a triangle.

  • Conclusion: Figure $ABC$ has $2$ sides.

  • Conditions Preventing the Use of the Law of Detachment:

• Problem 5:

  • Statement: If a road is icy, then driving conditions are hazardous.
  • Given: Driving conditions are hazardous.
  • Evaluation: Not possible to draw a conclusion.
  • Reason: The conclusion is given as true rather than the hypothesis. Driving conditions could be hazardous due to heavy rain, fog, or debris.

• Problem 6:

  • Statement: If you score a touchdown, then you receive $6$ points.
  • Given: The Cubs have $6$ points.
  • Evaluation: Not possible to draw a conclusion.
  • Reason: The conclusion is affirmed rather than the hypothesis; six points can be accumulated through methods other than a touchdown.

• Problem 7:

  • Statement: If an angle is right, then it is not obtuse.

  • Given: An angle is not right.

  • Evaluation: Not possible to draw a conclusion.

  • Reason: The hypothesis is negated rather than affirmed; an angle that is not right could be either acute or obtuse.

  • The Law of Syllogism:

• Definition: Allows a valid conclusion to be drawn from two true conditional statements when the conclusion of one statement matches the hypothesis of the other statement.

• Logical Form:

  • Statement 1: If $a$, then $b$.
  • Statement 2: If $b$, then $c$.
  • Resulting Conclusion: If $a$, then $c$.

• Introductory Scenario:

  • Statement 1: If you live in Monett, then you live in Missouri.

  • Statement 2: If you live in Missouri, then you live in the Midwest.

  • Conclusion: If you live in Monett, then you live in the Midwest.

  • Applications of the Law of Syllogism:

• Problem 1:

  • Statement 1: If you do your homework, then you will pass the test.
  • Statement 2: If you pass the test, then you will pass the class.
  • Conclusion: If you do your homework, then you will pass the class.

• Problem 2:

  • Statement 1: If a number ends in $0$, then it is divisible by $10$.
  • Statement 2: If a number is divisible by $10$, then it is divisible by $5$.
  • Conclusion: If a number ends in $0$, then it is divisible by $5$.

• Problem 3:

  • Statement 1: If a quadrilateral is a square, then it contains four right angles.
  • Statement 2: If a quadrilateral contains four right angles, then it’s a rectangle.
  • Conclusion: If a quadrilateral is a square, then it’s a rectangle.

• Problem 4:

  • Statement 1: If two lines are not parallel or skew, then they intersect.

  • Statement 2: If two lines intersect, then they intersect at a point.

  • Conclusion: If two lines are not parallel or skew, then they intersect at a point.

  • Combining the Law of Syllogism and the Law of Detachment:

• Introductory Multi-Step Problem:

  • Statement 1: If a river is more than $4000\,\text{miles}$ long, then it is longer than the Amazon.
  • Statement 2: If a river is longer than the Amazon, then it is the longest river in the world.
  • Fact: The Nile is $4132\,\text{miles}$ long.
  • Syllogism conclusion: If a river is more than $4000\,\text{miles}$ long, then it is the longest river in the world.
  • Detachment conclusion: The Nile is the longest river in the world.

• Problem 1:

  • Statement 1: If it is raining, then the temperature is greater than $32^{\circ}\text{F}$.
  • Statement 2: If the temperature is greater than $32^{\circ}\text{F}$, then it is not freezing outside.
  • Fact: It rained on Monday.
  • Law of Syllogism statement: If it is raining, then it is not freezing outside.
  • Law of Detachment conclusion: It was not freezing outside on Monday.

• Problem 2:

  • Statement 1: If you live in Providence, then you live in Rhode Island.
  • Statement 2: If you live in Rhode Island, then you live in the smallest state in the United States.
  • Fact: Sam lives in Providence.
  • Law of Syllogism statement: If you live in Providence, then you live in the smallest state in the United States.
  • Law of Detachment conclusion: Sam lives in the smallest state in the United States.

• Problem 3:

  • Statement 1: If two angles are vertical, then they are congruent.
  • Statement 2: If angles are congruent, then they have equal measures.
  • Fact: $\angle A$ and $\angle B$ are vertical.
  • Law of Syllogism statement: If two angles are vertical, then they have equal measures.
  • Law of Detachment conclusion: $\angle A$ and $\angle B$ have equal measures.

• Problem 4:

  • Statement 1: If you ate a Golden Delicious, then you ate an apple.
  • Statement 2: If you ate an apple, then you ate a fruit.
  • Fact: Cody ate a Golden Delicious.
  • Law of Syllogism statement: If you ate a Golden Delicious, then you ate a fruit.
  • Law of Detachment conclusion: Cody ate a fruit.