Geometry Study Notes: Proofs, Logic, and Reasoning

Conditional Statements and Related Conditionals

• Definitions and Concepts:

  • A conditional statement is a logical statement that has two parts: a hypothesis $p$ and a conclusion $q$. It is symbolized by $p \rightarrow q$ (read as "$p$ implies $q$").
  • In if-then form, the "if" part contains the hypothesis $p$ and the "then" part contains the conclusion $q$.
  • Example:
  • Statement: "If a polygon is a triangle, then the sum of its angle measures is $180^\circ$".
  • Hypothesis ($p$): A polygon is a triangle.
  • Conclusion ($q$): The sum of its angle measures is $180^\circ$.

• Negation:

  • The negation of a statement is the opposite of the original statement.
  • Symbolized by $\sim p$ (read as "not $p$").
  • Examples:
  • Statement: "The ball is red." $\rightarrow$ Negation: "The ball is not red."
  • Statement: "The cat is not black." $\rightarrow$ Negation: "The cat is black."
  • Statement: "$\angle A$ and $\angle B$ are supplementary." $\rightarrow$ Negation: "$\angle A$ and $\angle B$ are not supplementary."
  • Statement: "Triangle $ABC$ is not a right triangle." $\rightarrow$ Negation: "Triangle $ABC$ is a right triangle."

• Rewriting Statements in If-Then Form:

  • "You are in Florida if you are in Miami."
  • Hypothesis ($p$): You are in Miami.
  • Conclusion ($q$): You are in Florida.
  • If-Then Form: If you are in Miami, then you are in Florida.
  • "A polygon with five sides is a pentagon."
  • Hypothesis ($p$): A polygon has five sides.
  • Conclusion ($q$): It is a pentagon.
  • If-Then Form: If a polygon has five sides, then it is a pentagon.
  • "All $30^\circ$ angles are acute angles."
  • Hypothesis ($p$): An angle measures $30^\circ$.
  • Conclusion ($q$): It is an acute angle.
  • If-Then Form: If an angle measures $30^\circ$, then it is an acute angle.
  • "$2x + 7 = 1$, because $x = -3$".
  • Hypothesis ($p$): $x = -3$.
  • Conclusion ($q$): $2x + 7 = 1$.
  • If-Then Form: If $x = -3$, then $2x + 7 = 1$.

• Related Conditionals:

  • Conditional Statement: If $p$, then $q$ ($p \rightarrow q$).
  • Converse: Exchange the hypothesis and the conclusion. Words: If $q$, then $p$ ($q \rightarrow p$).
  • Inverse: Negate both the hypothesis and the conclusion. Words: If not $p$, then not $q$ ($\sim p \rightarrow \sim q$).
  • Contrapositive: First write the converse, then negate both the hypothesis and the conclusion. Words: If not $q$, then not $p$ ($\sim q \rightarrow \sim p$).

• Equivalent Statements and Truth Values:

  • When two statements are both true or both false, they are called equivalent statements.
  • A conditional statement and its contrapositive are equivalent statements (either both are true or both are false).
  • The converse and inverse of a conditional statement are equivalent statements (either both are true or both are false).
  • Truth Value Examples:
  • Example 1:
    • Conditional: If a number is even, then it is divisible by $2$. (True)
    • Converse: If a number is divisible by $2$, then it is even. (True)
    • Relationship: Equivalent statements.
  • Example 2:
    • Conditional: If a shape is a square, then it has exactly $5$ sides. (False)
    • Contrapositive: If a shape does not have exactly $5$ sides, then it is not a square. (False)
    • Relationship: Equivalent statements.
  • Example 3:
    • Conditional: If a figure is a rectangle, then it has four sides. (True)
    • Converse: If a figure has four sides, then it is a rectangle. (False)
    • Relationship: Non-equivalent statements.
  • Example 4: Let $p$ be "an angle measures $90^\circ$" and $q$ be "it is a right angle".
    • Conditional ($p \rightarrow q$): If an angle measures $90^\circ$, then it is a right angle. (True)
    • Converse ($q \rightarrow p$): If an angle is a right angle, then it measures $90^\circ$. (True)
    • Inverse ($\sim p \rightarrow \sim q$): If an angle does not measure $90^\circ$, then it is not a right angle. (True)
    • Contrapositive ($\sim q \rightarrow \sim p$): If an angle is not a right angle, then it does not measure $90^\circ$. (True)
  • Example 5: Statement: "If two angles are complementary, then the angles are acute."
    • Converse: If two angles are acute, then they are complementary. (False)
    • Inverse: If two angles are not complementary, then they are not acute. (False)
    • Contrapositive: If two angles are not acute, then they are not complementary. (True)
  • Example 6: Statement: "If two points lie on the same line, then the points are collinear."
    • Converse: If points are collinear, then they lie on the same line. (True)
    • Inverse: If two points do not lie on the same line, then the points are not collinear. (True)
    • Contrapositive: If points are not collinear, then they do not lie on the same line. (True)

• Biconditional Statements:

  • When a conditional statement and its converse are both true, they can be written as a single biconditional statement.
  • Contains the phrase "if and only if". Symbolized by $p \leftrightarrow q$ (read as "$p$ if and only if $q$").
  • Any definition can be written as a biconditional statement.
  • Examples:
  • Perpendicular Lines Definition:
    • Conditional: If two lines intersect to form a right angle, then they are perpendicular lines.
    • Biconditional: Two lines are perpendicular if and only if they intersect to form a right angle.
  • Quadrilateral Definition:
    • Biconditional: A polygon is a quadrilateral if and only if it has four sides.
    • Conditional: If a polygon is a quadrilateral, then it has four sides. (True)
    • Converse: If a polygon has four sides, then it is a quadrilateral. (True)
  • Perpendicular Bisector Definition:
    • Biconditional: A line is the perpendicular bisector of a segment if and only if it divides the segment into two congruent segments.
    • Conditional: If a line is the perpendicular bisector of a segment, then it divides the segment into two congruent segments. (True)
    • Converse: If a line divides a segment into two congruent segments, then it is the perpendicular bisector of the segment. (False - Counterexample: a line intersecting a segment's midpoint at a non-right angle).
  • Congruent Segments:
    • Conditional: If two line segments have the same length, then they are congruent segments. (True)
    • Converse: If two line segments are congruent segments, then they have the same length. (True)
    • Biconditional: Two line segments have the same length if and only if they are congruent segments.
  • Isosceles Triangles:
    • Conditional: If a triangle is isosceles, then it has at least two congruent sides. (True)
    • Converse: If a triangle has at least two congruent sides, then it is isosceles. (True)
    • Biconditional: A triangle is isosceles if and only if it has at least two congruent sides.

Inductive and Deductive Reasoning

• Inductive Reasoning and Conjectures:

  • Inductive Reasoning: The process of finding a pattern in specific cases and then writing a conjecture for the general case.
  • Conjecture: An unproven statement that is based on observations.
  • Counterexample: A specific case for which a conjecture is false. To show that a conjecture is true, it must be proven true for all cases. To show that a conjecture is false, finding only one counterexample is sufficient.

• Finding Counterexamples to Disprove Conjectures:

  • Conjecture: "The sum of two numbers is always more than the greater number."
  • Counterexample: $3 + (-5) = -2$ (or $-5 + (-8) = -13$). Because $-2 < 3$ and $-13 < -5$, the conjecture is false.
  • Conjecture: "An acute angle measures $30^\circ$".
  • Counterexample: An angle measuring $45^\circ$ (or $60^\circ$) is an acute angle, but its measure is not $30^\circ$.
  • Conjecture: "If two angles are complementary, then the angles measure $45^\circ$".
  • Counterexample: Angles measuring $30^\circ$ and $60^\circ$ are complementary ($30^\circ + 60^\circ = 90^\circ$), but neither angle measures $45^\circ$.
  • Conjecture: "Congruent angles are always vertical angles."
  • Counterexample: Base angles in an isosceles triangle, or alternate interior angles formed by parallel lines, are congruent but not vertical angles.
  • Conjecture: "If two angles are adjacent, then they are supplementary."
  • Counterexample: Two adjacent angles measuring $30^\circ$ and $40^\circ$ share a vertex and side, but their sum is $70^\circ \neq 180^\circ$.

Geometric Postulates and Diagram Interpretation

• Point, Line, and Plane Postulates:

  • Postulate 2.1 (Two Point Postulate): Through any two points, there exists exactly one line. (Through points $A$ and $B$, there is exactly one line $l$).
  • Postulate 2.2 (Line-Point Postulate): A line contains at least two points.
  • Postulate 2.3 (Line Intersection Postulate): If two lines intersect, then their intersection is exactly one point. (The intersection of line $m$ and line $n$ is point $C$).
  • Postulate 2.4 (Three Point Postulate): Through any three noncollinear points, there exists exactly one plane. (Through points $D$, $E$, and $F$, there is exactly one plane $R$ containing at least three noncollinear points).
  • Postulate 2.5 (Plane-Point Postulate): A plane contains at least three noncollinear points.
  • Postulate 2.6 (Plane-Line Postulate): If two points lie in a plane, then the line containing them lies in the plane.
  • Postulate 2.7 (Plane Intersection Postulate): If two planes intersect, then their intersection is a line. (The intersection of plane $S$ and plane $T$ is line $l$).

• Perpendicular Lines and Planes:

  • A line is perpendicular to a plane 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.
  • In geometric diagrams, a line perpendicular to a plane must be explicitly marked with a right-angle symbol ($90^\circ$ mark).

• Interpreting Geometric Diagrams:

  • Valid Assumptions:
  • Points that lie on a straight line drawn in the diagram are collinear (e.g., points $A$, $B$, and $F$).
  • Lines drawn intersecting at a point intersect at that point (e.g., line $\overleftrightarrow{AF}$ intersects line $\overleftrightarrow{BC}$ at point $B$).
  • Two planes shown intersecting meet along a line (e.g., plane $S$ and plane $T$ intersect at line $\overleftrightarrow{BC}$).
  • Invalid Assumptions (Cannot be Assumed without Markings):
  • Points are collinear if no drawn line connects them (e.g., points $E$, $B$, and $D$ cannot be assumed collinear).
  • Lines are perpendicular without a right-angle symbol present.
  • Segments or angles are congruent without explicit tick marks or arc symbols.

Algebraic Properties and Proofs

• Algebraic Properties of Equality (for real numbers $a$, $b$, and $c$):

  • Addition Property of Equality: If $a = b$, then $a + c = b + c$
  • Subtraction Property of Equality: If $a = b$, then $a - c = b - c$
  • Multiplication Property of Equality: If $a = b$, then $a \cdot c = b \cdot c$, where $c \neq 0$
  • Division Property of Equality: If $a = b$, then $\frac{a}{c} = \frac{b}{c}$, where $c \neq 0$
  • Substitution Property of Equality: If $a = b$, then $a$ can be substituted for $b$ (or $b$ for $a$) in any equation or expression.
  • Distributive Property:
  • Sum: $a(b + c) = ab + ac$
  • Difference: $a(b - c) = ab - ac$

• Justifying Steps in Solving Algebraic Equations:

  • Example 1:
  • Equation: Solve $3x + 2 = 23 - 4x$
  • Step 1: $3x + 2 = 23 - 4x$ | Reason: Given
  • Step 2: $3x = 21 - 4x$ | Reason: Subtraction Property of Equality
  • Step 3: $7x = 21$ | Reason: Addition Property of Equality
  • Step 4: $x = 3$ | Reason: Division Property of Equality
  • Example 2:
  • Equation: Solve $-5(7w + 8) = 30$
  • Step 1: $-5(7w + 8) = 30$ | Reason: Given
  • Step 2: $-35w - 40 = 30$ | Reason: Distributive Property
  • Step 3: $-35w = 70$ | Reason: Addition Property of Equality
  • Step 4: $w = -2$ | Reason: Division Property of Equality
  • Example 3:
  • Equation: Solve $3(3x + 14) = -3$
  • Step 1: $3(3x + 14) = -3$ | Reason: Given
  • Step 2: $9x + 42 = -3$ | Reason: Distributive Property
  • Step 3: $9x = -45$ | Reason: Subtraction Property of Equality
  • Step 4: $x = -5$ | Reason: Division Property of Equality
  • Example 4:
  • Equation: Solve $4 = -10b + 6(2 - b)$
  • Step 1: $4 = -10b + 6(2 - b)$ | Reason: Given
  • Step 2: $4 = -10b + 12 - 6b$ | Reason: Distributive Property
  • Step 3: $4 = -16b + 12$ | Reason: Combine Like Terms
  • Step 4: $-8 = -16b$ | Reason: Subtraction Property of Equality
  • Step 5: $b = 0.5$ | Reason: Division Property of Equality
  • Example 5:
  • Equation: Solve $-3(2r - 5) = -2(9 - 4r)$
  • Step 1: $-3(2r - 5) = -2(9 - 4r)$ | Reason: Given
  • Step 2: $-6r + 15 = -18 + 8r$ | Reason: Distributive Property
  • Step 3: $-14r + 15 = -18$ | Reason: Subtraction Property of Equality
  • Step 4: $-14r = -33$ | Reason: Subtraction Property of Equality
  • Step 5: $r = \frac{33}{14}$ | Reason: Division Property of Equality

• Real-Life Modeling Proof:

  • Formula: $p(r + 1) = n$, where $p$ is the previous wage, $r$ is the percent increase as a decimal, and $n$ is the new wage.
  • Solving the Formula for $r$:
  • Step 1: $p(r + 1) = n$ | Reason: Given
  • Step 2: $pr + p = n$ | Reason: Distributive Property
  • Step 3: $pr = n - p$ | Reason: Subtraction Property of Equality
  • Step 4: $r = \frac{n - p}{p}$ | Reason: Division Property of Equality
  • Application Problem:
  • An hourly wage increases from $\$15.25$ to $\$15.86$. Calculate the percent raise.
  • Calculation: $r = \frac{15.86 - 15.25}{15.25} = \frac{0.61}{15.25} = 0.04 = 4\%$

Equivalence Properties and Congruence Theorems

• Reflexive, Symmetric, and Transitive Properties:

  • Reflexive Property:
  • Real Numbers: $a = a$
  • Segment Lengths: $AB = AB$
  • Angle Measures: $m\angle A = m\angle A$
  • Symmetric Property:
  • Real Numbers: If $a = b$, then $b = a$.
  • Segment Lengths: If $AB = CD$, then $CD = AB$.
  • Angle Measures: If $m\angle A = m\angle B$, then $m\angle B = m\angle A$.
  • Transitive Property:
  • Real Numbers: If $a = b$ and $b = c$, then $a = c$.
  • Segment Lengths: If $AB = CD$ and $CD = EF$, then $AB = EF$.
  • Angle Measures: If $m\angle A = m\angle B$ and $m\angle B = m\angle C$, then $m\angle A = m\angle C$.

• Theorems of Congruence:

  • Theorem: A statement that can be proven. Once proven, it can be used as a reason in other proofs.
  • Theorem 2.1 (Properties of Segment Congruence):
  • Reflexive: For any segment $AB$, $\overline{AB} \cong \overline{AB}$.
  • Symmetric: If $\overline{AB} \cong \overline{CD}$, then $\overline{CD} \cong \overline{AB}$.
  • Transitive: If $\overline{AB} \cong \overline{CD}$ and $\overline{CD} \cong \overline{EF}$, then $\overline{AB} \cong \overline{EF}$.
  • Theorem 2.2 (Properties of Angle Congruence):
  • Reflexive: For any angle $A$, $\angle A \cong \angle A$.
  • Symmetric: If $\angle A \cong \angle B$, then $\angle B \cong \angle A$.
  • Transitive: If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$.

• Naming Properties Illustrating Statements:

  • Statement: If $\angle T \cong \angle V$ and $\angle V \cong \angle R$, then $\angle T \cong \angle R$. $\rightarrow$ Transitive Property of Angle Congruence.
  • Statement: If $\overline{JL} \cong \overline{YZ}$, then $\overline{YZ} \cong \overline{JL}$. $\rightarrow$ Symmetric Property of Segment Congruence.

Geometric Two-Column Proofs

• Structure of Two-Column Proofs:

  • A proof is a logical argument that uses deductive reasoning to show that a statement is true.
  • A two-column proof has numbered statements and corresponding reasons in logical order.
  • Left-hand column: Given information or statement derived from previous steps.
  • Right-hand column: Explanation/justification (definitions, properties, postulates, or theorems) for the statement.

• Proof 1: Symmetric Property of Segment Congruence:

  • Given: $\overline{LM} \cong \overline{NP}$
  • Prove: $\overline{NP} \cong \overline{LM}$
  • Proof Steps:
  1. $\overline{LM} \cong \overline{NP}$ | Reason: Given
  2. $LM = NP$ | Reason: Definition of congruent segments
  3. $NP = LM$ | Reason: Symmetric Property of Equality
  4. $\overline{NP} \cong \overline{LM}$ | Reason: Definition of congruent segments

• Proof 2: Midpoint Relations:

  • Given: $M$ is the midpoint of $\overline{AB}$
  • Prove: $AB = 2AM$ and $AM = \frac{1}{2}AB$
  • Proof Steps:
  1. $M$ is the midpoint of $\overline{AB}$ | Reason: Given
  2. $\overline{AM} \cong \overline{MB}$ | Reason: Definition of midpoint
  3. $AM = MB$ | Reason: Definition of congruent segments
  4. $AM + MB = AB$ | Reason: Segment Addition Postulate
  5. $AM + AM = AB$ | Reason: Substitution Property of Equality
  6. $2AM = AB$ | Reason: Combine Like Terms / Distributive Property
  7. $AM = \frac{1}{2}AB$ | Reason: Division Property of Equality

• Proof 3: Perpendicular Rays from Supplementary Congruent Angles:

  • Given: $\angle JKL \cong \angle MKL$
  • Prove: $\overleftrightarrow{JM} \perp \overleftrightarrow{LN}$
  • Proof Steps:
  1. $\angle JKL \cong \angle MKL$ | Reason: Given
  2. $m\angle JKL = m\angle MKL$ | Reason: Definition of Congruence
  3. $m\angle JKL + m\angle MKL = 180^\circ$ | Reason: Linear Pairs are supplementary
  4. $m\angle JKL + m\angle JKL = 180^\circ$ | Reason: Substitution Property of Equality
  5. $2(m\angle JKL) = 180^\circ$ | Reason: Combine Like Terms
  6. $m\angle JKL = 90^\circ$ | Reason: Division Property of Equality
  7. $\overleftrightarrow{JM} \perp \overleftrightarrow{LN}$ | Reason: Definition of Perpendicular

• Proof 4: Spotlight Beam Reflection:

  • Given: $m\angle 1 = m\angle 3$
  • Prove: $m\angle DBA = m\angle EBC$
  • Proof Steps:
  1. $m\angle 1 = m\angle 3$ | Reason: Given
  2. $m\angle DBA = m\angle 3 + m\angle 2$ | Reason: Angle Addition Postulate / Substitution
  3. $m\angle 1 + m\angle 2 = m\angle EBC$ | Reason: Angle Addition Postulate
  4. $m\angle DBA = m\angle EBC$ | Reason: Transitive Property of Equality / Substitution

• Proof 5: Complementary Angles in a Right Angle:

  • Given: $\angle LJC$ is a right angle
  • Prove: $\angle 1$ and $\angle 2$ are complementary
  • Proof Steps:
  1. $\angle LJC$ is a right angle | Reason: Given
  2. $m\angle LJC = 90^\circ$ | Reason: Definition of Right Angle
  3. $m\angle 1 + m\angle 2 = m\angle LJC$ | Reason: Angle Addition Postulate
  4. $m\angle 1 + m\angle 2 = 90^\circ$ | Reason: Substitution Property of Equality
  5. $\angle 1$ and $\angle 2$ are complementary | Reason: Definition of Complementary Angles

B.E.S.T. EOC Sample Assessment Items

• Sample Item 1 (Biconditional Logic):

  • Given Biconditional: "Two angles are complementary if and only if the sum of their measures is $90$ degrees."
  • True Statements:
  • If two angles are complementary, then the sum of their measures is $90$ degrees. (True - Conditional)
  • If the sum of two angles is $90$ degrees, then they are complementary. (True - Converse)
  • Two angles that add up to $90$ degrees are always complementary. (True)

• Sample Item 2 (Statement Relationships):

  • Statement I: If two angles are vertical angles, then they are congruent.
  • Statement II: If two angles are not congruent, then they are not vertical angles. $\rightarrow$ Contrapositive of Statement I.
  • Statement III: If two angles are not vertical angles, then they are not congruent. $\rightarrow$ Inverse of Statement I.

• Sample Item 3 (Counterexample Diagram):

  • Disprove statement: "If two angles are complementary, then they are adjacent."
  • Counterexample Diagram: Draw two separate, non-adjacent angles (such as an angle measuring $30^\circ$ and a separate angle measuring $60^\circ$) that sum to $90^\circ$.

• Sample Item 4 (Squares and Regular Polygons):

  • Given Conditional: "If a figure is a square, then the figure is a regular polygon."
  • Analysis:
  • The conditional statement is True.
  • The converse statement ("If a figure is a regular polygon, then it is a square") is False.
  • A biconditional statement cannot be written.
  • Supporting Reason: A biconditional statement cannot be written because a regular pentagon is a regular polygon, but it is not a square.