Geometry Units 2 and 3: Logic, Proofs, and Parallel lines

Inductive Reasoning and Conjectures

• Inductive Reasoning: The process of making a conclusion based on specific observations and identified patterns.

• Conjecture: A concluding statement reached specifically through the use of inductive reasoning.

• Sequence Examples and Conjectures:

  • Sequence: $38, 31, 24, 17, 10, 3, -4, -11, -18$

    • Conjecture: To generate the next term, subtract $7$ from the previous term.

  • Sequence: $2, 5, 11, 23, 41, 95, 191, 383, 767$

    • Conjecture: To generate the next term, double the previous term and add $1$.

  • Sequence: $1, 4, 9, 16, 25, 36, 49, 64, 81$

    • Conjecture: Each term is a perfect square. To find the "$n^{th}$ term", calculate $n^2$.

  • Sequence: $A, D, G, J, M, P, S, V, Y$

    • Conjecture: To generate the next term, skip two letters after the previous term.

  • Sequence: $7:30, 7:55, 8:20, 8:45, 9:10, 9:35, 10:00, 10:25$

    • Conjecture: To generate the next term, add $25\,\text{minutes}$ to the previous term.

  • Sequence: $3, 1, 4, 1, 5, 9, 2, 6, 5, 3$

    • Conjecture: Each term is a digit of $\pi$. To find the "$n^{th}$ term", find the "$n^{th}$ digit" of $\pi$.

Logic and Compound Statements

• Statement: A sentence that is either true or false. Statements are typically represented using lowercase letters such as $p$ or $q$.

• Truth Value: The truth or falsity of a statement.

  • Example: $p:$ Supplementary angles have a sum of $180^\circ$. (Truth Value: $T$).

• Negation: A statement that has the opposite truth value of the original statement. It is shown by the symbol $\sim$.

  • Read as: "not $p$".

  • Example: $\sim p:$ Supplementary angles do not have a sum of $180^\circ$. (Truth Value: $F$).

• Compound Statement: Formed when two or more statements are joined by the words "and" or "or".

• Conjunction: A compound statement joined by the word "and".

  • Written as: $p \wedge q$.

  • Truth Conditions: A conjunction is true only when both individual statements are true ($T+T=T$). If one or both statements are false, the conjunction is false ($T+F=F, F+F=F$).

• Disjunction: A compound statement joined by the word "or".

  • Written as: $p \vee q$.

  • Truth Conditions: A disjunction is true when at least one statement is true ($T+T=T, T+F=T$). It is only false when both statements are false ($F+F=F$).

Truth Tables for Logic Operations

• Negation Table:

  • If $p$ is $T$, $\sim p$ is $F$.

  • If $p$ is $F$, $\sim p$ is $T$.

• Conjunction Table ($p \wedge q$):

  • $T \wedge T = T$

  • $T \wedge F = F$

  • $F \wedge T = F$

  • $F \wedge F = F$

• Disjunction Table ($p \vee q$):

  • $T \vee T = T$

  • $T \vee F = T$

  • $F \vee T = T$

  • $F \vee F = F$

• Constructing Truth Tables:

  1. Include columns for each statement involved ($p, q, r$, etc.).

  2. Include columns for any required negations ($\sim p, \sim q$, etc.).

  3. Include columns for the final compound statement(s).

Conditional and Biconditional Statements

• Conditional Statement: A statement that can be written in "if-then" form.

  • Symbolic Form: $p \rightarrow q$ (Read as "if $p$, then $q$" or "$p$ implies $q$").

  • Hypothesis: The phrase immediately following the word "if".

  • Conclusion: The phrase immediately following the word "then".

  • Example: If you live in Nashville ($p$), then you live in Tennessee ($q$).

• Biconditional Statement: The conjunction of a conditional and its converse.

  • Symbolic Form: $(p \rightarrow q) \wedge (q \rightarrow p)$, or simply $p \leftrightarrow q$.

  • Read as: "$p$ if and only if $q$".

  • Truth Value: A biconditional is true only when both the conditional and the converse are true.

  • Example: Two angles are supplementary if and only if the sum of their measures is $180^\circ$. (True because both directions are true).

Laws of Logic and Venn Diagrams

• Law of Syllogism: Allows for a valid conclusion when the conclusion of one statement is the hypothesis of the next.

  • Symbolic Map: If $p \rightarrow q$ and $q \rightarrow r$, then the valid conclusion is $p \rightarrow r$.

  • Example: If it is Saturday, then Jake has a tournament. If Jake has a tournament, then he packs lunch. Conclusion: If it is Saturday, then Jake packs lunch.

• Venn Diagrams: Used to show relationships between sets of elements.

  • Nested Circles: All elements of $p$ are elements of $q$.

  • Overlapping Circles: Some elements of $p$ are elements of $q$.

  • Disjoint Circles: No relationship between $p$ and $q$ (no shared elements).

Algebraic Proof Properties of Equality

• Addition Property: If $a = b$, then $a + c = b + c$.

• Subtraction Property: If $a = b$, then $a - c = b - c$.

• Multiplication Property: If $a = b$, then $a \times c = b \times c$.

• Division Property: If $a = b$, then $\frac{a}{c} = \frac{b}{c}$.

• Distributive Property: $a(b + c) = ab + ac$.

• Substitution Property: If $a = b$, then $a$ may be replaced by $b$ in any expression or equation.

• Reflexive Property: For any real number $a$, $a = a$ (A value always equals itself).

• Symmetric Property: If $a = b$, then $b = a$.

• Transitive Property: If $a = b$ and $b = c$, then $a = c$.

Geometric Segment and Angle Proof Definitions

• Definition of Congruence: Segments or angles are congruent if and only if they have the same measure. ($AB = CD \leftrightarrow AB \cong CD$; $m\angle A = m\angle B \leftrightarrow \angle A \cong \angle B$).

• Definition of Midpoint: The midpoint of a segment divides the segment into two congruent parts. If $M$ is the midpoint of $AB$, then $AM = MB$.

• Segment Addition Postulate: If $A, B$, and $C$ are collinear and $B$ is between $A$ and $C$, then $AB + BC = AC$.

• Definition of a Right Angle: An angle measures $90^\circ$ if and only if it is a right angle.

• Definition of Complementary Angles: Two angles are complementary if and only if the sum of their measures is $90^\circ$.

• Definition of Supplementary Angles: Two angles are supplementary if and only if the sum of their measures is $180^\circ$.

• Definition of Angle Bisector: Divides an angle into two equal parts.

• Definition of Perpendicular: Perpendicular lines form right angles.

• Angle Addition Postulate: $m\angle ABD + m\angle DBC = m\angle ABC$.

Geometric Theorems and Angle Relationships

• Vertical Angles Theorem: If two angles are vertical, then they are congruent.

• Complement Theorem: If two angles form a right angle, they are complementary.

• Linear Pair (Supplement) Theorem: If two angles form a linear pair, they are supplementary.

• Congruent Complements Theorem: If two angles are complementary to the same angle, then they are congruent.

• Congruent Supplements Theorem: If two angles are supplementary to the same angle, then they are congruent.

• Parallel Lines Cut by a Transversal:

  • Corresponding Angles: Congruent.

  • Alternate Interior Angles: Congruent.

  • Alternate Exterior Angles: Congruent.

  • Consecutive (Same-Side) Interior Angles: Supplementary.

  • Consecutive (Same-Side) Exterior Angles: Supplementary.

Proving Lines are Parallel (Converses)

• Corresponding Angles Converse: If corr. angles are $\cong$, then lines are parallel.

• Alternate Interior Angles Converse: If alt. int. angles are $\cong$, then lines are parallel.

• Alternate Exterior Angles Converse: If alt. ext. angles are $\cong$, then lines are parallel.

• Consecutive Interior Angles Converse: If cons. int. angles are supplementary (sum to $180^\circ$), then lines are parallel.

Geometry Mock Exam Review Content

• Coordinate Geometry Formulas:

  • Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

  • Midpoint Formula: $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.

  • Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Parallel vs. Perpendicular Slopes:

  • Parallel lines have the same slope.

  • Perpendicular lines have slopes that are negative reciprocals (product is $-1$).

• Polygon and Triangle Rules:

  • Triangle Inequality: The sum of the two smaller sides must be greater than the largest side.

  • Polygon Interior Angle Sum: $(n - 2) \times 180^\circ$. (e.g., Hexagon sum is $720^\circ$).

  • Exterior Angle Sum: Always $360^\circ$ for any convex polygon.

• Similarity and Multi-Step Trig:

  • Geometric Mean (Altitude): $h = \sqrt{(\text{part}_1) \times (\text{part}_2)}$.

  • Special Right Triangles: $45-45-90$ (sides $x, x, x\sqrt{2}$) and $30-60-90$ (sides $x, x\sqrt{3}, 2x$).

  • Trigonometric Ratios: $\tan(\theta) = \frac{\text{opp}}{\text{adj}}$, $\sin(\theta) = \frac{\text{opp}}{\text{hyp}}$, $\cos(\theta) = \frac{\text{adj}}{\text{hyp}}$.

Questions & Discussion

• Scenario: Find the measure of angles when $m\angle 8 = 23^\circ$ and lines are parallel.

  • Response: $m\angle 1 = 157^\circ$ (Consecutive exterior to $\angle 8$), $m\angle 2 = 23^\circ$ (Alternate exterior to $\angle 8$), $m\angle 5 = 157^\circ$ (Linear pair to $\angle 8$).

• Scenario: Given $X$ is the midpoint of $WY$ and $WX = XZ$, prove $XY \cong XZ$.

  • Step 1: Identify $X$ is midpoint of $WY$ (Given).

  • Step 2: $WX = XY$ (Def. of Midpoint).

  • Step 3: $WX = XZ$ (Given).

  • Step 4: $XY = XZ$ (Substitution Property).

  • Step 5: $XY \cong XZ$ (Def. of Congruence).