Math 1+ Semester 1 Comprehensive Study Guide

Chapter 1: Solving Linear Equations

• 1.1 Solving Simple Equations

  • Goal: The primary objective is to isolate the variable using inverse operations.
  • Inverse Operations: Operations that undo each other.
    • Addition $↔$ Subtraction
    • Multiplication $↔$ Division
  • Steps: Undo operations in the reverse order of PEMDAS (Order of Operations).
  • Example: Solve $3x + 5 = 14$
    • Subtract 5: $3x = 9$
    • Divide by 3: $x = 3$

• 1.2 Solving Multi-Step Equations

  • Standard Procedure:
    1. Simplify each side: Distribute coefficients and combine like terms.
    2. Move Variables: Use addition or subtraction to collect variable terms on one side of the equation.
    3. Move Constants: Use addition or subtraction to collect constant terms on the other side.
    4. Isolate Variable: Use multiplication or division.
    5. Verification: Always check your solution by substituting it back into the original equation.
  • Example: Solve $2(x + 3) - 5 = 11$
    • Distribute: $2x + 6 - 5 = 11$
    • Combine Like Terms: $2x + 1 = 11$
    • Subtract 1: $2x = 10$
    • Divide by 2: $x = 5$

• 1.3 Solving Equations with Variables on Both Sides

  • Steps:
    1. Simplify both sides of the equation.
    2. Move variables to one side by collecting like terms.
    3. Move constants to the opposite side.
    4. Solve for the isolated variable.
  • Special Cases:
    • Identity: An equation that is always true for any value, resulting in infinite solutions (e.g., $2x + 3 = 2x + 3$).
    • Contradiction: An equation that is never true, resulting in no solution (e.g., $x + 2 = x + 5$→$2 = 5$).

• 1.4 Solving Absolute Value Equations

  • Fundamental Rule: The equation $|x| = a$ has two solutions if $a \ge 0$
    • $x = a$ OR $x = -a$
  • Steps:
    1. Isolate the absolute value expression algebraically.
    2. Set up two separate equations (the positive case and the negative case).
    3. Solve both resulting equations.
    4. Check both solutions for validity.
  • Example: Solve $|2x - 1| = 5$
    • Case 1: $2x - 1 = 5 \rightarrow 2x = 6 \rightarrow x = 3$
    • Case 2: $2x - 1 = -5 \rightarrow 2x = -4 \rightarrow x = -2$
    • Solutions: $x = 3$ or $x = -2$
  • Condition for No Solution: If isolating the absolute value results in $|x| = \text{negative number}$, there is no solution because absolute value represents distance and cannot be negative.

• 1.5 Rewriting Equations and Formulas

  • Goal: Rearranging a formula to solve for a specific variable of interest.
  • Method: Apply the same algebraic inverse operations used in solving standard equations, treating all other variables as if they were constants.
  • Common Geometric and Physical Formulas:
    • Distance: $d = rt$; solve for $r$: $r = \frac{d}{t}$
    • Area of a Trapezoid: $A = \frac{1}{2}h(b_1 + b_2)$; solve for $h$: $h = \frac{2A}{b_1 + b_2}$
    • Temperature conversion: $F = (\frac{9}{5})C + 32$; solve for $C$: $C = (\frac{5}{9})(F - 32)$

Chapter 2: Solving Linear Inequalities

• 2.1 Writing and Graphing Inequalities

  • Inequality Symbols and Graphing Notation:
    • $<$ (less than): Open circle (\u25cb)
    • $>$ (greater than): Open circle (\u25cb)
    • $\le$ (less than or equal to): Closed circle (\u25cf)
    • $\ge$ (greater than or equal to): Closed circle (\u25cf)
  • Graphing on a Number Line:
    • Use an open circle for strictly less than or greater than.
    • Use a closed circle when the value is included ($\le$ or $\ge$).
    • Shade the number line in the direction that contains the solutions.
  • Key Verbal Phrases:
    • "at most" $\rightarrow \le$
    • "at least" $\rightarrow \ge$
    • "no more than" $\rightarrow \le$
    • "no less than" $\rightarrow \ge$

• 2.2 Solving Inequalities Using Addition or Subtraction

  • Principle: Adding or subtracting the same quantity from both sides does not alter the direction of the inequality symbol.
  • Examples:
    • $x + 3 < 7 \rightarrow x < 4$
    • $x - 5 \ge 2 \rightarrow x \ge 7$

• 2.3 Solving Inequalities Using Multiplication or Division

  • The Critical Rule: When multiplying or dividing both sides of an inequality by a NEGATIVE number, you must FLIP the direction of the inequality symbol.
  • Examples:
    • $3x < 12 \rightarrow x < 4$ (divided by positive 3, no flip).
    • $-2x < 10 \rightarrow x > -5$ (divided by negative 2, FLIP).
    • $\frac{x}{-3} \ge 4 \rightarrow x \le -12$ (multiplied by negative 3, FLIP).

• 2.4 Solving Multi-Step Inequalities

  • Procedure: Mirror the steps for equations (Simplify, Collect Variables, Collect Constants, Isolate) while meticulously applying the negative multiplication/division flip rule.
  • Example: Solve $3(x - 2) \le 2x + 5$
    • Distribute: $3x - 6 \le 2x + 5$
    • Subtract 2x: $x - 6 \le 5$
    • Add 6: $x \le 11$

• 2.5 Solving Compound Inequalities

  • AND (Conjunction): Represents an intersection where both conditions must be satisfied simultaneously.
    • Forms: $a < x < b$ or $a \le x \le b$
    • Graph: A segment connecting two values.
    • Example: $-3 < x \le 5$ indicates $x$ is between $-3$ and $5$, including $5$ but not $-3$.
    • Solving: Solve both parts and identify the overlap/intersection.
  • OR (Disjunction): Represents a union where at least one condition must be true.
    • Form: $x < a \text{ OR } x > b$
    • Graph: Two separate rays extending in opposite directions.
    • Example: $x < -2 \text{ OR } x \ge 4$
    • Solving: Solve both separate inequalities and combine the solution sets into a union.

• 2.6 Solving Absolute Value Inequalities

  • Less Than Case ($|x| < a$): Creates an AND compound inequality.
    • $|x| < a \rightarrow -a < x < a$
    • Example: $|x - 3| < 5 \rightarrow -5 < x - 3 < 5 \rightarrow -2 < x < 8$
  • Greater Than Case ($|x| > a$): Creates an OR compound inequality.
    • $|x| > a \rightarrow x < -a \text{ OR } x > a$
    • Example: $|x + 2| > 4 \rightarrow x + 2 < -4 \text{ OR } x + 2 > 4 \rightarrow x < -6 \text{ OR } x > 2$

Chapter 3: Graphing Linear Functions

• 3.1 Functions

  • Definition: A relation where every input ($x$) is paired with EXACTLY ONE output ($y$).
  • Domain: The complete set of all possible input values ($x$-values).
  • Range: The complete set of all possible output values ($y$-values).
  • Tests for Functions:
    • Vertical Line Test: If any vertical line intersects a graph in more than one point, the relation is NOT a function.
    • Mapping Diagram: Check ensuring each input originates only one mapping arrow to an output.
  • Types of Functions:
    • Discrete: Represented as distinct, unconnected points.
    • Continuous: Represented as a connected line or curve.

• 3.2 Linear Functions

  • Definition: A function that possesses a constant rate of change, resulting in a straight-line graph.
  • Equation Form: $y = mx + b$
  • Rate of Change (Slope): Defined as $\frac{\text{change in } y}{\text{change in } x}$.
    • Positive Slope: The line rises from left to right.
    • Negative Slope: The line falls from left to right.
    • Zero Slope: A horizontal line.
    • Undefined Slope: A vertical line (Note: Vertical lines are NOT functions).

• 3.3 Function Notation

  • Notation: $f(x)$ is read as "function of $x$" or refers to the value of "$y$".
    • $f(x) = 2x + 3$ is equivalent to $y = 2x + 3$
    • $f(2)$ represents the specific output value when the input $x = 2$
  • Evaluating Functions: Substitute the given numerical value for every instance of $x$ in the expression.
    • Example: If $f(x) = 3x - 1$, then $f(4) = 3(4) - 1 = 11$
  • Operational Laws:
    • Addition: $(f + g)(x) = f(x) + g(x)$
    • Subtraction: $(f - g)(x) = f(x) - g(x)$

• 3.4 Graphing Linear Equations in Standard Form

  • Standard Form: $Ax + By = C$, where $A, B, C$ are integers and $A \ge 0$
  • Method of Intercepts:
    1. x-intercept: Set $y = 0$ and solve for $x$. Point is $(x, 0)$.
    2. y-intercept: Set $x = 0$ and solve for $y$. Point is $(0, y)$.
    3. Plot both intercepts and connect them with a straight line.
  • Example: $2x + 3y = 12$
    • $x$-int: $2x = 12 \rightarrow x = 6 \rightarrow (6, 0)$
    • $y$-int: $3y = 12 \rightarrow y = 4 \rightarrow (0, 4)$

• 3.5 Graphing Linear Equations in Slope-Intercept Form

  • Form: $y = mx + b$
    • $m = \text{slope}$ (rise over run).
    • $b = y\text{-intercept}$ (point where line crosses the vertical axis).
  • Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$
  • Graphing Procedure:
    1. Initial point: Plot the $y$-intercept at $(0, b)$.
    2. Next point: Use the slope (rise/run) to navigate from the $y$-intercept to a second point.
    3. Draw the line extending through both points.
  • Example: $y = 2x - 3$
    • $y$-int: $(0, -3)$
    • Slope: $2 = \frac{2}{1}$ (rise 2, run 1).
    • Second point calculated: $(-3 + 2, 0 + 1) = (-1, 1)$.

Chapter 4: Writing Linear Functions

• 4.1 Writing Equations in Slope-Intercept Form

  • Scenario 1: Given Slope ($m$) and y-intercept ($b$)
    • Plug directly into $y = mx + b$.
    • Example: $m = 3, b = -2 \rightarrow y = 3x - 2$
  • Scenario 2: Given Slope ($m$) and one point ($x_1, y_1$)
    1. Substitute $m, x_1,$ and $y_1$ into $y = mx + b$.
    2. Solve for the unknown value $b$.
    3. State the final equation using both $m$ and $b$.
  • Scenario 3: Given Two Points ($x_1, y_1$) and ($x_2, y_2$)
    1. Calculate slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
    2. Use the calculated slope and one of the points to determine $b$.
    3. Construct the equation.

• 4.2 Writing Equations in Point-Slope Form

  • Form: $y - y_1 = m(x - x_1)$
  • Application: Best used when the slope and any single point $(x_1, y_1)$ are known.
  • Steps:
    1. Determine $m$ and identify the point $(x_1, y_1)$.
    2. Substitute values into the formula.
    3. Optional: Isolate $y$ to convert to slope-intercept form.
  • Example: Slope = 2, Point (3, 5)
    • $y - 5 = 2(x - 3) \rightarrow y - 5 = 2x - 6 \rightarrow y = 2x - 1$

• 4.3 Writing Equations of Parallel and Perpendicular Lines

  • Parallel Lines:
    • Have identical slopes ($m_1 = m_2$).
    • Are always equidistance and never intersect.
    • Example: $y = 2x + 1$ and $y = 2x - 5$
  • Perpendicular Lines:
    • Slopes are negative reciprocals ($m_1 \times m_2 = -1$).
    • They intersect at a exact $90^\circ$ angle.
    • Example: $y = 2x + 1$ and $y = -\frac{1}{2}x + 3$
  • Equations Steps:
    1. Parallel: Keep the source slope, calculate a new intercept ($b$) using the target point.
    2. Perpendicular: Invert and negate the source slope, then solve for the target intercept ($b$).
  • Special Horizontal/Vertical Cases:
    • Horizontal lines ($y = k$): Slope is 0.
    • Vertical lines ($x = h$): Slope is undefined.
    • Horizontal and vertical lines always meet perpendicularly.

• 4.4 Scatter Plots and Lines of Fit

  • Scatter Plot: A graphical display mapping the relationship between two numerical variables.
  • Correlation Types:
    • Positive: $y$ increases as $x$ increases (upward trend).
    • Negative: $y$ decreases as $x$ increases (downward trend).
    • None: The points exhibit no discernible linear pattern.
  • Line of Fit: A modeled line used to represent data trends.
    • Must pass close to the majority of points.
    • Should have roughly equal distribution of points above and below the line.
  • Prediction Terminology:
    • Interpolation: Predicting values within the existing range of data points.
    • Extrapolation: Predicting values outside the established data range (considered less reliable).

• 4.5 Analyzing Lines of Fit

  • Residuals: The vertical gap between an observed data point and the modeled line of fit.
    • $\text{Residual} = \text{actual } y\text{-value} - \text{predicted } y\text{-value}$
    • Positive: Point resides above the line.
    • Negative: Point resides below the line.
  • Residual Plot: A diagnostic tool plotting residuals on a graph.
    • Random scatter: Indicates the linear model is a good fit.
    • Specific Pattern: Suggests the linear model is inappropriate for the data.
  • Correlation Coefficient ($r$): Defines the strength and direction of the linear relationship.
    • Range: $-1 \le r \le 1$
    • $r \approx 1$: Strong positive correlation.
    • $r \approx -1$: Strong negative correlation.
    • $r \approx 0$: Minimal or no linear correlation.

• 4.6 Arithmetic Sequences

  • Definition: A numerical sequence where the difference between successive terms is a constant value.
  • Common Difference ($d$): $d = a_n - a_{n-1}$
  • Explicit Formula: $a_n = a_1 + (n - 1)d$
    • $a_1$: The first term of the sequence.
    • $n$: The position/number of the term.
    • $d$: Common difference.
  • Example: Sequence 3, 7, 11, 15…
    • $d = 4; a_1 = 3$
    • Formula: $a_n = 3 + (n - 1)(4) = 4n - 1$
  • Recursive Formula: $a_n = a_{n-1} + d$ (must specify the initial term $a_1$).

Chapter 5: Solving Systems of Linear Equations

• 5.1 Solving Systems by Graphing

  • System of Equations: A set of two or more equations featuring identical variables.
  • Solution: An ordered pair $(x, y)$ that makes every equation in the system true simultaneously; visualized as the point of intersection.
  • Types of Solution Sets:
    1. One Solution: Slopes are different; lines cross at exactly one coordinate.
    2. No Solution: Slopes are identical but $y$-intercepts differ; lines are parallel and never meet.
    3. Infinite Solutions: Equations are equivalent; lines are identical and overlap everywhere.

• 5.2 Solving Systems by Substitution

  • Best Use: When one equation is already isolated for a variable (e.g., $y = ...$) or contains a variable with a coefficient of 1.
  • Step-by-Step:
    1. Solve one equation for a selected variable.
    2. Plug the resulting expression into the second equation for that variable.
    3. Solve the resulting single-variable equation.
    4. Substitute the found value back to find the value of the other variable.
  • Example:
    • $y = 2x + 1$
    • $3x + y = 11$
    • Substitute: $3x + (2x + 1) = 11 \rightarrow 5x + 1 = 11 \rightarrow 5x = 10 \rightarrow x = 2$
    • Back-sub: $y = 2(2) + 1 = 5$
    • Solution: $(2, 5)$

• 5.3 Solving Systems by Elimination

  • Best Use: When terms are already lined up vertically and coefficients can be easily matched as opposites.
  • Steps:
    1. Align like terms in vertical columns.
    2. Multiply equations by constants so that the coefficients of one variable are additive opposites (e.g., $5x$ and $-5x$).
    3. Add the equations to eliminate that variable.
    4. Solve for the remaining variable.
    5. Substitute found value to determine the final variable.
  • Example:
    • $2x + 3y = 8$
    • $3x - 3y = 7$
    • Add: $5x = 15 \rightarrow x = 3$
    • Back-sub: $2(3) + 3y = 8 \rightarrow 6 + 3y = 8 \rightarrow 3y = 2 \rightarrow y = \frac{2}{3}$
    • Solution: $(3, \frac{2}{3})$

• 5.4 Solving Special Systems

  • No Solution (Inconsistent): Algebraically results in a false statement like $0 = 5$.
  • Infinite Solutions (Dependent): Algebraically results in an identity like $0 = 0$ or $x = x$.
  • Standard Form Analysis ($Ax + By = C$):
    • If the ratio of $A_1/B_1 = A_2/B_2$ but involves different $C$ values, there is no solution.
    • If all ratios ($A_1/B_1$ and $C_1/C_2$) are the same, there are infinite solutions.

• 5.5 Solving Equations by Graphing

  • Technique 1 (Roots Method): Set the entire equation to equal zero. Graph the function $y = \text{expression}$. The solutions are the $x$-intercepts.
  • Technique 2 (System Method): Graph each side of the equation ($y = \text{left side}$ and $y = \text{right side}$) as separate functions. The solutions are the $x$-coordinates of the intersection points.

• 5.6 Graphing Linear Inequalities in Two Variables

  • Boundary Lines:
    • Dashed: Used for strictly $<$ or $>$ (points on line are not solutions).
    • Solid: Used for $\le$ or $\ge$ (points on line are solutions).
  • Shading:
    1. Select a test point not on the line (typically $(0,0)$).
    2. Evaluate the inequality with the test point.
    3. Shade the region containing the point if it is true; otherwise, shade the opposite side.

• 5.7 Systems of Linear Inequalities and Linear Programming

  • System Solution: The region where all individual shaded regions overlap.
  • Linear Programming Goal: To maximize or minimize a specific quantity (the objective function) within a bounded set of constraints (the feasible region).
  • Components:
    • Variables: $x$ and $y$ (the decision factors).
    • Objective Function: $C = ax + by$
    • Constraints: Inequalities defining the limits of the variables.
    • Feasible Region: The polygon area formed by the overlap of constraint inequalities.
  • Optimization Rules: The maximum and minimum values typically occur at the vertices (corner points) of the feasible region.
  • Example Optimization: Maximize $P = 3x + 2y$ given $x \ge 0, y \ge 0, 2x + y \le 10, x + 2y \le 8$.

Chapter 6: Exponential Functions and Sequences

• 6.1 Exponential Functions

  • General Form: $y = ab^x$ (where $a \ne 0, b > 0, b \ne 1$)
    • $a$: Initial amount/value (corresponding to the $y$-intercept).
    • $b$: Base (the factor of growth or decay).
    • $x$: Exponent.
  • Growth vs. Decay:
    • Growth: $b > 1$. The graph curves upward infinitely.
    • Decay: $0 < b < 1$. The graph curves downward towards the asymptote.
  • Key Attributes:
    • Domain: All real numbers.
    • Range: $y > 0$ (if $a > 0$) or $y < 0$ (if $a < 0$).
    • Asymptote: A horizontal line that the graph approaches but never touches (base case: $y = 0$).

• 6.2 Exponential Growth and Decay Models

  • Growth: $y = a(1 + r)^t$ ($1+r$ is the growth factor).
  • Decay: $y = a(1 - r)^t$ ($1-r$ is the decay factor).
  • Compound Interest: $A = P(1 + \frac{r}{n})^{nt}$
    • $P$: Principal (initial balance).
    • $r$: Annual interest rate (must be a decimal).
    • $n$: Number of compoundings per year (e.g., quarterly $n=4$, monthly $n=12$).
    • $t$: Time in years.
  • Example Calculation: $1000 principal at 5% interest compounded quarterly for 3 years.
    • $A = 1000(1 + \frac{0.05}{4})^{4 \times 3} = 1000(1.0125)^{12} \approx \$1160.75$
  • Half-Life / Doubling Time: The time required for a specific quantity to either reduce to half its size or increase to twice its original value.

• 6.3 Comparing Linear and Exponential Functions

  • Linear: Characterized by adding the same amount (constant first differences).
  • Exponential: Characterized by multiplying by the same amount (constant ratio between outputs).
  • Table Identification: Equal $x$-intervals that lead to equal ratios in $y$ identify an exponential function.

• 6.4 Solving Exponential Equations

  • Same Base Method: If $b^x = b^y$, then it implies $x = y$.
  • Steps:
    1. Rewrite both sides to have an identical numerical base.
    2. Set the resulting exponents equal to each other.
    3. Solve the remaining equation.
  • Example: Solve $4^{x+1} = 16$
    • $16 = 2^4, 4 = 2^2$
    • $(2^2)^{x+1} = 2^4 \rightarrow 2^{2x+2} = 2^4$
    • $2x + 2 = 4 \rightarrow x = 1$

• 6.5 Geometric Sequences

  • Definition: A sequence where each term is the product of the previous term and a common ratio ($r$).
  • Common Ratio Calculation: $r = \frac{a_n}{a_{n-1}}$
  • Explicit Formula: $a_n = a_1 \cdot r^{n-1}$
  • Example: 2, 6, 18, 54…
    • $r = 3; a_1 = 2$
    • $a_n = 2 \cdot 3^{n-1}$

• 6.6 Recursively Defined Sequences and Transformations

  • Recursive Rules: Define the current term based on the preceding term (e.g., Fibonacci: $a_n = a_{n-1} + a_{n-2}$).
  • Transformations of Exponential Functions ($g(x) = a \cdot b^{x-h} + k$):
    • $a$: Vertical stretch (if $|a| > 1$) or shrink (if $0 < |a| < 1$). Negative reflections over $x$-axis.
    • $h$: Horizontal shift (Right if positive, Left if negative).
    • $k$: Vertical shift; shifts the asymptote to the line $y = k$.

Chapter 7: Statistics & Data Analysis

• 7.1 Measures of Center and Variation

  • Measures of Center:
    1. Mean ($\bar{x}$): The arithmetic average (Sum divided by count). Use for symmetric data; highly sensitive to outliers.
    2. Median: The middle value of an ordered set. Use for skewed data; resistant to outliers.
    3. Mode: Most frequent element. Best for categorical data.
  • Measures of Variation:
    1. Range: Max - Min.
    2. Interquartile Range (IQR): $Q_3 - Q_1$. Represents the middle 50% of data.
    3. Standard Deviation ($\sigma$): Represents average distance from the mean.
       • Formula: $\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}$
       • Steps: 1. Find Mean. 2. Subtract Mean from each value. 3. Square differences. 4. Sum squares. 5. Divide by count. 6. Square root results.

• 7.2 Box-and-Whisker Plots

  • Five-Number Summary: Minimum, First Quartile ($Q_1$), Median ($Q_2$), Third Quartile ($Q_3$), and Maximum.
  • Outlier Calculation: Values less than $Q_1 - 1.5(IQR)$ or greater than $Q_3 + 1.5(IQR)$.

• 7.3 Shapes of Distributions

  • Symmetric: Mean $\approx$ Median $\approx$ Mode; mirror image.
  • Skewed Left: Tail is longer on the left; Mean $<$ Median.
  • Skewed Right: Tail is longer on the right; Mean $>$ Median.

• 7.4 Two-Way Tables

  • Joint Frequency: Values within the interior cells representing specific categories.
  • Marginal Frequency: Totals found along the outer row/column.
  • Relative Frequency: $\text{Frequency} / \text{Total}$.
  • Conditional Relative Frequency: Ratio comparing a cell value to the specific row or column total.

Chapter 8: Basics of Geometry

• 8.1 Undefined and Defined Terms

  • Point: Zero dimensions, no size.
  • Line: One dimension, infinite in length.
  • Plane: Two dimensions, infinite flat surface.
  • Segment: Finite part of a line with two endpoints.
  • Collinear: Points residing on the same line.
  • Coplanar: Points residing on the same plane.

• 8.2 Measuring and Constructing Segments

  • Distance AB: $|x_2 - x_1|$
  • Segment Addition Postulate: If point B lies between A and C, then $AB + BC = AC$.

• 8.3 Midpoint and Distance Formulas

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

• 8.4 Perimeter and Area

  • Triangle: Area = $\frac{1}{2}bh$
  • Rectangle: Area = $\ell w$
  • Polygons: Tri (3), Quad (4), Pent (5), Hex (6), Oct (8).

• 8.5 Measuring and Constructing Angles

  • Angle Addition Postulate: The sum of adjacent partial angles equals the total angle.
  • Types: Acute ($< 90^\circ$), Right ($90^\circ$), Obtuse ($> 90^\circ, < 180^\circ$), Straight ($180^\circ$).

• 8.6 Pairs of Angles

  • Complementary: Add to $90^\circ$.
  • Supplementary: Add to $180^\circ$.
  • Vertical Angles: Congruent angles directly opposite across an intersection.
  • Linear Pair: Adjacent supplementary angles forming a straight line.

Chapter 9: Reasoning and Proofs

• 9.1 Conditional Statements

  • Logic Structure: "If p, then q" ($p \rightarrow q$).
  • Truth Table: True except when $p$ is true and $q$ is false.
  • Related Statements:
    • Converse: $q \rightarrow p$
    • Inverse: $\sim p \rightarrow \sim q$
    • Contrapositive: $\sim q \rightarrow \sim p$ (Logically equivalent to original conditional).
  • Biconditional: "p if and only if q"; true when conditional and converse are both true.

• 9.2 Reasoning Types and Laws

  • Inductive Reasoning: Observing patterns and making a conjecture (Specific $\rightarrow$ General).
  • Deductive Reasoning: Starting with facts/rules to prove a specific case (General $\rightarrow$ Specific).
  • Law of Detachment: If $p \rightarrow q$ is true and $p$ is provided as true, therefore $q$ must be true.
  • Law of Syllogism: If $p \rightarrow q$ and $q \rightarrow r$, then $p \rightarrow r$.

• 9.3 Postulates and Diagrams

  • Key Assumptions: Through any two points is exactly one line; through three noncollinear points is exactly one plane.
  • Assumptions Avoidance: Never assume segments are congruent, angles are right, or lines are parallel from a diagram unless symbols are present.

• 9.4 Properties of Equality

  • Reflexive: $a = a$
  • Symmetric: If $a = b$, then $b = a$
  • Transitive: If $a = b$ and $b = c$, then $a = c$
  • Addition/Subtraction: Adding/subtracting same value to both sides.

Chapter 10: Parallel and Perpendicular Lines

• 10.1 Angle Pairs

  • Corresponding Angles: Same relative position.
  • Alternate Interior: Between the lines, opposite sides of the transversal.
  • Consecutive Interior: Between the lines, same side of transversal (supplementary if lines are parallel).

• 10.5 Slopes and Coordinate Geometry

  • Parallel Slopes: Equal ($m_1 = m_2$).
  • Perpendicular Slopes: Negative reciprocals ($m_1 \times m_2 = -1$).
  • Distance from Point to Line: Find the perpendicular line through the point, locate the intersection, and apply distance formula.