Mathematics

Lines, Angles, and Shapes 1. Lines and Angles Basic Definitions Parallel Lines: Lines in a plane that never meet, no matter how far they extend. Perpendicular Lines: Lines that intersect at a right angle ($90^\circ$). Types of Angles: Acute: $0^\circ < \theta < 90^\circ$ Right: $\theta = 90^\circ$ Obtuse: $90^\circ < \theta < 180^\circ$ Reflex: $180^\circ < \theta < 360^\circ$ Angle Relationships Angles on a straight line: Sum to $180^\circ$. Angles around a point: Sum to $360^\circ$. Vertically opposite angles: Are equal. Parallel Line Angles: Corresponding Angles: Equal (F-shape). Alternate Angles: Equal (Z-shape). Co-interior Angles: Sum to $180^\circ$ (C-shape). ...

Making Sense of Algebra 1. Using Letters for Unknowns In algebra, letters (variables) are used to represent numbers that are either unknown or can change. Expression: A mathematical phrase without an equals sign (e.g., $3x + 5$). Equation: A statement that two expressions are equal (e.g., $3x + 5 = 11$). Formula: A rule showing the relationship between different variables (e.g., $A = lw$). 2. Substitution Substitution is the process of replacing variables with specific numerical values to evaluate an expression. ...

Review of Number Concepts 1. Types of Numbers Classifications Natural Numbers ($\mathbb{N}$): Counting numbers starting from 1 (1, 2, 3, …). Integers ($\mathbb{Z}$): Whole numbers, including negatives and zero (…, -2, -1, 0, 1, 2, …). Prime Numbers: Natural numbers greater than 1 with exactly two factors: 1 and themselves (2, 3, 5, 7, 11, …). Note: 1 is not prime; 2 is the only even prime. Square Numbers: Result of multiplying an integer by itself ($1, 4, 9, 16, 25, \dots$). Cube Numbers: Result of multiplying an integer by itself twice ($1, 8, 27, 64, 125, \dots$). Rational Numbers ($\mathbb{Q}$): Numbers that can be written as a fraction $\frac{a}{b}$ where $a, b$ are integers and $b \neq 0$. Includes terminating and recurring decimals. Irrational Numbers: Numbers that cannot be written as simple fractions (e.g., $\pi, \sqrt{2}, \sqrt{3}$). Their decimal expansions are non-terminating and non-recurring. Reciprocals: The reciprocal of a number $x$ is $\frac{1}{x}$. The product of a number and its reciprocal is always 1. ...