Math Basics: Angles, Geometry, and Algebra

1. Angles and Algebra

Angles play a crucial role in mathematics, particularly in geometry and algebra. Defined as the union of two rays with a common endpoint, angles can be measured in degrees or radians. A positive angle is measured counterclockwise from the initial side to the terminal side, while a negative angle is measured clockwise. An angle is in standard position if its vertex is at the origin and its initial side extends along the positive x-axis.

1.1. Angle Names

Angles can be named using a point on each ray and the vertex, such as angle DEF, or in symbol form ∠DEF. Greek letters are often used as variables for the measure of an angle.

1.2. Angle Measurement

The measure of an angle is the amount of rotation from the initial side to the terminal side. The most common unit of angle measurement is the degree, with a complete circular rotation containing 360 degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol °.

1.3. Graphing Angles in Standard Position

To draw angles in standard position on a coordinate plane, the initial side is drawn along the positive x-axis, and the terminal side is placed according to the fraction of a full rotation. For example, an angle of 30° is drawn by dividing a quarter rotation into thirds, while an angle of −135° is drawn by moving clockwise one full quarter and one-half of another quarter.

2. Basic Geometry

2.1. Triangles

A triangle is a polygon with three sides and three angles. The sum of the angles in a triangle is always 180 degrees. Triangles can be classified based on their sides and angles, such as equilateral, isosceles, and scalene.

2.2. Circles

A circle is a set of points in a plane at a constant distance from a central point, called the center. The distance from the center to any point on the circle is called the radius. The longest distance across the diameter of the circle is the diameter.

2.3. Perimeter and Area

The perimeter of a shape is the distance around the outer edge, while the area is the amount of space inside the shape. In a rectangle, the perimeter is the sum of the lengths of all four sides, while the area is the product of the length and width. In a circle, the perimeter is called the circumference, and the area is called the area of the circle.

3. Basic Algebra

Algebra is the branch of mathematics that deals with equations and their manipulation. It is a powerful tool for solving problems and understanding relationships between quantities.

3.1. Linear Equations

A linear equation is an equation in which the highest power of the variable is 1. It represents a straight line on the coordinate plane. For example, the equation y = 2x + 1 represents a line with a slope of 2 and a y-intercept of 1.

3.2. Quadratic Equations

A quadratic equation is an equation in which the highest power of the variable is 2. It represents a parabola on the coordinate plane. For example, the equation x^2 + 4x + 4 = 0 represents a parabola opening upwards with a vertex at (2, 0).

3.3. Solving Equations

To solve an equation, one must isolate the variable on one side of the equation. This can be done by adding or subtracting the same quantity from both sides, dividing or multiplying both sides by the same non-zero quantity, or using the distributive property.