Geometry Basics and Properties

Study Notes

Definition of Geometry

Study of shapes, sizes, and properties of space.
Branch of mathematics concerned with spatial relationships.

Basic Concepts

Points: Exact location in space, no size.
Lines: Straight path extending in both directions with no endpoints.
Line Segments: Part of a line with two endpoints.
Rays: Part of a line that starts at a point and extends infinitely in one direction.

Types of Angles

Acute Angle: Less than 90 degrees.
Right Angle: Exactly 90 degrees.
Obtuse Angle: More than 90 degrees but less than 180 degrees.
Straight Angle: Exactly 180 degrees.

Triangles

Types by Sides:
- Equilateral: All sides equal.
- Isosceles: Two sides equal.
- Scalene: All sides different.
Types by Angles:
- Acute: All angles acute.
- Right: One angle is a right angle.
- Obtuse: One angle is obtuse.

Triangle Properties

Sum of Angles: Interior angles sum to 180 degrees.
Pythagorean Theorem: In a right triangle, a² + b² = c² (where c is the hypotenuse).

Polygons

Definition: Closed figure with straight sides.
Types:
- Quadrilateral: Four sides (e.g., square, rectangle, trapezoid).
- Pentagon: Five sides.
- Hexagon: Six sides.
Interior Angle Sum: (n-2) × 180 degrees, where n = number of sides.

Circles

Radius: Distance from center to any point on the circle.
Diameter: Distance across the circle through the center (2 × radius).
Circumference: The distance around the circle, given by C = πd or C = 2πr.
Area: The amount of space inside the circle, given by A = πr².

3D Shapes

Prisms: Two parallel bases connected by rectangular faces.
Pyramids: Base is a polygon, triangular faces converge at a point (apex).
Cylinders: Circular bases connected by a curved surface.
Spheres: Round, three-dimensional shape, all points equidistant from the center.

Theorems and Formulas

Congruence: Two shapes are congruent if they are the same shape and size.
Similarity: Two shapes are similar if they have the same shape but different sizes.
Area & Volume Formulas:
- Rectangle: Area = length × width.
- Triangle: Area = 0.5 × base × height.
- Circle: Area = πr².
- Cylinder: Volume = πr²h.
- Sphere: Volume = (4/3)πr³.

Coordinate Geometry

Coordinates: System for defining points in space using ordered pairs (x, y).
Slope: Measure of steepness of a line, calculated as (y2 - y1) / (x2 - x1).
Distance Formula: Distance between two points (x₁, y₁) and (x₂, y₂) is d = √((x₂ - x₁)² + (y₂ - y₁)²).

Definition of Geometry

Concerned with spatial relationships, shapes, sizes, and properties of space.

Basic Concepts

Points are exact locations without size.
Lines extend infinitely in both directions, no endpoints.
Line segments have two endpoints.
Rays have one endpoint and extend infinitely in one direction.

Types of Angles

Acute: Less than 90 degrees.
Right: Exactly 90 degrees.
Obtuse: More than 90 degrees but less than 180 degrees.
Straight: Exactly 180 degrees.

Triangles

By Sides:
- Equilateral: All sides are equal.
- Isosceles: Two sides are equal.
- Scalene: All sides are different.
By Angles:
- Acute: All angles are acute.
- Right: One angle is a right angle.
- Obtuse: One angle is obtuse.

Triangle Properties

The sum of interior angles is always 180 degrees.
Pythagorean Theorem: In a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b): a² + b² = c².

Polygons

Closed figures with straight sides.
Types:
- Quadrilateral: Four sides (e.g., square, rectangle, trapezoid).
- Pentagon: Five sides.
- Hexagon: Six sides.
Interior Angle Sum Formula: (n-2) × 180 degrees, where n is the number of sides.

Circles

Radius: The distance from the center to any point on the circle.
Diameter: The distance across the circle through the center, twice the radius.
Circumference: The distance around the circle, calculated as C = πd or C = 2πr.
Area: The space inside the circle, calculated as A = πr².

3D Shapes

Prisms: Two parallel bases connected by rectangular faces.
Pyramids: Base is a polygon, triangular faces meet at a point (apex).
Cylinders: Circular bases connected by a curved surface.
Spheres: Round, 3D shapes, all points equidistant from the center.

Theorems and Formulas

Congruence: Two shapes are congruent if they have the same shape and size.
Similarity: Two shapes are similar if they have the same shape but different sizes.
Area & Volume Formulas:
- Rectangle: Area = length × width.
- Triangle: Area = 0.5 × base × height.
- Circle: Area = πr².
- Cylinder: Volume = πr²h.
- Sphere: Volume = (4/3)πr³.

Coordinate Geometry

Coordinates: System of ordered pairs (x, y) to define points in space.
Slope: Measure of a line's steepness, (y2 - y1) / (x2 - x1).
Distance Formula: Distance between points (x₁, y₁) and (x₂, y₂): d = √((x₂ - x₁)² + (y₂ - y₁)²).

Description

Explore the fundamental concepts of geometry, including points, lines, and angles. This quiz covers various types of triangles and their properties, providing a comprehensive introduction to spatial relationships in mathematics.