Introduction to Geometry

• Geometry is a branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs
• It is one of the oldest sciences

Euclidean Geometry

• Euclidean geometry is based on a set of axioms formulated by the Greek mathematician Euclid around 300 BC
• It deals with space and figures in it

Non-Euclidean Geometry

• Non-Euclidean geometry includes geometries that differ from Euclidean geometry in their axioms regarding the nature of parallel lines
• These geometries include hyperbolic and elliptic geometry

Basic Elements

• Point: A point is a location in space, having no dimension
• Line: A line is a straight, one-dimensional figure extending infinitely in both directions
• Plane: A plane is a flat, two-dimensional surface that extends infinitely

Angles

• An angle is formed by two rays sharing a common endpoint, called the vertex

Types of Angles

• Acute Angle: An angle measuring less than 90 degrees
• Right Angle: An angle measuring exactly 90 degrees
• Obtuse Angle: An angle measuring greater than 90 degrees but less than 180 degrees
• Straight Angle: An angle measuring exactly 180 degrees
• Reflex Angle: An angle measuring greater than 180 degrees but less than 360 degrees

Lines and Their Relationships

• Parallel Lines: Lines in a plane that do not intersect
• Perpendicular Lines: Lines that intersect at a right angle (90 degrees)
• Intersecting Lines: Lines that cross each other at a point

Triangles

• A triangle is a polygon with three edges and three vertices
• The sum of the angles in a triangle is always 180 degrees

Types of Triangles

• Equilateral Triangle: A triangle with all three sides equal in length and all three angles equal to 60 degrees
• Isosceles Triangle: A triangle with two sides of equal length and two equal angles
• Scalene Triangle: A triangle with all sides of different lengths and all angles of different measures
• Right Triangle: A triangle with one angle measuring 90 degrees

Parts of a Right Triangle

• Hypotenuse: The side opposite the right angle
• Legs: The two sides that form the right angle

Pythagorean Theorem

• In a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): a² + b² = c²

Quadrilaterals

• A quadrilateral is a polygon with four edges and four vertices
• The sum of the angles in a quadrilateral is 360 degrees

Types of Quadrilaterals

• Square: A quadrilateral with all four sides equal in length and all four angles equal to 90 degrees
• Rectangle: A quadrilateral with opposite sides equal in length and all four angles equal to 90 degrees
• Parallelogram: A quadrilateral with opposite sides parallel and equal in length
• Rhombus: A quadrilateral with all four sides equal in length
• Trapezoid: A quadrilateral with at least one pair of parallel sides

Polygons

• A polygon is closed two-dimensional shape with straight sides

Types of Polygons

• Regular Polygon: A polygon with all sides and angles equal
• Irregular Polygon: A polygon with sides and angles not all equal
• Convex Polygon: A polygon with all interior angles less than 180 degrees
• Concave Polygon: A polygon with at least one interior angle greater than 180 degrees

Circles

• A circle is a set of all points in a plane that are at a given distance from a center point

Parts of a Circle

• Radius: The distance from the center of the circle to any point on the circle
• Diameter: The distance across the circle through the center (twice the radius)
• Circumference: The distance around the circle
• Area: The amount of space inside the circle
• Chord: A line segment connecting two points on the circle
• Tangent: A line that touches the circle at only one point
• Secant: A line that intersects the circle at two points
• Arc: A portion of the circumference of the circle

Formulas for Circles

• Circumference: C = 2πr (where r is the radius)
• Area: A = πr² (where r is the radius)

Solid Geometry

• Solid geometry is the geometry of three-dimensional space
• It deals with shapes such as cubes, prisms, pyramids, spheres, and cylinders

Three-Dimensional Shapes

• Cube: A solid with six square faces
• Rectangular Prism: A solid with six rectangular faces
• Sphere: A solid consisting of all points that are at a given distance from a center point
• Cylinder: A solid with two parallel circular bases connected by a curved surface
• Cone: A solid with a circular base and a curved surface that tapers to a single point (vertex)
• Pyramid: A solid with a polygonal base and triangular faces that meet at a single point (vertex)

Volume and Surface Area

• Volume: The amount of space occupied by a three-dimensional object
• Surface Area: The total area of the surfaces of a three-dimensional object

Transformations

• Geometric transformations are operations that change the position, size, or shape of a geometric figure

Types of Transformations

• Translation: Sliding a figure along a straight line without changing its size or orientation
• Rotation: Turning a figure around a fixed point (center of rotation)
• Reflection: Creating a mirror image of a figure across a line (axis of reflection)
• Dilation: Enlarging or reducing the size of a figure by a scale factor

Congruence and Similarity

• Congruent Figures: Figures that have the same size and shape
• Similar Figures: Figures that have the same shape but different sizes
• Corresponding Parts: Sides or angles in congruent or similar figures that are in the same relative position

Coordinate Geometry

• Coordinate geometry uses a coordinate system to represent geometric figures and solve geometric problems

Coordinate Plane

• The coordinate plane is formed by two perpendicular number lines (x-axis and y-axis) that intersect at a point called the origin

Coordinates

• Each point in the coordinate plane is identified by an ordered pair of numbers (x, y), where x is the x-coordinate (abscissa) and y is the y-coordinate (ordinate)
• Distance Formula: The distance between two points (x₁, y₁) and (x₂, y₂) in the coordinate plane is given by √((x₂ - x₁)² + (y₂ - y₁)²).
• Midpoint Formula: The coordinates of the midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) are ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope of a Line

• The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by m = (y₂ - y₁) / (x₂ - x₁), where x₁ ≠ x₂
• Parallel Lines: Parallel lines have the same slope
• Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other (m₁ * m₂ = -1)

Equations of Lines

• Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept
• Point-Slope Form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line
• Standard Form: Ax + By = C, where A, B, and C are constants

Conic Sections

• Conic sections are curves formed by the intersection of a plane and a double cone

Types of Conic Sections

• Circle: A set of all points equidistant from a center point
• Ellipse: A set of all points such that the sum of the distances from two fixed points (foci) is constant
• Parabola: A set of all points equidistant from a fixed point (focus) and a fixed line (directrix)
• Hyperbola: A set of all points such that the difference of the distances from two fixed points (foci) is constant