Geometry Basics: Points, Lines, and Planes

Geometry

Points, Lines, and Planes

• A point is a location in space, represented by a set of coordinates (x, y, z).
• A line is a set of points extending infinitely in two directions.
• A plane is a flat surface that extends infinitely in all directions.

Angles and Measurements

• An angle is formed by two rays sharing a common endpoint (vertex).
• Angles can be classified as:
  • Acute: less than 90°
  • Right: exactly 90°
  • Obtuse: greater than 90° but less than 180°
  • Straight: exactly 180°
• Measurements:
  • Perimeter: distance around a shape
  • Area: size of a shape's interior
  • Volume: amount of space inside a 3D shape

Properties of Shapes

• Properties of 2D Shapes:
  • Congruent: same size and shape
  • Similar: same shape but different size
  • Regular: all sides and angles equal
• Properties of 3D Shapes:
  • Faces: flat surfaces
  • Edges: lines where faces meet
  • Vertices: corners where edges meet

Theorems and Postulates

• Theorems: proven statements about geometric relationships
• Postulates: assumed statements about geometric relationships
• Examples:
  • Pythagorean Theorem: a^2 + b^2 = c^2 (right triangles)
  • Parallel Postulate: through a point not on a line, there is exactly one line parallel to the original line

Geometry

Fundamentals

• A point has a set of coordinates (x, y, z) and represents a location in space.
• A line is a set of points extending infinitely in two directions.
• A plane is a flat surface that extends infinitely in all directions.

Angles and Measurements

• An angle is formed by two rays sharing a common endpoint (vertex).
• Angles can be classified as:
  • Acute: less than 90°
  • Right: exactly 90°
  • Obtuse: greater than 90° but less than 180°
  • Straight: exactly 180°
• Measurements in geometry include:
  • Perimeter: the distance around a shape
  • Area: the size of a shape's interior
  • Volume: the amount of space inside a 3D shape

Properties of Shapes

2D Shapes

• Congruent shapes are the same size and shape.
• Similar shapes have the same shape but different sizes.
• Regular shapes have all sides and angles equal.

3D Shapes

• Faces are the flat surfaces of a 3D shape.
• Edges are the lines where faces meet.
• Vertices are the corners where edges meet.

Theorems and Postulates

• Theorems are proven statements about geometric relationships.
• Postulates are assumed statements about geometric relationships.
• Examples of theorems and postulates include:
  • Pythagorean Theorem: a^2 + b^2 = c^2 for right triangles
  • Parallel Postulate: through a point not on a line, there is exactly one line parallel to the original line