Geometry Basics

Angles And Triangles

• Angle Properties:
  • Angles are measured in degrees (°)
  • Angles can be classified as acute (< 90°), right (90°), obtuse (> 90°), or straight (180°)
  • Sum of interior angles in a triangle is 180°
• Triangle Properties:
  • Types of triangles: equilateral, isosceles, scalene, right, oblique
  • Pythagorean theorem: a² + b² = c² (for right triangles)
  • Triangle inequality theorem: a + b > c (for all triangles)

Points And Lines

• Point Properties:
  • A point has no size, only location
  • Points are represented by coordinates (x, y) in a Cartesian plane
• Line Properties:
  • A line extends infinitely in two directions
  • Lines can be represented by equations (e.g., y = mx + b) or by two points
  • Lines can be parallel, perpendicular, or intersecting

Quadrilaterals And Polygons

• Quadrilateral Properties:
  • Types of quadrilaterals: rectangle, square, rhombus, trapezoid, kite
  • Sum of interior angles in a quadrilateral is 360°
• Polygon Properties:
  • A polygon is a closed shape with three or more sides
  • Types of polygons: triangle, quadrilateral, pentagon, hexagon, etc.
  • Sum of interior angles in a polygon is (n-2) × 180° (where n is the number of sides)

Three Dimensional Geometry

• 3D Shapes:
  • Types of 3D shapes: cube, cuboid, sphere, cylinder, cone, pyramid
  • Surface area and volume formulas for each shape
• Coordinate Geometry:
  • Points in 3D space are represented by coordinates (x, y, z)
  • Distance and midpoint formulas for 3D points

Circles And Circles Theorems

• Circle Properties:
  • A circle is a set of points equidistant from a central point (center)
  • Circle formulas: circumference (C = 2πr), area (A = πr²)
• Circle Theorems:
  • Inscribed angle theorem: ∠AOB = ½ × ∠ACB (where O is the center)
  • Alternate segment theorem: ∠AOB = ∠ACB (where O is the center)
  • Circle tangent theorem: a line is tangent to a circle if it intersects the circle at a single point

Angles And Triangles

• Angles are measured in degrees (°)
• Angles can be classified as acute (< 90°), right (90°), obtuse (> 90°), or straight (180°)
• Sum of interior angles in a triangle is 180°

Triangle Properties

• Equilateral triangles have all sides equal
• Isosceles triangles have at least two sides equal
• Scalene triangles have all sides unequal
• Right triangles have one right angle (90°)
• Oblique triangles are not right triangles
• Pythagorean theorem: a² + b² = c² (for right triangles)
• Triangle inequality theorem: a + b > c (for all triangles)

Points And Lines

• A point has no size, only location
• Points are represented by coordinates (x, y) in a Cartesian plane
• A line extends infinitely in two directions
• Lines can be represented by equations (e.g., y = mx + b) or by two points
• Lines can be parallel, perpendicular, or intersecting

Quadrilaterals And Polygons

• Rectangles are quadrilaterals with all sides equal and all angles right (90°)
• Squares are quadrilaterals with all sides equal and all angles right (90°)
• Rhombuses are quadrilaterals with all sides equal
• Trapezoids are quadrilaterals with two pairs of opposite sides unequal
• Kites are quadrilaterals with two pairs of adjacent sides equal
• Sum of interior angles in a quadrilateral is 360°
• A polygon is a closed shape with three or more sides
• Types of polygons: triangle, quadrilateral, pentagon, hexagon, etc
• Sum of interior angles in a polygon is (n-2) × 180° (where n is the number of sides)

Three Dimensional Geometry

• A cube is a 3D shape with all edges equal and all angles right (90°)
• A cuboid is a 3D shape with all edges unequal and all angles right (90°)
• A sphere is a 3D shape where every point on the surface is equidistant from the center
• A cylinder is a 3D shape with two parallel and circular bases connected by a curved surface
• A cone is a 3D shape with a circular base and a curved surface tapering to a point
• A pyramid is a 3D shape with a polygon base and triangular faces meeting at a point
• Surface area and volume formulas for each 3D shape
• Points in 3D space are represented by coordinates (x, y, z)
• Distance and midpoint formulas for 3D points

Circles And Circles Theorems

• A circle is a set of points equidistant from a central point (center)
• Circle formulas: circumference (C = 2πr), area (A = πr²)
• Inscribed angle theorem: ∠AOB = ½ × ∠ACB (where O is the center)
• Alternate segment theorem: ∠AOB = ∠ACB (where O is the center)
• Circle tangent theorem: a line is tangent to a circle if it intersects the circle at a single point