Geometry Basics

Study Notes

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)

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

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)

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

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 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°

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)

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

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