Geometry, Triangles, and 3D Shape Properties Notes

1. Assessment Overview

2. Key Geometric Theorems

Quadrilateral Angle Sum Theorem:
- The sum of all four interior angles inside a quadrilateral equals $360^\circ$ .
- Equation: $a + b + c + d = 360^\circ$
Exterior Angle Theorem of a Triangle:
- An exterior angle of a triangle equals the sum of the two opposite interior angles.
- Equation: $a + b = c$ (where $a$ and $b$ are opposite interior angles and $c$ is the exterior angle)

3. Classification of Triangles

By Angle Measures:
- Acute Triangles: All interior angles are acute.
- Right Triangles: Exactly one right angle ( $90^\circ$ ) and two acute angles.
- Obtuse Triangles: Exactly one obtuse angle and two acute angles.
By Side Lengths:
- Isosceles Triangles: Have two equal side lengths.
- Scalene Triangles: All sides have different side lengths.

4. Components and Representations of 3D Shapes

3D Shape Components:
- Vertices (Points): Point intersections on a geometric shape (Singular: Vertex, Plural: Vertices).
- Edges (Sides): Straight line segments connecting vertices.
- Faces (Planes): Flat planar surfaces constituting a 3D figure.
Nets:
- A $2\text{D}$ flat pattern that can be folded to make a $3\text{D}$ solid.
- Examples:
  - Cube: Folded from a net of $6$ square faces.
  - Square Pyramid: Folded from a net with a square base and $4$ triangular faces.

5. Position and Scale

Position (Directions):
- Cardinal directions: North, East, South, West.
- Intercardinal directions: NorthEast, SouthEast, SouthWest, NorthWest.
Scale:
- Used to represent real-life dimensions from drawing measurements.
- Example: If $1\,\text{cm}$ in drawing represents $10\,\text{km}$ in real life:
  - $2\,\text{cm} = 20\,\text{km}$
  - $3.5\,\text{cm} = 35\,\text{km}$

6. Terminology and Action Items

Cultural / Contextual Terms:
- Kōkiri Whakamua
- Kāo - No
- Muri
Workflow Items:
- Pātai - Ask
- Hui - Meet
- Mahi - Work