Classifying 2D Figures

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🔵Hierarchy of Quadrilaterals

What is a Quadrilateral?

A quadrilateral is a polygon with 4 sides and 4 angles.

Quadrilateral Family Tree:

                        QUADRILATERAL
                      (4 sides, 4 angles)
                             │
            ┌────────────────┼────────────────┐
            │                │                │
        TRAPEZOID        PARALLELOGRAM     (Other: kite, etc.)
      (1 pair of    (2 pairs of parallel
       parallel           sides)
         sides)              │
                ┌────────────┼────────────┐
                │            │            │
            RECTANGLE     RHOMBUS       SQUARE?
            (4 right   (4 equal sides,   (special case!)
             angles)    not right angles)
                │            │
                └──────┬─────┘
                       │
                    SQUARE
            (4 right angles + 4 equal sides)

Quadrilateral Hierarchy:

Shape	Properties
Trapezoid	1 pair of parallel sides
Parallelogram	2 pairs of parallel sides, opposite sides equal
Rectangle	Parallelogram + 4 right angles
Rhombus	Parallelogram + 4 equal sides
Square	Rectangle + Rhombus (4 right angles + 4 equal sides)

QUADRILATERAL HIERARCHY DIAGRAM:

 ┌─────────────────┐
 │ QUADRILATERAL │
 │ (4 sides) │
 └────────┬────────┘
 │
 ┌────────────────┼────────────────┐
 │ │ │
 ▼ ▼ ▼
 ┌───────────┐ ┌───────────┐ ┌───────────┐
 │ TRAPEZOID │ │PARALLELO- │ │ KITE │
 │1 pair ∥ │ │ GRAM │ │2 pairs │
 └───────────┘ │2 pairs ∥ │ │adj = sides│
 └─────┬─────┘ └───────────┘
 │
 ┌──────────────┼──────────────┐
 │ │ │
 ▼ ▼ ▼
 ┌───────────┐ ┌───────────┐ ┌───────────┐
 │ RECTANGLE │ │ RHOMBUS │ │ SQUARE? │
 │4 right ∠s │ │4 = sides │ │(needs both│
 └─────┬─────┘ └─────┬─────┘ │ properties│
 │ │ └───────────┘
 └────────┬───────┘
 │
 ▼
 ┌───────────┐
 │ SQUARE │
 │4 right ∠s │
 │4 = sides │
 └───────────┘


VISUAL EXAMPLES OF EACH:

 TRAPEZOID PARALLELOGRAM RECTANGLE
 /‾‾‾‾‾ /‾‾‾‾‾ ┌─────┐
 / / │ │
 / / │ │
 /___________ /___________ └─────┘
 
 RHOMBUS SQUARE
 ◇ ┌───┐
 / │ │
 / │ │
 /______ └───┘

👀 See a worked example

A shape has 2 pairs of parallel sides and 4 right angles. What is it?

2 pairs of parallel sides = parallelogram
4 right angles = rectangle
Could also be square if sides equal, but rectangle is correct
Answer: Rectangle

Draw the hierarchy from widest to narrowest: square, quadrilateral, rectangle, parallelogram

Widest: Quadrilateral
Then: Parallelogram
Then: Rectangle
Narrowest: Square
Answer: Quadrilateral → Parallelogram → Rectangle → Square

A trapezoid has how many pairs of parallel sides?

Definition of trapezoid: exactly 1 pair of parallel sides
Answer: 1 pair

🤖 Vidi's Key Points

Quadrilateral = any 4-sided polygon
Trapezoid = 1 pair of parallel sides
Parallelogram = 2 pairs of parallel sides
Rectangle = parallelogram + 4 right angles
Rhombus = parallelogram + 4 equal sides
Square = rectangle + rhombus (all properties!)

🟢Classifying Triangles by Sides

What is a Triangle?

A triangle is a polygon with 3 sides and 3 angles. Sum of angles = 180°.

Classification by Sides:

Type	Description	Example
Equilateral	All 3 sides EQUAL	3 cm, 3 cm, 3 cm
Isosceles	2 sides EQUAL	5 cm, 5 cm, 3 cm
Scalene	NO sides equal	4 cm, 5 cm, 6 cm

Classification by Angles:

Type	Description	Example
Acute	All angles < 90°	60°, 60°, 60°
Right	One angle = 90°	30°, 60°, 90°
Obtuse	One angle > 90°	30°, 30°, 120°

TRIANGLES BY SIDES:

 EQUILATERAL ISOSCELES SCALENE
 / / /
 / / / 
 / / / 
 /______ /______ /______
 
 All sides = 2 sides = All sides
 3 equal marks 2 equal marks different


TRIANGLES BY ANGLES:

 ACUTE RIGHT OBTUSE
 / | /
 / | / 
 / | / 
 /______ |___ /______
 
 All angles One 90° One angle
 < 90° angle > 90°


COMBINING CLASSIFICATIONS:

 Triangle can have TWO names:
 
 ┌─────────────────────────────────────┐
 │ │
 │ Acute Equilateral = 60°,60°,60° │
 │ Right Isosceles = 90°,45°,45° │
 │ Obtuse Scalene = 120°,30°,30° │
 │ Acute Scalene = 50°,60°,70° │
 │ │
 └─────────────────────────────────────┘


MARKING CONVENTIONS:

 Equal sides = hash marks:
 
 /
 / Side with one mark = equal
 / Side with two marks = equal
 /______
 /
 /

 Right angle = square in corner:
 
 |
 | 
 |__

👀 See a worked example

A triangle has sides 6 cm, 6 cm, and 8 cm. How would you classify it by sides?