Properties of Squares
A square is both a rectangle AND a rhombus. It inherits every property of both — making it the most special parallelogram.
Introduction
Square: equal sides, right angles, ≅ ⊥ diagonals
Past Knowledge
Rectangles (9.2.3). Rhombi (9.2.4). 45-45-90 triangles (8.2.1).
Today's Goal
Combine rectangle and rhombus properties to understand squares.
Future Success
Trapezoids & kites (9.3), area & perimeter, coordinate proofs.
Key Concepts
Square = Rectangle ∩ Rhombus
From Rectangle:
- Four right angles
- Congruent diagonals
From Rhombus:
- Four congruent sides
- Perpendicular diagonals
- Diagonals bisect angles
Combined: Diagonals are congruent, perpendicular, bisect each other, and bisect the vertex angles (creating 45° angles).
Formulas
- Diagonal: (from 45-45-90)
- Area: or
- Perimeter:
Quadrilateral Hierarchy
Every square is a rectangle, a rhombus, and a parallelogram — but not vice versa.
Worked Examples
Diagonal of a Square
A square has side 8. Find the diagonal.
Side from Diagonal
A square has diagonal 14. Find its side and area.
Area: or
Side = , Area = 98
Classification Check
A quadrilateral has four congruent sides and diagonals of 10 and 10. Is it a square?
Four congruent sides → rhombus → parallelogram.
Congruent diagonals → rectangle (since it's already a parallelogram).
Rhombus + Rectangle → Square.
Yes — it's a square.
Common Pitfalls
“A Rectangle is Not a Square” — Wrong Direction!
Every square IS a rectangle. But not every rectangle is a square. The hierarchy matters: square ⊂ rectangle ⊂ parallelogram ⊂ quadrilateral.
Forgetting to Check Both Conditions
To prove a quadrilateral is a square, you need BOTH equal sides AND right angles (or rectangle + one pair of adj. sides equal, or rhombus + one right angle).
Real-Life Applications
Chess Boards
Every cell on a chess board is a square. The 8×8 grid has 64 unit squares, but counting all possible square sizes gives 204 total squares!
Floor Tiles
Square tiles are the most common floor covering because they tessellate perfectly with no gaps or overlaps, and their 90° angles align with room corners.
Practice Quiz
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