Properties of Rectangles
A rectangle is a parallelogram with four right angles. It inherits all 5 parallelogram properties — plus one more: congruent diagonals.
Introduction
Rectangle: four right angles, congruent diagonals
Past Knowledge
Parallelogram properties (9.2.1). Right angles. Pythagorean Theorem (8.1.1).
Today's Goal
State and prove the special property of rectangles: congruent diagonals.
Future Success
Rhombus (9.2.4), square (9.2.5), coordinate proofs (9.3.4).
Key Concepts
Rectangle = Parallelogram + Right Angles
A rectangle has ALL the properties of a parallelogram, plus:
- All four angles are 90°
- Diagonals are congruent (this is the rectangle's unique property)
- Diagonals still bisect each other (inherited from parallelogram)
Diagonal Length (via Pythagorean Theorem)
For a rectangle with length and width :
Theorem & Proof
Two-Column Proof: Diagonals of a Rectangle Are Congruent
Given: Rectangle
Prove:
| # | Statement | Reason |
|---|---|---|
| 1 | Reflexive Property | |
| 2 | Opposite sides of a parallelogram are congruent | |
| 3 | Definition of rectangle | |
| 4 | SAS Congruence (steps 2, 3, 1) | |
| 5 | CPCTC |
∎ The right angles plus congruent sides make the two triangles formed by the diagonals congruent.
Worked Examples
Diagonal Length
A rectangle is 12 by 5. Find the diagonal.
d = 13 (5-12-13 Pythagorean triple!)
Using Congruent Diagonals
Rectangle ABCD has AC = 3x + 1 and BD = x + 13. Find the diagonals.
Diagonals are congruent: →
AC = BD = 19
Is It a Rectangle?
Parallelogram PQRS has diagonals PR = 20 and QS = 20. Is it a rectangle?
It's a parallelogram with congruent diagonals → YES, it's a rectangle.
Converse: if a parallelogram has congruent diagonals, it is a rectangle.
Yes — parallelogram + congruent diagonals = rectangle.
Common Pitfalls
Thinking Diagonals Are Perpendicular
In a rectangle, diagonals are congruent but NOT perpendicular (unless it's a square). Perpendicular diagonals are a rhombus property.
Congruent Diagonals ≠ Rectangle (Without Parallelogram)
An isosceles trapezoid also has congruent diagonals. You must first establish it's a parallelogram.
Real-Life Applications
Construction — Checking Corners
Before pouring a foundation, builders measure both diagonals. If they're equal, the foundation is rectangular (has true 90° corners). This is faster than checking each angle.
Screens & Displays
Every TV screen, phone, and monitor is a rectangle. The “screen size” is the diagonal length: a 55" TV has a 55-inch diagonal.
Practice Quiz
Loading...