The 5 Properties of Parallelograms
A parallelogram has two pairs of parallel sides — and from that one property, five powerful consequences follow.
Introduction
Parallelogram ABCD with diagonals intersecting at E
Past Knowledge
Parallel lines (3.1). Alt. int. angles (3.1.4). Triangle congruence (5.2).
Today's Goal
Learn and apply the 5 properties of parallelograms.
Future Success
Proving parallelograms (9.2.2), rectangles (9.2.3), rhombi (9.2.4).
Key Concepts
The 5 Properties
- Property 1: Both pairs of opposite sides are parallel (by definition)
- Property 2: Both pairs of opposite sides are congruent
- Property 3: Both pairs of opposite angles are congruent
- Property 4: Consecutive angles are supplementary (sum to 180°)
- Property 5: The diagonals bisect each other
Theorem & Proof
Two-Column Proof: Opposite Sides of a Parallelogram Are Congruent
Given: Parallelogram (, )
Prove: and
| # | Statement | Reason |
|---|---|---|
| 1 | Draw diagonal | Through any two points, exactly one line exists |
| 2 | Alt. interior angles (, transversal ) | |
| 3 | Alt. interior angles (, transversal ) | |
| 4 | Reflexive Property | |
| 5 | ASA Congruence (steps 2, 4, 3) | |
| 6 | and | CPCTC |
∎ The key move: a diagonal turns the parallelogram into two congruent triangles via ASA.
Worked Examples
Opposite Sides
ABCD is a parallelogram. AB = 3x + 2, DC = 5x − 6. Find AB.
Opposite sides are congruent: → →
AB = DC = 14
Consecutive Angles
In parallelogram PQRS, ∠P = (4x + 10)° and ∠Q = (2x + 50)°. Find all angles.
Consecutive angles supplementary:
→
,
All 90° — this parallelogram is a rectangle!
Diagonal Bisection
Diagonals of parallelogram ABCD meet at E. AE = 2x + 3, EC = x + 7. Find AC.
Diagonals bisect each other:
→
, so
AC = 22
Common Pitfalls
Diagonals Are NOT Congruent (in general)
In a generic parallelogram, the diagonals bisect each other but are NOT equal in length. Congruent diagonals is a special property of rectangles.
Confusing Opposite and Consecutive
Opposite angles are congruent. Consecutive (adjacent) angles are supplementary. Don't mix them up.
Real-Life Applications
Scissor Lifts
Scissor lift mechanisms use interconnected parallelograms. As the angle changes, opposite sides stay parallel, keeping the platform level while it rises.
Pantographs
A pantograph (drawing tool that copies and scales images) uses parallelogram linkages so that opposite sides stay parallel during movement.
Practice Quiz
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