Parallelogram theorems (opposite sides, angles, diagonals)
Parallelogram Theorems
Parallelogram Theorems is the opening topic of Quadrilateral Theorems in the Common Core State Standards. It is aligned to Standard G-CO.11, which requires students to prove theorems about parallelograms, including properties of their sides, angles, and diagonals.
In a parallelogram, opposite sides and opposite angles are equal and the diagonals bisect each other.
Theory
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. That single condition forces several properties:
- Opposite sides are congruent.
- Opposite angles are congruent.
- Consecutive angles are supplementary (sum \(180^\circ\)).
- The diagonals bisect each other.
The parallelogram properties:
How to solve parallelogram problems
- Identify the relationship needed (opposite, consecutive, or diagonal).
- Set up an equation: equal parts, sum \(180^\circ\), or equal halves of a diagonal.
- Solve for the unknown.
Opposite sides of a parallelogram are congruent.
| \(CD\) | \(=\) | \(AB=12\) |
Opposite angles are congruent.
| \(\text{opposite angle}\) | \(=\) | \(70^\circ\) |
Consecutive angles of a parallelogram are supplementary (co-interior angles).
| \(180^\circ-70^\circ\) | \(=\) | \(110^\circ\) |
Diagonals bisect each other, so \(AM=MC\).
| \(2x-1\) | \(=\) | \(x+4\) |
| \(x\) | \(=\) | \(5\) |
Common pitfalls
Frequently asked questions
What are the properties of a parallelogram?
Opposite sides congruent and parallel, opposite angles congruent, consecutive angles supplementary, and diagonals that bisect each other.
Are the diagonals of a parallelogram equal?
Not in general — they bisect each other but are equal only in a rectangle (or square).
What does it mean that diagonals bisect each other?
They cross at their common midpoint, so each diagonal is cut into two equal pieces.
Are consecutive angles of a parallelogram equal?
No, they are supplementary (sum to \(180^\circ\)). Only opposite angles are equal.