Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for geometry parallelograms special quadrilaterals? Then this Geometry lesson on Quadrilateral Properties: Parallelograms to Kites explains each idea step by step, with worked examples.
Quick answer: A parallelogram has two pairs of parallel sides, so opposite sides and opposite angles are congruent, consecutive angles are supplementary, and the diagonals bisect each other. A quadrilateral is a parallelogram if its opposite sides or angles are congruent, its diagonals bisect each other, or one pair of sides is parallel and congruent. Rectangles, rhombuses, trapezoids and kites add properties.
Standards: G-CO.C.11 · Level: Grade 10 Geometry
What is true in every parallelogram?
A parallelogram is a quadrilateral with two pairs of parallel sides. That definition guarantees four more facts.
- Opposite sides are congruent.
- Opposite angles are congruent.
- Consecutive angles are supplementary.
- The diagonals bisect each other.
How do you prove that a quadrilateral is a parallelogram?
The converses work, and one more test is available. A quadrilateral is a parallelogram if any one of these holds:
- both pairs of opposite sides are congruent;
- both pairs of opposite angles are congruent;
- the diagonals bisect each other;
- one pair of opposite sides is both parallel and congruent.
Beware: one pair of congruent sides and one pair of parallel sides is not enough, because an isosceles trapezoid has that pattern.
How are special quadrilaterals related?
| Shape | Extra properties |
|---|---|
| Rectangle | Four right angles; diagonals are congruent |
| Rhombus | Four congruent sides; diagonals are perpendicular and bisect the angles |
| Square | Both a rectangle and a rhombus |
| Isosceles trapezoid | One pair of parallel sides; legs congruent; diagonals congruent |
| Kite | Two pairs of adjacent congruent sides; diagonals are perpendicular |
- Name the shape and list its properties.
- Pick the property that connects the given and unknown parts.
- Write an equation and solve it.
- Check against a second property, such as angles adding to 180°.
In parallelogram ABCD, m∠A = (3x + 10)° and m∠B = (5x − 30)°. Consecutive angles are supplementary, so 8x − 20 = 180 and x = 25. Then ∠A = 85° and ∠B = 95°.
Do not assume that the diagonals of every parallelogram are congruent or perpendicular. Those facts belong only to rectangles and rhombuses.
Think of the family tree: whatever is true for a parallelogram is true for rectangles, rhombuses and squares below it.
- Parallelograms: opposite sides and angles congruent, diagonals bisect each other.
- Four tests prove a quadrilateral is a parallelogram, including diagonals that bisect each other.
- Rectangle diagonals are congruent; rhombus diagonals are perpendicular.
- A trapezoid midsegment equals the average of the two bases.
- A kite has two pairs of adjacent congruent sides and perpendicular diagonals.
Quadrilateral Properties: Parallelograms to Kites: frequently asked questions
How do you know a quadrilateral is a parallelogram?
Show any one test: both pairs of opposite sides congruent, both pairs of opposite angles congruent, diagonals that bisect each other, or one pair of sides that is parallel and congruent.
Is a square a rectangle?
Yes. A square has four right angles, so it is a rectangle. It also has four congruent sides, so it is a rhombus as well.
Why are consecutive angles of a parallelogram supplementary?
Consecutive angles are same-side interior angles formed by parallel sides and a transversal side. Same-side interior angles of parallel lines add to 180°.
What is the midsegment of a trapezoid?
It joins the midpoints of the two legs. It is parallel to the bases, and its length equals half the sum of the base lengths.
What distinguishes a kite from a rhombus?
A kite has two pairs of adjacent congruent sides, while a rhombus has four congruent sides. Every rhombus is a kite, but most kites are not rhombuses.