Geometry Problem Solver Are squares parallelograms?
Yes. A parallelogram is any quadrilateral with both pairs of opposite sides parallel, and a square satisfies that. A square is in fact a parallelogram, a rectangle and a rhombus all at once. The statement does not reverse: most parallelograms are not squares.
The family tree, in one direction
Start with a quadrilateral: four sides. Make both pairs of opposite sides parallel and it is a parallelogram. Add four right angles and it is a rectangle. Instead add four equal sides and it is a rhombus. Do both and it is a square.
Every step keeps the previous conditions, so the labels accumulate downward. That is why a square answers yes to parallelogram, rectangle and rhombus, and why a parallelogram answers no to square unless it happens to qualify.
How to test a shape
For a parallelogram you can check any one of these, and each implies the rest: both pairs of opposite sides parallel, both pairs of opposite sides equal, both pairs of opposite angles equal, or the diagonals cutting each other in half.
That last one is the quickest with a ruler: if the diagonals bisect each other, the quadrilateral is a parallelogram, whatever it looks like.
Which statements are true
| Statement | True? |
|---|---|
| Every square is a parallelogram | Yes |
| Every parallelogram is a square | No |
| Every rectangle is a parallelogram | Yes |
| Every rhombus is a parallelogram | Yes |
| Every square is a rhombus | Yes |
| Every trapezoid is a parallelogram | No, it needs only one pair of parallel sides |
Questions
Can a rectangle be a parallelogram?
Yes, always. A rectangle has both pairs of opposite sides parallel, which is exactly the definition. Rectangles are the parallelograms whose angles are all 90°.
Is a rhombus a square?
Only if its angles are right angles. A rhombus has four equal sides but can lean; a square is the upright case.
Is a trapezoid a parallelogram?
No. A trapezoid needs only one pair of parallel sides. A parallelogram needs both, so a trapezoid with both would simply be a parallelogram.
More geometry reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the theorem named at each one.
