So a square is a special case of a rhombus. Just to remind ourselves, a rhombus, the opposite sides are parallel to each other. You have two sets of parallel sides.
A square has two sets of parallel sides, and it has the extra condition that all of the angles are right angles. So a square is definitely going to be a rhombus. Now, all rhombuses have four sides.
So all rhombuses are quadrilaterals. But not all quadrilaterals are rhombuses. You could have a quadrilateral where none of the sides are parallel to each other. So we won't click this one.
Once again, a parallelogram. So all rhombuses are parallelograms. They have two sets of parallel sides, two sets of parallel line segments representing their sides. But all parallelograms are not rhombuses. So I would say that if someone gives you square, you can say, look, a square is always going to be a rhombus. A quadrilateral isn't always going to be a rhombus, nor is a parallelogram always going to be a rhombus. The four vertices of a quadrilateral may be concyclic , i.
In this case, the quadrilateral is known as circumscritptible or, simpler, cyclic. If a quadrilateral admits an incircle that touches all four of its sides or more generally, side lines , it is known as inscriptible. A quadrilateral, both cyclic and inscriptible, is bicentric. The diagram below which is a modification of one from wikipedia.
The applet below illustrates the properties of various quadrilaterals. In the applet, one can drag the vertices and the sides of the quadrilateral. You can display its diagonals, angle bisectors and the perpendicular bisectors of its sides. With these props, it's a simple matter to observe every single kind of quadrilateral, with a possible exception of bicentric. Which, too, is not overly difficult if you first get an isosceles trapezoid.
This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. As in the classification of triangles , the definitions may be either inclusive or exclusive.
For example, trapezoid may be defined inclusively as a quadrilateral with a pair of parallel opposite sides, or exclusively as a quadrilateral with exactly one such pair.
In the former case, parallelogram is a trapezoid, in the latter, it is not. Similarly, a square may or may not be a rectangle or a rhombus.
My preference is with the inclusive approach. For, I'd like to think of a square as a rhombus with right angles, or as a rectangle with all four sides equal. Here is a list of all the properties of quadrilaterals that we have mentioned along with the classes of the quadrilaterals that possess those properties:. Orthodiagonal or inscriptible parallelogram is a rhombus; cyclic parallelogram is a rectangle. In particular, a parallelogram with equal diagonals is necessarily a rectangle.
And not to forget, every simple quadrilateral tiles the plane. A simple quadrilateral with two pairs of equal opposite angles is a parallelogram. Because then the opposite sides are parallel. So all other quadrilaterals are irregular. This may seem odd, as in daily life we think of a square as not being a rectangle Oh Yes! A quadrilateral is a polygon.
In fact it is a 4-sided polygon, just like a triangle is a 3-sided polygon, a pentagon is a 5-sided polygon, and so on. Now that you know the different types, you can play with the Interactive Quadrilaterals. Example: A parallelogram with: all sides equal and angles "A" and "B" as right angles is a square! Example: a square is also a rectangle.
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