Definition and properties ofall types of quadrilateral
Answers
A quadrilateral is a closed figure with four straight sides. You can make a quadrilateral by taking (or imagining) anything straight and thin you might have handy: pens, toothpicks, chopsticks, etc. A square is one type of a special quadrilateral.
However, just to make it interesting, try to use four things that are not all the same length. Place your objects so that each end of one touches the end of another. I used pencils to form my quadrilateral, as you can see here. I placed each pencil tip so that it touches the eraser of another pencil. That way, I get a closed figure, meaning there are no gaps between sides and no side extends past the end of another side. Now we have our quadrilateral with four straight sides.
Let's simplify the figure by replacing the pencils with segments:
Each point where two sides touch is called a vertex. We name each vertex with a capital letter. Let's name our quadrilateral with the four vertices: P, N, C, and L. It also has four sides: the segments PN, NC, CL, and LP. To name the whole quadrilateral, we choose any vertex as a starting point and list all of the vertices going around either clockwise or counterclockwise. There are many possible names, including PNCL, LCNP, or CLPN.
If you play around a little with the objects that form your quadrilateral, it might be possible that you get a shape that looks something like this:
This is still a quadrilateral because it follows the definition; it has four straight sides that form a closed figure. Again, we can simplify the figure with segments and name the vertices.
This quadrilateral has four vertices: A, R, O, and W. It has four sides: segments AR, RO, OW, and WA. The whole quadrilateral could be named AWOR, ROWA, or AROW.
The two quadrilaterals, PNCL and AROW, are examples of two different types of quadrilaterals. There are several other special quadrilaterals, such as parallelograms, trapezoids, and kites, but we won't get into the properties specific to each of those types in this lesson. We are only going to discuss two categories of quadrilaterals. Let's look at how to tell the difference between those two types in the next section.
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Properties
A quadrilateral has:
four sides (edges)four vertices (corners)interior angles that add to 360
rhombus is a four-sided shape where all sides have equal length.
Also opposite sides are parallel and opposite angles are equal.
Another interesting thing is that the diagonals (dashed lines in second figure) meet in the middle at a right angle. In other words they "bisect" (cut in half) each other at right angles.
A rhombus is sometimes called a rhomb or adiamond.
The Square
A square has equal sides and every angle is a right angle (90°)
Also opposite sides are parallel.
A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).
The Parallelogram
A parallelogram has opposite sides parallel and equal in length. Also opposite angles are equal (angles "a" are the same, and angles "b" are the same).
NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!
trapezoid (called a trapezium in the UK) has a pair of opposite sides parallel.
And a trapezium (called a trapezoid in the UK) is a quadrilateral with NO parallel sides:
TrapezoidTrapeziumIn the US:a pair of parallel sidesNO parallel sides In the UK:NO parallel sidesa pair of parallel sides(the US and UK definitions are swapped over!)
An Isosceles trapezoid, as shown above, has left and right sides of equal length that join to the base at equal angles.
The Kite
it has adjacent sides equal
it looks like a kite (usually).
It has two pairs of sides:
Each pair is made of two equal-length sides that join
Irregular Quadrilaterals
The only regular (all sides equal and all angles equal) quadrilateral is a square. So all other quadrilaterals are irregular.
Example: a square is also a rectangle.
So we include a square in the definition of a rectangle.
(We don't say "Having all 90° angles makes it a rectangle except when all sides are equal then it is a square.")
This may seem odd, as in daily life we think of a square as not being a rectangle ... but in mathematics it is.
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