Professor Dave again, let’s learn some more shapes.
We’ve been talking about triangles for a while now, so let’s talk about shapes with
more than three sides.
Any shape with four sides is called a quadrilateral, and there are many different types.
With four sides there must be four angles, and if we place various constraints on these
sides and angles, we get specific types of quadrilaterals.
For example, if a quadrilateral has precisely one pair of parallel sides, it is a trapezoid.
The other pair of sides must not be parallel, which can produce a variety of different trapezoids.
A kite is a quadrilateral that has two pairs of congruent consecutive sides.
That means these two are the same length, and these two are the same length.
Kites have diagonals that are perpendicular.
In this other quadrilateral, these opposite sides are parallel and of equal length, or
congruent, and so are these two.
Because of this, opposite angles are also congruent.
Any quadrilateral that meets these constraints is called a parallelogram.
The diagonals of a parallelogram always bisect one another.
Now from here we can narrow things down further.
Any parallelogram that has four congruent sides is called a rhombus.
Any parallelogram that has four congruent angles, and therefore four right angles, is
a rectangle.
And lastly, any parallelogram that has four congruent sides and four right angles is a
square.
Therefore, a square qualifies as a rectangle, rhombus, parallelogram, and quadrilateral,
it is just a specific subset of these different shapes, which we can see in this diagram where
different bubbles represent different shapes.
Squares sit all the way in the middle, which demonstrates how they also qualify as some
of the other shapes that have fewer constraints.
So we can think of each new constraint as generating a new and more specific type of
quadrilateral.
This is what we mean when we say that a square is a rectangle but a rectangle is not necessarily
a square.
Any square must have four right angles, so it is a rectangle, but it must also have four
equal sides, and not all rectangles have four equal sides, so most rectangles are not squares.
This is a syllogism, a form of deductive reasoning that we can use in everyday arguments.
But we won’t get into logic and philosophy here, that will have to wait for another day.
Now that we know the various quadrilaterals, what about shapes with more than four sides?
These are called polygons.
To be specific, a polygon is any shape formed from three or more line segments, so triangles
and quadrilaterals are technically polygons, but since we have specific names for those,
we usually only use the word polygon to refer to shapes with five sides are more.
We must understand that only shapes with straight lines for sides count as polygons, so circles
are not polygons.
We have prefixes that indicate how many sides and therefore also how many angles are in
a polygon.
Starting with five sides we can have pentagons, hexagons, heptagons, octagons, nonagons, and
decagons.
To qualify as these shapes, they simply have to have these numbers of sides, which can
be of any length, and form any angles with one another.
If a polygon has all of its sides and angles equal to one another, then it is a regular
polygon, and these are the shapes that are most familiar to us, like the regular octagonal
stop-sign.
Each shape will have its angles add up to a specific number.
As we said, the angles of a triangle add up to one eighty.
A quadrilateral can be cut up into two triangles, so two times one eighty means that the angles
of a quadrilateral will add up to three sixty.
A pentagon can be split up into three triangles, so its angles add up to five forty.
A hexagon can be split up into four triangles, so its angles add up to seven twenty.
We could go on and on in this fashion, and every shape will contain a number of triangles
that is two less than its number of sides.
So to find out what the angles of a polygon add up to, take the number of sides it has,
subtract two, and multiply by one eighty.
Now that we know about a variety of shapes, let’s check comprehension.