It’s Professor Dave, let’s discuss angles.
Now we know what lines are, so let’s draw a few.
Here are two lines, and they cross at this point.
But in what way do they cross?
Do they approach one another slowly and then diverge slowly, or is it more rapid?
In asking this question, we are asking about the angle between the lines.
If a door is closed, it’s just closed.
But if it’s open, how open is it?
Is it slightly ajar, is it wide open?
The angle between the door and the wall simply represents a way of reporting how open the
door is without having to make any reference to any spatial dimensions.
Angles are measurements that are independent of the dimensions of objects, and they are
ubiquitous in our every day life.
To begin examining angles, we can just look at two rays that start at the same point,
which we can call the vertex, and the distance between the rays, denoted by this curved line
segment, is an angle.
Moving these rays closer together and further apart, while their endpoints are anchored
together, will generate different angles.
While there are a few ways to measure angles, we will first learn how to use degrees.
There are three hundred sixty degrees in a circle, so we can take this tiny angle, watch
it increase up to ninety degrees, and then continue all the way until one hundred eighty,
at which point we just have a line.
If we continue in this direction, it will then go down from one hundred eighty all the
way back to zero, since the angle will be getting smaller, just on the other side.
Let’s learn about some types of angles.
If an angle is less than ninety degrees, it’s called an acute angle.
We can remember this because an acute angle is cute and tiny.
If an angle is exactly ninety degrees, this is a right angle, and this will be denoted
by a square.
If the angle is between ninety and one hundred eighty degrees, this is called an obtuse angle.
So we should now be able to look at any angle and label it as acute, right, or obtuse.
Now instead of two rays, let’s use two lines.
If these lines cross, we generate four angles.
What are the relationships of these angles?
First we can examine the angles that are directly opposite one another, like these two.
These are called vertical angles, and they are always equal to one another.
This seems intuitive by looking at the diagram, but geometry is all about rigorously proving
things that might seem obvious, though we don’t know why.
We also have adjacent angles, which are the ones that are next to each other, and if these
are the result of two lines crossing like this, these are also called supplementary
angles.
We can also have complementary angles, which add up to ninety degrees.
With these rules alone, we should be able to look at a diagram like this with some unknown
angles, and solve for the unknown angles on the basis of these rules.
Let’s also recall some relationships between lines.
If two lines within a plane intersect at a right angle, they are called perpendicular
lines.
If two lines within a plane do not intersect, they are called parallel lines.
If we draw two parallel lines and then a third line that crosses both of them, we can define
some more terms.
These four angles on the inside are called interior angles.
These four angles on the oustide are called exterior angles.
Because of the relationships between vertical angles and supplementary angles, we can see
that there are two sets of four equivalent angles here.
First, each angle is equal to the one across from it.
Then, these angles here are called alternate interior angles, and if these lines are parallel,
they must be equal.
The same goes for these alternate exterior angles, they must also be equal.
This is stated by the alternate interior angles theorem and the alternate exterior angles
theorem.
There is also the same-side interior angles theorem, which tells us that interior angles
on the same side of the transversal are supplementary.
That makes sense because these are supplementary, and this one is equal to this one, so these
two must also add up to one eighty.
Lastly, angles in the same position with respect to one another are called corresponding angles.
We will see diagrams like this quite a lot in geometry, and we may even have do algebra
with them.
Like if we set these complementary angles equal to three X and X plus ten.
How do we find the measure of these angles?
We just use the definition of complementary angles.
They must add up to ninety degrees, so we make an equation, combine like terms to get
four X plus ten equals ninety, subtract ten and divide by four, which gives us twenty
degrees for X. Plugging in twenty, that makes the angles thirty and sixty degrees.
Now that we’ve worked a bit with lines, it’s time to look at shapes, which each
have some number of line segments for sides.
If we are talking about straight lines, we can’t have a shape with one line segment,
nor two, so it would seem that three line segments are the minimum required to produce
a shape, and this simplest shape is called a triangle.
Let’s move forward and learn about all different kinds of triangles, but first, let’s check comprehension.