Professor Dave here, let’s learn about different types of triangles.
We know that if we connect three line segments, we can create a simple shape called a triangle.
But in what way will these lines connect, and what are the different types of triangles
that will result?
Let’s go through a list of the most common types of triangles now.
First, a few general definitions.
Since we know what acute, obtuse, and right angles are, we can then learn about acute,
obtuse, and right triangles.
Acute triangles have angles that are all acute, or less than ninety degrees.
An obtuse triangle has one obtuse angle, or more than ninety degrees.
And a right triangle has one right angle.
As we can see by looking at the angles in these triangles, and every other triangle
we could possibly draw, the angles in a triangle will always add up to one hundred eighty degrees.
This is why any triangle can have at most one right or obtuse angle, because two of
them would add up to 180 or more, and then there is nothing left for the third angle.
This fact will help us algebraically determine the measures of certain angles later.
Triangles can be classified not just by angles, but also by side lengths.
A scalene triangle is defined as having three sides of different lengths.
An isosceles triangle has two sides out of the three that are exactly the same length,
and therefore two angles that are precisely the same measure, specifically the two angles
that are opposite the two identical sides.
And an equilateral triangle has all three sides of the same length, and therefore three
identical angles.
Since the three identical angles must add up to one hundred eighty degrees, each angle
must therefore be sixty degrees in an equilateral triangle.
Every triangle must be one of these three types, because if they have three sides, it
must either be the case that they are all the same, two are the same, or none are the
same, as those are all the options we have.
It’s worth our time to also quickly mention some special kinds of right triangles.
One is the forty-five forty-five ninety triangle.
This will have two legs of equal length, X, and a third side, a hypotenuse, of length
X root two.
There are also thirty-sixty-ninety triangles, and these have sides of length X, X root three,
and two X.
Now that we are equipped with all of this information, we should be able to look at
a diagram like this and fill in all the missing angles.
We just fill them in one at a time by using the rules we know.
If we have one of two vertical angles, the other will be the same as the first.
If we have two angles in a triangle, then the third will be one eighty minus the other
two.
If we have one of two adjacent angles, the other will be one eighty minus the first,
as these are supplementary, such as with the case of an exterior angle like this one.
The exterior angle will also be equal to the sum of the other two angles in the triangle,
since this one plus the exterior angle equals one eighty, and this one plus the other two
equals one eighty, so these values must be equal.
This kind of reasoning is the essence of geometry.
All we are really doing is drawing pictures in the sand like the Greeks did, and asking
questions about them.
What can and can’t be true about these shapes?
It’s like a playground for the brain, and if we follow the rules of the playground,
we get meaningful answers to questions.
These kinds of rules, like the ones that dictate angle relationships, will be applied constantly
during our study of geometry, so let’s check comprehension.