Exterior Angles Theorem
Objective
Triangle Sum Theorem
So, according Wikipedia, the triangle sum theorem is: “ In a Euclidean space, the sum of angles of a triangle equals a straight angle, or in other words, 180 degrees.”
And we know this, right?
We know that if you add all of the angles of a triangle together, they will equal 180 degrees.
But how do we know that?
(Other than because Burch told you).
Well, here’s a way that we can see it.
Really looking at the Triangle Sum Theorem
So, to understand the triangle sum theorem, we’re gonna need a triangle.
And since we’re trying to figure things out about its angles, let’s take a look at its angles:
Now, let’s look back at what the theorem said.
It said, a triangles angles will add up to straight angle.
But we know a straight angle is a line.
So, what we’re saying then, is that all of these angles should be able to be manipulated to form a line.
Well, let’s look at that:
But, that doesn’t really work, since all of the angles aren’t adjacent to each other.
So, let’s rotate the top angle:
And now, we can see that the angles can add up to a straight line:
So, now we’ve seen it, but is that enough?
Well, no.
Even though we can see that it works, we need to actually prove it before we can say with absolute certainty that all angles to a triangle add up to 180 degrees.
So, let’s go ahead and start that.
The Triangle Sum Conjecture Proof
Prove: The sum of the angles of
FEG = 180 degrees
Statements | Reason |
| |
| |
| |
| |
| |
| |
Definition of angles residing on a line
Alternate interior Angles
Alternate interior Angles
Substitution
Thus:
The sum of the angles of
FEG = 180 degrees
Given
🐙
Whoop whoop whoop whoop whoop whoop
NOW WE KNOW
So we’ve seen this proof before, but we really didn’t understand it until now.
Mainly because we now know what angles we were looking at to make this all make sense.
So, what does this mean for us?
Well, it means we can now use this to find the missing angles of a triangle, as well as many other polygons.
But we’ll get to that later.
So, let’s look at an example problem we may see.
Example
Solve for x:
6x + 15
Well, as we can see, we have three angles.
The top angle is 55 degrees
The right angle is 50 degrees
And the left angle is 6x + 15
So, to find x, we need to add all of these together and set them equal to 180
(since we just proved the theorem).
So:
So, then we know that x = 10.
But how do we check?
By plugging it in, and seeing if it adds to 180.
So:
So we were right!
The Exterior Angles Theorem
So, again, according to Wikipedia, the Exterior Angles Theorem is:
“The exterior angle theorem is an Euclidian theorem which states that the measure of an exterior angle of a triangle is equal to the sum of the remote interior angles of the triangle.”
There seems to be a bit to unpack there, so let’s take a look at what it means.
To do this, let’s start with a triangle, and its angles.
Now let’s add an exterior angle:
So, let’s take a second and look at this.
We can see that: red angle + orange angle = 180 degrees, right?
We also know that: orange angle + blue angle + green angle = 180 degrees as well, right?
So, doesn’t it make sense that: red angle + blue angle + green angle = 180?
But again, it’s not enough to just show that this works
We need to prove it.
(Your favorite!)
Proving the Exterior Angles Theorem
Statement | Reason |
| |
| |
| |
| |
| |
Given
Angles Residing on a Line/Linear Pair
Triangle Sum Theorem
Substitution Property
Subtraction Property
NOW WE KNOW
Now, like before, since we’ve proven the theorem, we can use it.
So, what does this mean for us?
Well, it means we can now use this to find the exterior angle of a triangle.
So, let’s look at an example problem we may see.
Example 1
Solve for x:
12x+1
7x + 7
Well, according to the exterior angles theorem, we know that: exterior angle = top angle + right angle
So:
122 = 12x + 1 + 7x + 7
122 = 19x + 8
-8 - 8
114 = 19x
________
19 19
x = 6
So, how do we check?
Again, we plug x back in to see if it’s equal.
Like so:
122 = 12(6) + 1 + 7(6) + 7
122 = 72 + 1 + 42 + 7
122 = 73 + 49
122 = 122
So we were right!
Example 2
Solve for x:
7 + 12x
29x - 1
Well, according to the exterior angles theorem, we know that: exterior angle = top angle + right angle
So:
29x - 1 = 7 + 12x + 60
29x - 1 = 12x + 67
+1 + 1
29x = 12x + 68
-12x -12x
17x = 68
So, how do we check?
Again, we plug x back in to see if it’s equal.
Like so:
29(4) - 1 = 7 + 12(4) + 60
116 - 1 = 7 + 48 + 60
115 = 55 + 60
115 = 115
So we were right!
________
17 17
x = 4
Example 3
Solve for x:
3x+13
11x + 1
Well, according to the exterior angles theorem, we know that: exterior angle = top angle + right angle
So:
So, how do we check?
Again, we plug x back in to see if it’s equal.
Like so:
11(9) + 1 = 60 + 3(9) + 13
99 + 1 = 60 + 27 + 13
100 = 87 + 13
100 = 100
So we were right!
11x + 1 = 60 + 3x + 13
11x + 1 = 73 + 3x
-1 - 1
11x = 72 + 3x
-3x - 3x
8x = 72
________
8 8
x = 9