Types of Triangles

Objective

• Review over the Triangle Sum Theorem
• Review over the Exterior angles Theorem
• Go over Isosceles Triangles
• Go over Equilateral Triangles
• Briefly go over Right Triangles
• Do some examples
• Homework

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

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

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

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

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!

Different types of triangles

So far, we’ve studied all sorts of things about triangles.

We know that all the angles in a triangle add up to 180 degrees

We know that the exterior angle of a triangle is equal to the sum of the two opposite angles of the triangle.

Now we’re going to look at specific triangles, because each of them has their own special properties.

So, why do we care?

Well, mainly because we can find many of these special triangles in nature, and it helps us to predict things like:

• The distance between planets, stars, comets, and meteors (that may possibly hit Earth)
• Weather patterns

It also helps us to create things digitally, like:

• Pictures from the camera lens on your phone
• Filters for pictures
• Video games
• Basically any digital photo or creation

So, let’s look at some of these

Isosceles Triangles

A triangle is considered isosceles only if: two sides of the triangle are congruent.

So why is this important?

Well, mainly because since two of its sides are congruent, that means that the angles opposite of those sides are congruent.

It’s a weird relationship that sides and angles share, but if two sides of a triangle are congruent, then their opposite angles are congruent as well.

For example, let’s say we have a triangle:

And we measured all of the distances of the triangle:

5’

5’

4’

Now if we measure the angles of the triangle, we’ll find that the two angles that are opposite of the equal sides will be the same measurement.

So:

As we can see, the yellow angle and the red angle are the same

And, well, that’s about it.

Which again, are the angles opposite of the equal sides:

But, again, because this is math, we can’t actually use this until we have proven it.

So, here is the proof.

The proof of the Isosceles Triangle Theorem

A

B

C

Given

Construct an angle bisector for angle A

Every angle has 1 angle bisector

D

Definition of a segment bisector

Reflexive Property

SSS congruence

CPCTC

Application of the Isosceles Triangle Theorem

So, why is this important?

Mainly because it helps us find a missing side, or a missing angle, depending on what we are given.

So, for example, suppose we are given something like:

7

7

2

3x - 11

And we are asked to find x, how do we do it?

Well, as we can see, the triangle has two sides that are equal:

So, we know that we have an isosceles triangle.

Since we have an isosceles triangle, we know that the angles across from the equal sides are also equal.

So, to find x, we need to set the two angles equal to each other (since they’re the same measurement).

So:

73 = 3x - 11

+11 +11

84 = 3x

_______

3 3

28 = x

So, x = 28.

Finding the remaining Side as well

So, why is this important?

Mainly because it helps us find a missing side, or a missing angle, depending on what we are given.

So, for example, suppose we are given something like:

4x

2x + 28

2

And we are asked to find x, how do we do it?

Well, as we can see, the triangle has two angles that are equal:

So, we know that we have an isosceles triangle.

Since we have an isosceles triangle, we know that the sides across from the equal angles are also equal.

So, to find x, we need to set the two sides equal to each other (since they’re the same measurement).

So:

4x = 2x + 28

-2x -2x

2x = 28

_______

2 2

x = 14

So, x = 14.

WHY CAN’T WE JUST USE THE PYTHAGOREAN THEOREM INSTEAD?

Because the Pythagorean Theorem only works on right triangles.

It won’t work on any other triangles, so we need to use other ways to find sides and angles.

Now let’s look at another type of triangle.

EQUILATERAL TRIANGLES

Equilateral Triangles are triangles whose sides are all equal in length.

Because their sides are all equal, their angles are all equal as well.

So why is this useful?

Well, one special trait about an equilateral triangle is that each angle is equal to 60 degrees.

This also helps just in case we’re presented with this special triangle, then we know how to solve for whatever it is that we need.

But, as you know, we can’t use this until we can prove it

So, here’s the proof.

THE PROOF OF THE EQUILATERAL TRIANGLE THEOREM

A

B

C

Given

Isosceles Triangle Theorem

Isosceles Triangle Theorem

Transitive Property

Transitive Property

APPLICATION OF THE EQUILATERAL TRIANGLE THEOREM

So let’s say we have something like:

And we are asked to find x.

So how do we do it?

2x + 12

4x - 4

Well, we know that this triangle is equilateral, which means all of the sides are equal.

So, we can set these equal to each other to find x.

So:

2x + 12 = 4x - 4

+ 4 + 4

2x + 16 = 4x

-2x - 2x

16 = 2x

_______

2 2

8 = x

So x = 8

Finding the angle

So let’s say we have something like:

And we are asked to find x.

So how do we do it?

Well, we know that this triangle is equilateral, which means all of the angles are equal to 60 degrees.

So all we need to do is set our equation equal to 60.

So:

3x - 21 = 60

+21 + 21

3x = 81

_______

3 3

x = 27

So x = 27

3x - 21

Right Triangles

The last triangle we will go over is the right triangle.

Basically, a triangle is a right triangle if it has a right angle.

Since we know that all of the angles in a triangle add up to 180, and a triangle needs to have three angles, tri (meaning three) and angle, then a triangle mathematically can only have one right angle.

So why do we care?

Well, mainly because with right triangles we can use the Pythagorean Theorem to find the left over side.

So, let’s go over the Pythagorean Theorem.

​

The Pythagorean Theorem

The Pythagorean theorem states:

“The area of the square whose side is the hypotenuse (the side opposite of the right angle) is equal to the sum of the areas of the squares on the other two sides.”

Which basically boils down to this.

If you square the hypotenuse, it’s equal to the square of the other two sides.

So, let’s show that this is the case.

​

​

1. First we need to start off with four equal copies of the same triangle.

(These are rotated 90, 180 and 270 degrees from the original as they need to be).

2. Next let’s set some sides (they will be labeled in a second).

Green – side c.

Red – side a

Blue – side b.

Now let’s maneuver them so it fits what we would like to show!

​

So now we have a square with a few properties

1. We have a square with a measurement of c on all sides

2. We have a smaller square in the middle with sides (a-b)

3. We know each triangle has an area of ½ ba

Now for some calculations.

We know the area of each triangle is ½ ba, so adding it up for all 4 we have: 4( ½ ab) which we know as 2ab

We know that the area of the little white square is (a-b) (a-b)

Example 1

Find the remaining side:

20

15

So, as we can see, we have two sides, and they want us to find the third.

First things first, we need to know if any of these sides is the hypotenuse of the triangle.�

If it is, then we need to make sure we put it as c.

In this case it isn’t, so we can just plug and chug!

So:

A = 15

B = 20

(No, it doesn’t matter what order, I just chose).

So, what we have is:

c = 25

Example 2:

Find the remaining side:

2

So, as we can see, we have two sides, and they want us to find the third.

First things first, we need to know if any of these sides is the hypotenuse of the triangle.�

If it is, then we need to make sure we put it as c.

In this case we can see that 2 is the hypotenuse, so we need to make sure we put that in its right place.

So:

So, what we have is: