Triangle Inequality Theorem

Objective

• Review over Isosceles Triangles
• Review over Equilateral Triangle
• Review briefly over right triangles and the Pythagorean theorem
• Go over the Triangle Inequality Theorem
• Show how it is useful
• Do some examples
• Homework

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

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

Triangle Inequality Theorem

So, we know how to tell if 3 sides make a right triangle by using the Pythagorean theorem.

But what happens when the triangle isn’t a right triangle?

How do we tell if three sides make a triangle, regardless of the type of triangle?

Well, this is where the triangle inequality theorem comes in.

So, according to Wikipedia, “The triangle inequality theorem states that the sum of any two sides of a triangle is greater than or equal to the third side.”

So basically, if we have three sides of a supposed triangle, we add them together, and the sum is greater than or equal to the other side, then we have a triangle.

Otherwise, we don’t have a triangle.

Sounds more complicated than it is, so here’s an example:

EXAMPLE

Determine whether the following lengths will create a triangle or not:

​

14, 3, 12

So, to determine whether or not these lengths will create a triangle, we need to pick two, add them together, and see if their sum is bigger than the remaining side.

So:

But we know that 14 + 3 = 17, which is bigger than 12

So it seems to be okay so far.

Now we need to change it up some

But we know that 3 + 12 = 15, which is bigger than 14

So it seems to be okay so far.

Now we need to change it up once more

And again, we know that 12 + 14 = 26, which is bigger than 3

Since we determined all of these sides do add up correctly, then we know these sides add together to make a triangle

EXAMPLE 2

Determine whether the following lengths will create a triangle or not:

​

1, 3, 12

So, to determine whether or not these lengths will create a triangle, we need to pick two, add them together, and see if their sum is bigger than the remaining side.

So:

But we know that 1 + 12 = 13, which is bigger than 3

So it seems to be okay so far.

Now we need to change it up some

But we know that 3 + 12 = 15, which is bigger than 1

So it seems to be okay so far.

Now we need to change it up once more

But, we know that 1 + 3 = 4, which is smaller than 12

So, these three sides will not make a triangle.

EXAMPLE 3

Determine what x may be to create a triangle:

​

x , 5, 22

This is a little harder to do, but we can do it.

To begin with, we need to start setting up the inequalities appropriately

That’ll tell us the smallest number x can be

So:

Now, we know how to handle inequalities.

We solve them like they are an equation

So in this instance, we need to -5 to both sides.

-5 -5

Now let’s mix it up some and look at another inequality.

Finally, let’s mix it up one last time to look at the last inequality.

Then we’ll put all of these together to figure out what x needs to be.

-22 -22

Now let’s look at this.

Well, we know x must be bigger than 17 and -17

But any number bigger than 17 is also bigger than -17, so:

But we can also see that x must be smaller than 27.

So:

So any number between 17 and 27, including 17 and 27, will work.

EXAMPLE 4

Determine what x may be to create a triangle:

​

14 , x, 22

This is a little harder to do, but we can do it.

To begin with, we need to start setting up the inequalities appropriately

That’ll tell us the smallest number x can be

So:

Now, we know how to handle inequalities.

We solve them like they are an equation

So in this instance, we need to -14 to both sides.

-14 -14

Now let’s mix it up some and look at another inequality.

Finally, let’s mix it up one last time to look at the last inequality.

Then we’ll put all of these together to figure out what x needs to be.

-22 -22

Now let’s look at this.

Well, we know x must be bigger than 8 and -8

But any number bigger than 8 is also bigger than -8, so:

But we can also see that x must be smaller than 36.

So:

So any number between 8 and 36, including 8 and 36, will work.