As you can see in the picture below, it's not possible to create a triangle that has side lengths of 4 , 10 , and 3 .

It turns out that there are some rules about the side lengths of triangles. You can's just make up 3 random numbers and have a triangle! You could end up with 3 lines like those pictured above that cannot be connected to form a triangle.

Video Tutorial on the Triangle Inequality Theorem

The Formula

Show Formula

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side.

Note: This rule must be satisfied for all 3 conditions of the sides.

In other words, as soon as you know that the sum of 2 sides is less than (or equal to ) the measure of a third side, then you know that the sides do not make up a triangle .

You only need to see if the two smaller sides are greater than the largest side!

Look at the eample above, the problem was that
4 + 3 (sum of smaller sides) is not greater than 10 (larger side)

Demonstrations illustrating the Triangle Inequality Theorem

The interactive demonstration below shows that the sum of the lengths of any 2 sides of a triangle must exceed the length of the third side. The demonstration also illustrates what happens when the sum of 1 pair of sides equals the length of the third side--you end up with a straight line! You can't make a triangle!

Otherwise, you cannot create a triangle from the 3 sides.

Demonstration 1: When the sum of 1 pair of sides exactly equals the measure of a 3^{rd} side.

Demonstration 2 When the sum of 1 pair of sides is less than the measure of a 3^{rd} side.

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Practice Problems

Problem 1) Could a triangle have side lengths of
Side 1: 4
Side 2: 8
Side 3: 2

Answer

No

Use the triangle inequality theorem and examine all 3 combinations of the sides. As soon as the sum of any 2 sides is less than the third side then the triangle's sides do not satisfy the theorem.

problem 2)
Could a triangle have side lengths of
Side 1: 5
Side 2: 6
Side 3: 7

Answer

Yes

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side .

small + small > large

because 5 + 6 > 7

problem 3)
Could a triangle have side lengths of
Side 1: 6
Side 2: 8
Side 3: 15

Answer

No
Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side .

small + small > large

because 6 + 8 $$\color{Red}{ \ngtr } $$ 16

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problem 4)
Do the side lengths below satisfy the triangle inequality theorem (ie can you make a triangle from the side lengths below?)
Side 1: 5
Side 2: 5
Side 3: 10

Answer

No

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side .