In this lesson, we will look at triangle constructions as we focus on two very important theorems regarding sides and angles.

Let us begin with the **Triangle Inequality Theorem** which states that:

the sum of the lengths of any two sides of a triangle

is greater than the length of the third side.

### Example

We can use this theorem to determine if the following lengths are legs of triangles.

4, 9, 5 and 9, 5, 5

*Strategy – Choose the smallest two of the three sides and add them together. Then, compare that sum to the third side.

4, 9, 5

4 + 5 > 9 (THIS IS **FALSE**)

Since the sum is not greater than the third side, this is NOT a triangle.

9, 5, 5

5 + 5 > 9 (THIS IS **TRUE**)

Sine the sum IS greater than the third side, this is a triangle.

### Example

A triangle has side lengths of 6 and 12; what are the possible lengths of the third side?

Think about it.

6 + 12 = 18 (for the side to make the inequality theorem work, 6 + 12 > x)

12 – 6 = 6 (for the side to make the inequality theorem work, 6 + x > 12)

This means that the third side must be between 6 and 18. (Difference and sum of the given sides)

Solution: 6 < x < 18

Here are two more Triangle Theorems involving side lengths of triangles.

1. If one side of a triangle is **longer** than another side,

then the *angle opposite* the longer side is the **larger** **angle**.

2. If one angle of a triangle is **larger** than another angle,

then the *side opposite* the larger angle is the **longer side**.

Watch this video to see examples of what is behind the Triangle Inequality Theorem.

The triangle inequality theorem deals with the side lengths. Another theorem, called the **Triangle Angle Sum Theorem** states that:

the sum of the angles in any triangle is 180°.

No triangles can have two obtuse angles.

Is it possible to construct a triangle with angles measuring 61°, 33°, and 86°?

Use the Triangle Angle Sum Theorem to test these angle measures. Add the angle measures: 61 + 33 + 86 = 180. A triangle with these measures is possible.

## Practice

#### Are these a triangle or not?

- Triangle with side lengths 3, 2, and 1
- Triangle with side lengths 10, 12, 14
- Triangle with angles 20, 70, and 90
- Triangle with angles 60, 60, and 70

(source)