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.

Now, using this information, can you determine if these angle measures will create one unique angle or many angles?

If you use a protractor to draw triangles with these measures, you would see that it is possible to draw many different triangles and they would all be similar. The side lengths would be proportional and the angles would be congruent.

There are other theorems used to construct unique triangles.

**SAS – Side, Angle, Side**(This theorem involves two sides and the included angle.)**ASA- Angle, Side, Angle**(This theorem has 2 angles and the side between them)**SSS – Side, Side, Side**(This theorem has 3 sides with specified lengths that meet the triangle inequality theorem.)

Time for a short video demonstrating the Angle Sum Theorem.

It is time to use the Angle Sum Theorem to solve for the missing angle in a triangle.

Look at the triangle below. We know the measurements of two of the angles, but the third one is a mystery. How can we use the Triangle Angle Sum Theorem to find the measurement of that third angle?

The Triangle Angle Sum Theorem states that the sum of all angles in a triangle equal 180°

You can write an equation to solve for the missing side:

86 + 53 + x = 180

139 + x= 180

-139 -139

x = 41

41 + 86 + 53 = 180

The missing angle = 41°

Here is another type of problem you can solve using this theorem. Instead of solving for the missing angle, solve for x.

Write an equation using the triangle angle sum theorem.

X + 12 + 32 + 40= 180 (Combine terms)

X + 84 = 180

-84 – 84

x = 96 If you plug this into the angle, you can also use x to identify the measure of the angle!

## Practice

#### Are these: many triangles, not a triangle, unique triangle

- Triangle with side lengths 3, 2, and 1
- Triangle with side lengths 7, 7, 14
- Triangle with side lengths 10, 12, 14
- Triangle with side lengths 3, 4, and 5
- Triangle with angles 80, 30, and 70
- Triangle with angles 30, 30, and 120
- Triangle with angles 60, 60, and 70

(source)