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Triangle Definition

We’ve all encountered triangles in our lives. From the Great Pyramid of Giza to fun Sunday puzzles: we’ve all seen what a triangle or pyramid looks like. The question is, how do we define what a triangle is?
cone_triangle

 

Let's take a look at the most basic definition of a triangle.

 

Tri- Prefix that means three
Angle The bed that forms between two lines
Triangle A polygon that has three sides, whose corners form three angles

 

As you can see, the definition of triangle can be found within the word itself. No matter what kind of triangle you’re dealing with, a triangle will always have three sides and three angles.

Triangle Properties

Now that we know the definition of a triangle, let’s see some of the properties of triangles.

 

Property Name Description
1 Angle Sum Property All 3 interior angles have a sum of 180 degrees
2 Length Sum Property The sum of two sides is greater than the length of the last side
3 Largest Angle Property The angle opposite the largest side is the largest angle
4 Exterior Angle Property The exterior angle of any interior angle of the triangle is equal to the sum of the other two interior angles

As you can see, these properties can get kind of tricky - what is an exterior angle and what is an ‘opposite’ side? Let’s take a look at these properties in depth.

Angle Sum Property

The angle sum property states that all three interior angles of a triangle have to add up to 180 degrees. Let’s take a look at what interior angles are.

Definition
Interior Angle The interior angle of a triangle is any angle that is formed by the sides of the triangle and is inside the triangle

Let’s take a look at some examples.

 

interior_angles

 

Can you tell which ones are exterior and interior triangles? Take a look at the table below and see if you answered correctly.

Interior?
A No
B Yes
C Yes
D Yes
E No

Length Sum Property

The length sum property states that the length of two sides is always greater than the last side. We can understand this property by picking the smallest length you can think of for all sides. Take a length of 1 cm for example.

 

triangle_properties

 

AB + BC 1+1 = 2
BC + CA 1+1 = 2
CA + AB 1+1 = 2

As you can see, the sum of any of the sides is always greater than the last side.

Largest Angle Property

The largest angle property states that the largest angle is opposite the largest side of a triangle. The proof for this property is quite complex, however it boils down to the fact that the larger the angle is, the longer of a line it projects.

 

angle_projection

 

A B
The angle of 80 degrees projects a longer line The angle of 10 degrees projects a smaller line

Exterior Angle Property

The exterior angle property states that the exterior angle of any interior angle of a triangle is equal to the sum of the other two interior angles. Let’s take a look at this in practice.

 

exterior_angles

 

Exterior angle 109 degrees
Angle A + Angle C 38 + 71 = 109

Equilateral Triangle

There are four different types of triangles. Let’s start by looking at an equilateral triangle. An equilateral triangle can be defined by the table below.

 

Equilateral Triangle Definition
1 A triangle
2 Three equal sides
3 Three equal angles
4 Which are all 60 degrees

It’s quite easy to identify the different types of triangles. Every triangle has a different notation. You should familiarize yourself with this notation so that you can understand what type of triangle you’re dealing with.

 

same_angle_notation

 

A Three same angles
B Three same sides

 

In order to find the perimeter or area of an equilateral triangle, simply follow the formulas below.

 

Perimeter P = 3*s
Area A =

Scalene Triangle

Out of the four triangles, let’s take a look at the next type of triangle: a scalene triangle. A scalene triangle can be defined by the table below.

 

Scalene Triangle Definition
1 A triangle
2 No equal angles
3 No equal sides

As you can see, a scalene triangle is quite special. This is the only type of triangle where at least two sides or angles are not equal. Remember that when two sides are the same length, they automatically form two angles of the same length. Let’s take a look at the notation for scalene triangles.

 

angle_side_notation

 

A No angles are the same length, as market by the curves
B No sides are the same length, as marked by the lines

 

In order to find the perimeter or area of a scalene triangle, simply follow the formulas below.

 

isosceles_triangle

 

Perimeter P = a+b+c
Area A = ½ *b*h

Isosceles Triangle

The third most common triangle is the isosceles triangle. Let’s take a look at the properties of an isosceles triangle.

 

Isosceles Triangle Definition
1 A triangle
2 Two equal angles
3 Two sides are the same

As mentioned in the previous section, when two sides of a triangle are the same length, they form two angles that are the same size. Let’s take a look at the notation for this type of triangle.

Perimeter P = 2*s+c
Area A = ½ *b*h

Right Triangle

The last type of triangle is a right triangle. This type of triangle is the triangle with the most special properties. Let’s take a look at these properties.

 

Property Description
1 Has one right angle (90 degrees)
2 The side opposite the right angle is the hypotenuse
3 The right angle is the largest angle of a right triangle

 

A right triangle notation can be seen below.

 

right_angle

 

Pythagorean Theorem

The Pythagorean theorem applies specifically to right triangles. It states that if you know that the length of the hypotenuse squared is equal to the sum of the squares of the other two sides. Let's take a look at what this actually means.

 

pythagorean_theorem

 

Formula
c hypotenuse
a One side
b The other side

 

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Emma

Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.