Triangles are particularly important because arbitrary polygons (with 4, 5, 6, or sides) can be decomposed into triangles. Thus, understanding the basic properties of triangles allows for deeper study of these larger polygons as well. Interestingly, the triangle is the only rigid polygon formed out of straight line segments, meaning that if the three lengths of the sides are given, the measurements correspond to a unique triangle. Because of this, it is often possible, given some information about a triangle (e.g. some side lengths and some angles), to determine additional facts about the triangle.
If is a triangle, then In other words, the sum of the internal angles in any triangle is
Draw line that is parallel to and passes through as in the above figure.
Since , applying the principle of alternate interior angles shows that and
Since angles in a line sum up to 180 degrees, .
Thus, we conclude that .
Main article: Triangle Inequality
Triangles have the property that the sum of any two sides of the triangle is always strictly greater than the third side. This property, known as the triangle inequality, is explored in the wiki linked above.
Main article: Classification of Triangles
Triangles can be classified into different categories based on their sides and angles. For example, a triangle with one angle of measure is known as a right triangle, while a triangle with sides of all equal length is known as an equilateral triangle. These classifications and many others are explored in the wiki linked above.
Main article: Area of a Triangle
When determining the area of a triangle, note that a triangle can be thought of as half of a parallelogram. The following picture should make this point clear:
Because the area of a parallelogram is equal to the product of its base and height, the area of a triangle is simply half of that area.
The area of a triangle is , where is the length of the base and is the height.
What is the area of a triangle with base 10 and height 6?
The area of this triangle is
For more advanced methods of finding the area of a triangle, such as Heron's formula, see the wiki linked at the top of this section.
The measure of an exterior angle of a triangle is the sum of its two remote interior angles. Remote interior angles are the interior angles of a triangle that are opposite to the exterior angle under consideration.
For regular polygons, the formula to find the exterior angle of a polygon is where is the number of sides.
From the property of triangles-angle sum, the sum of measures of all angles in a triangle is . So, in , it follows that .
Since and form a straight line, It follows that both sets of angles are equal to