Triangle rules and theorems allow us to understand the properties of this shape. As one of the most central elements of trigonometry, triangles have many geometric rules. Among other things, these help us to distinguish right triangles from equilateral triangles and isosceles triangles.
Let's review some of the most notable trigonometric triangle rules.
Interior Angles Rule
The interior angles rule states that the three angles of a triangle must equal 180°. As you can see below, the three angle measurements of obtuse triangle ABC add to 180°.
Sides of a Triangle
The sides of a triangle rule asserts that the sum of the lengths of any two sides of a triangle has to be greater than the length of the third side. See the side lengths of the acute triangle below. The sum of the lengths of the two shortest sides, 6 and 7, is 13. That length is greater than the length of the longest side, 8.
Triangle Congruence Rules
Congruent triangles are triangles whose corresponding sides and angles are equal. In trigonometric fashion, equal sides and equal angles are proven congruent through the four triangle rules of congruence. We’ll go through these one at a time.
#1: SSS Rule
The side-side-side (SSS) rule says that when the three side measurements of a triangle match the three side measurements of another triangle, these two shapes are congruent.
See the right-angled triangles below. The sides of the triangle DEF are the same exact lengths as triangle GHI, making them congruent.
#2: ASA Rule
The angle-side-angle (ASA) rule states that when two angles and one side of a triangle are equal to that of another triangle, they are congruent triangles.
See triangles JKL and MNO. Angles J and M, K and N (the opposite angles to the length of the hypotenuse), and the hypotenuse-legs of both triangles are all equal. Therefore, triangles JKL and MNO are congruent.
#3: AAS Rule
The angle-angle-side (AAS) rule asserts that when two triangles have the following matching properties, they must be congruent:
- Two angles
- One opposite side length with no vertices
#4: SAS Rule
The side-angle-side (SAS) rule states that if the included angle and the two included side lengths of a triangle are equal to that of another triangle, then the two are congruent. See below triangles CDE and FGH. Right angle C and angle F, the length of d and g, and the hypotenuse length of c and f are equal. Therefore, triangle CDE=FGH.
The Importance of Triangle Rules
Expanding your knowledge of triangle rules will make it easier to learn other trigonometric ideas like Pythagoras theorem and cosine, tangent, and sine rules. This knowledge will also help you master the area of a triangle and polygon.