Triangles
Contents
Overview
A Triangle is a polygon with three edges and three vertices.
Triangles are named shapes, just like most polygons. In their case, each vertex is named (e.g., A, B, and C), so the triangle would be called: $\triangle ABC$
Triangles will always have exactly three edges and exactly three vertices. For flat geometries, a triangle’s interior angles will always add up to $180^\circ$.
Types of Triangles
Defined by Side Length
A Scalene Triangle is a triangle such that no side length (and thus angle) is identical in measure.
A Isosceles Triangle is a triangle such that exactly two side lengths (and their complementary angles) are equal in measure.
A Equilateral Triangle is a regular triangle. All three side lengths are the same length, and thus all angles are $60^\circ$.
Defined by Angles
You may also classify a triangle based on its internal angles.
A Right Triangle is a triangle such that one of its internal angles is $90^\circ$. Right triangles have unique terminologies:
- Hypotenuse denotes the longest side of the triangle, which is opposite the right angle.
- Leg refers to the sides of the triangle that are not the hypotenuse.
- 45-45-90 refers to a special right triangle where the other angles are $45^\circ$.
- If the triangle’s legs’ lengths are defined as $a$, then the hypotenuse will be $a\sqrt{2}$.
- 30-60-90 refers to a special right triangle where the other angles are $30^\circ$ and $60^\circ$.
- The side lengths of this triangle will always be in a ratio of $1 : \sqrt{3} : 2$.
An oblique triangle is any triangle that does not have an angle measuring $90^\circ$.
- Obtuse Triangle describes a triangle with an interior angle larger than $90^\circ$.
- Acute Triangle describes a triangle with an interior angle less than $90^\circ$.