What is a Triangle? (Definition & Properties)
Triangle definition
A triangle is a three-sided polygon that closes in a space. It uses lines, line segments or rays (in any combination) to form the three sides. When three sides form and meet, they create three vertices, or corners.
The corners inside the triangle are interior angles. The corners outside the triangle are exterior angles.
The word "triangle" literally means three angles, "tri" being a Latin prefix for three, like tricycle (three wheels), trio (three members of a group), or triceps (three muscles in a group).
Picture a wall with leaning on it. The wall is 20 feet tall and forms one of the two sides of the right triangle. The ladder is 30 feet tall and forms the leaning hypotenuse. The distance from the wall to the ladder is the base of our triangle.
Our right triangle is named △ESP. It has three sides:
ES - 20 feet (the wall)
SP - 22.36 feet (distance from the bottom of the wall to the ladder's feet)
PE - 30 feet (the hypotenuse; the ladder itself)
It has three angles:
∠E, an acute angle measuring around 41°
∠S, a right angle, measuring exactly 90°
∠P, another acute angle measuring around 48°
Triangle base
Every side of a triangle can be its base. You single out the base only when you are planning to construct an altitude, or height, for your triangle. In most cases, the base is presented horizontally to you, but that is not necessary. Wherever the altitude is constructed, the side it intersects is the base.
Triangle height (altitude)
Remember the escaping prisoners' leaning ladder? The wall of the prison was the height; the ladder was the hypotenuse, which is longer than the right triangle's height. The height or altitude of a triangle is found by constructing a perpendicular line from one side of a triangle to the opposite angle.
In a right triangle, you have two ready-made altitudes, the two sides that are not the hypotenuse.
In △ESP, side ES is the altitude for the way the triangle looks now. If we turn the entire picture 90°, side SP is now the altitude. If we turned the triangle around so the hypotenuse (the ladder side) was horizontal, we could construct an altitude from that hypotenuse up to ∠S. We would find that altitude to be 14.91 feet in height.
The altitude, or height, is always perpendicular to the base and always intersects the opposite angle. Every triangle has three altitudes. Only in an equilateral triangle will all three altitudes be congruent.
Opposite angle
Each side has its opposite angle. The hypotenuse, PE, has an opposite ∠S; side ES has the opposite ∠P, and side SP has the opposite ∠E.
This also means every angle has an opposite side. ∠S has opposite side PE, the hypotenuse, and so on.
Adjacent sides
Pick any side of the triangle. The two sides touching it are adjacent, which means they are touching. So for side PE, sides ES and SP are adjacent.
Pick any angle of the triangle. The two sides forming it are its adjacent sides. So for ∠E, sides ES and PE are adjacent.
Lesson summary
Now that you have carefully read the lesson and studied the video and drawings, you are able to recall and identify characteristics of a triangle, identify a triangle's three sides and three angles, recognize and locate a triangle's base and height (or altitude), and locate the opposite angle for a given side, and locate adjacent sides for a given side or angle.