# 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.