Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

## Triangle (Definition, Properties, & Examples)

A 30-foot ladder can come in mighty handy. In 1923, six prisoners used a handmade, 30-foot ladder to escape from Eastern State Penitentiary in Philadelphia, PA. That may not sound remarkable, but the walls inside the prison are only 20 feet tall, allowing the ladder to form a hypotenuse of a right triangle at a comfortable, climbable angle standing far out from the base of the thick, stone wall.

Once atop the wall, the escaping prisoners pulled the ladder up to lower to the outside for what they thought would be an easy climb down. The outside walls are 30 feet high. Their ladder was nearly useless. It could not form any kind of a triangle since the wall and hypotenuse (the ladder) were the same length.

The prison's designers made the inside exercise yards 10 feet higher than street level, making escape nearly impossible. It was a powerful design. Triangles are powerful, too, but what exactly are they?

## What you'll learn:

After working your way through this lesson and video, you will be 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)
• Locate the opposite angle for a given side, and locate adjacent sides for a given side or angle

## 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).

[insert semi-realistic cartoon drawing of a stone wall 20 feet tall, a leaning ladder 30 feet tall, and a base distance of 22.36 feet from the wall to the ladder's bottom]

Here we have a rough version of Eastern State Penitentiary's mighty wall. It is 20 feet tall and forms one of the two sides of the right triangle. The prisoners' escape ladder was 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:

1. $ES$ -- 20 feet (the wall)
2. $SP$ -- 22.36 feet (distance from the bottom of the wall to the ladder's feet)
3. $PE$ -- 30 feet (the hypotenuse; the ladder itself)

It has three angles:

1. $\angle E$, an acute angle measuring around $41°$
2. $\angle S$, a right angle, measuring exactly $90°$
3. $\angle P$, 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 $\angle S$. 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 $\angle S$; side $ES$ has the opposite $\angle P$, and side $SP$ has the opposite $\angle E$.

This also means every angle has an opposite side. $\angle S$ has opposite side $PE$, the hypotenuse, and so on.

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 $\angle E$, 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.

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