Properties of triangles

If you join three line segments together in a way that encloses an area, then you have made a triangle. You probably already knew that. But had you ever thought of it that way before? We’re going to be doing that a lot: telling you stuff you know or mostly know, along with some new ways to think about it.

So… a triangle is:

• Three line segments joined together
• Three dots, connected pairwise by line segments
• A three-sided polygon
• A figure with three angles

What other ways can you think of to describe a triangle?

Properties of triangles

Here are some properties that triangles share with some other types of figures:

• They are made of line segments
• They have corners (aka vertices)
• They have an area
• They have interior angles
• They have exterior angles that add up to 360 degrees (please see the picture below so that you understand how exterior angles are measured, and how they are not measured!)

An exterior angle is made by extending one side of the figure. This means that an interior angle and its exterior angle add up to 180 degrees – a straight line – and not a full 360 degrees!

Here are some properties that only triangles have:

• They have exactly three sides
• They have exactly three corners
• They have exactly three angles
• The degree measurements of their interior angles add up to exactly 180 degrees
• The largest side is opposite the largest angle
• The smallest side is opposite the smallest angle
• The medium side is opposite the medium angle
• If there is a tie for angle size, then there is a tie for side length
  • …and vice-versa
• Two sides always add up to more than the third side
  • Or to turn it around, if any two sides add up to less than the third side, then it isn’t a triangle at all! This is called the Triangle Inequality Theorem. (Fancy name, easy idea!)

Other definitions

• A median is a line segment from a vertex (corner) of a triangle to the midpoint of the opposite side.
• An altitude is a line segment from a vertex of the triangle to the opposite side (or to any line containing the opposite side), provided that the segment is perpendicular to that line.
  • An altitude is also called a height.
  • The side that is perpendicular to the height is called the base.
  • Therefore, a base and a height are always perpendicular to each other, i.e. they make a 90° angle.

Uses of triangles (i.e. why you should care)

Every polygon can be divided into triangles. So if you want to know pretty much anything about any polygon, then you can just divide it into triangles, then use your triangle knowledge.

More reading

…can be found at the Wikipedia entry for triangles.