In a triangle, we can order the lengths of the legs from shortest to longest if we know what the three angles of the triangle are.  There is a nice rule that says, in a triangle, the shortest leg is opposite the smallest angle, the longest leg is opposite the largest angle, and the middle-length leg is opposite the middle-sized angle.

If you want to see an original proof of this law that is relatively easy to understand, you can check out Proposition 18 and 19 from Book I of Euclid’s Elements.  Born thousands of years ago, Euclid was the “original geometer” and he discovered most of the basic properties of geometry that we still use to this day.

Let’s use the rule to help us order the sides of triangles from shortest to longest.

Example:  Order the sides of the triangle from shortest to longest.

SOLUTION:  By our rule, we know that the shortest side must be \(\overline {EF} \), since it is opposite the smallest angle.  Next, we know that the largest side must  be \(\overline {EG} \), since it is opposite the largest angle.  Finally, the middle side will be \(\overline {FG} \).

Then the sides of the triangle, from shortest to longest, are \(\overline {EF} \), \(\overline {FG} \), \(\overline {EG} \).

EXAMPLE:  Order the sides of the triangle from shortest to longest.

SOLUTION:  By our rule, we know that the shortest side is \(\overline {UW} \), since it is opposite the smallest angle.  Similarly, we know that the largest side must be \(\overline {VW} \), since it is opposite the largest angle.  Then the middle sized side will be \(\overline {UV} \).

Then the sides of the triangle, in order from shortest to longest, are \(\overline {UW} \), \(\overline {UV} \), \(\overline {VW} \).

Below you can download some free math worksheets and practice.