Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 13: Triangles 169

In 'RST, m’S = 43r and m’R = 28r. The measure of the exterior angle of the triangle at T is
equal to the sum of the two remote interior angles, ’S and ’R. Adding the measures of those
two angles tells you the exterior angle is 43 + 28 = 71r. You could calculate the measure of ’STR
(180 – 43 – 28 = 109r), and since ’STV is supplementary to ’STR, it will be 180 – 109 = 71r.

So for any triangle, you know what the angles add up to, but you don’t know what the individual
angles measure. When you look at sides, you can again say something that’s true all the time, but
there’s a limit to how specific you can be. In any triangle, the sum of the lengths of any two sides
will be greater than the length of the third. Imagine your home is at one vertex, A, of a triangle,
and your best friend’s home is at another vertex, C. To get to your friend’s home, you could walk
down the street from A to B and then turn the corner and follow the road from B to C. Or you
could take the shortcut through the backyards straight from A to C. The shortcut is, obviously,
shorter. AB + BC > AC.

Put another way, the length of any side of a triangle is less than the sum of the other two sides.
Suppose you know a triangle has sides 4 inches long and 7 inches long. Those add to 11 inches.
The third side must be less than 11 inches. But turn things around a little bit. The unknown
side plus the 4-inch side must be more than 7 inches, so the unknown side must be more than 3
inches. It has to be more than 3 but less than 11 inches. The length of any side of a triangle is less
than the sum of the other two sides but more than the difference between them.

The longest side of a triangle is always opposite the largest angle. Remember that angles are
measured by the rotation between the sides. The wider an angle opens, the longer the line
segment it will take to connect the ends of the sides. The longest side is opposite the largest angle,
and the shortest side is opposite the smallest angle. That, of course, means that the medium-sized
side is opposite the medium-sized angle. It also means that whenever a triangle has congruent
angles, the sides opposite the congruent angles are the same length.

Shortcut

A

B

C