Relationships between Sides and
Angles in Triangles
There is a special relationship between the lengths of the sides of a triangle
and the relative measures of each side’s opposite angle. The longest side of a tri-
angle is always opposite the largest angle, the shortest side is always opposite the
smallest angle, and the middle-length side is always opposite the middle-sized angle.
In the case that two angles are equal, then the two sides that are opposite
them will also be equal, and vice versa. Therefore, it is impossible for the
largest angle of a triangle to be opposite anything but the largest side. For
example, look at the following triangle, ΔPDX. Since PXis the largest side,
∠D will be the largest; since PDis the shortest side, ∠X will be the short-
est; and lastly, since DXis the middle side, ∠P will be the middle in size.
We can use the angles in ΔMAE below to get similar information. Since
∠M = ∠E, then you can be certain that MA= EA.
The last fact to keep in mind about triangles is that sum of any two sides
of a triangle must be greater than the length of the remaining third side. For
example, if two sides of a triangle are 8 and 10 the third side must be shorter
than 18. If it were longer than 18, the other two sides would not be able to
connect to the side’s endpoints. This is illustrated below in ΔAUG; notice
that when AG18, it is too long for the triangle to close.
MA
E65 °65 °DP X172240501 Geometry Questions