1962 AHSME Problems/Problem 20
Problem
The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be:
Solution
If the angles are in an arithmetic progression, they can be expressed as
,
,
,
, and
for some real numbers
and
.
Now we know that the sum of the degree measures of the angles of a pentagon is
.
Adding our expressions for the five angles together, we get
.
We now divide by 5 to get
. It so happens that
is one of the angles we defined earlier, so that angle must have a measure of
.
(In fact, for any arithmetic progression with an odd number of terms,
the middle term is equal to the average of all the terms.)