Question

The angles of a pentagon are in arithmetic progression
Which is the middle term of this progression?

Answer 1

Step-by-step explanation:

If the angles are in an arithmetic progression, they can be expressed as $a$, $a+n$, $a+2n$, $a+3n$, and $a+4n$ for some real numbers $a$ and $n$. Now we know that the sum of the degree measures of the angles of a pentagon is $180(5-2)=540$. Adding our expressions for the five angles together, we get $5a+10n=540$. We now divide by 5 to get $a+2n=108$. It so happens that $a+2n$ is one of the angles we defined earlier, so that angle must have a measure of $\boxed{108\textbf{ (A)}}$. (In fact, for any arithmetic progression with an odd number of terms, the middle term is equal to the average of all the terms.)