Intro to trees

$4$ Consider the following graph:

$($a$)$ Show that this is a tree by explaining how it satisfies Definition $10\mathrm{.}1\mathrm{.}2.$

$($b$)$ Verify that it satisfies the tree condition that $|E|=n-1.$

$($c$)$ Another property of trees is that adding any single edge will create a cycle. Add an edge

and find the resulting cycle.