What logic rule is used to show $(p \lor q) \land (p)=p~$? 
What logic rule is used to show: $(p \vee q) \wedge (p)=p~$?

This is obvious, because if p is true the whole expression is true, and if p is false, the whole expression is false, but I'm not sure which logic rule is used to draw this conclusion.

The statement to be proved is that
$$p \equiv p \land (p \lor q),$$
which from the definition of the biconditional can be broken into 2 statements
\begin{equation} \label{1} \tag{1}  p \rightarrow  p \land (p \lor q) \end{equation}
and
\begin{equation} \label{2} \tag{2} p \land (p \lor q) \rightarrow p. \end{equation}
$\ref{2} \ $ follows very simply, we just use the rule of simplification or conjunction elimination to conclude $p$ immediately.
For $ \ \ref{1} \ $ we start with $p$ and use addition or disjunction introduction to write $p \lor q$, and then combine the 2 expressions using conjunction introduction to conclude $ p \land (p \lor q)$
