How to show a bound quiver algebra of a finite, connected and acyclic quiver is a directed algebra? Let $A$ be an algebra over a field K. A path of length $s > 1$ in mod(A) is a tuple $(X_1,X_2, \dots ,X_s)$ of indecomposable $A$-modules such that for each $1 \leq i \leq s-1$ there exists a non-zero and non-invertible homomorphism $X_i \rightarrow X_{i+1}$. The algebra $A$ is called a directed algebra if there is no path $(X_1,X_2, \dots ,X_s)$ of length $s > 1$ in mod(A) with $X_1 \cong X_s$. 
Now suppose $A$ is a bound quiver algebra of a finite, connected and
acyclic quiver, how to get $A$ is a directed algebra?
 A: What you ask for is not true. For example consider the path algebra of the Kronecker quiver. The Kronecker quiver has two vertices $1,2$ and two arrows $1\to 2$. Then, this is not a directed algebra. For example, there is the sequence $(1,\lambda)\to \left(\begin{pmatrix}1&0\\0&1\end{pmatrix},\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}\right)\to (1,\lambda)$ of non-zero homomorphisms. Here, I give a representation in terms of the maps $(f,g)$ on the arrows $1\to 2$ (and $k$, resp $k^2$ on the vertices). 
In fact what you call directed (and is sometimes also called representation-directed since some people also call everything Morita equivalent to the bound quiver of a finite acyclic quiver directed, other call that triangular).
Even more, every finite dimensional representation-directed algebra is in fact representation-finite, so there are plenty of counterexamples. What is true is the converse: Every representation-finite algebra (over an algebraically closed field) is Morita equivalent to the path algebra of a finite acyclic quiver with relations.
Possible definitions of directed, in the sense of triangular, include that the Gabriel quiver of the algebra is acyclic, or that the algebra is quasi-hereditary with simple standard modules. One can equivalently characterise this as there exists no path $P_1\to P_2\to \dots\to P_n\to P_1$ between indecomposable projectives, where the maps are non-zero non-isomorphisms. The abstract argument for this is that the projective modules also form an exceptional collection (they are the costandard modules for the quasi-hereditary algebra). Thus, only path of length smaller than $n$ can exist. More concretely a non-zero non-isomorphism $P_i\to P_{i+1}$ is left multiplication by a linear combination of paths in the quiver. But since the quiver is acyclic, such sequence can never go in the opposite direction.
