aas: angle, angle, side
Given
\(\angle A\), \(\angle C\), \(a\) - two angles and the non-included side
Analysis
We can locate the first vertex of the triangle, \(A\) or \(C\), by constructing one of the given angles, \(\angle A\) or \(\angle C\) correspondingly. Let us assume that we have picked the given angle \(C\), constructed it and located the vertex \(C\). We can now use this vertex as a center of a circle built with the given side \(a\) as its radius. The intersection of that circle with one of the sides of the constructed angle \(C\) will locate the second vertex of the triangle, \(B\).
To locate the third vertex of the triangle we need to construct the remaining given angle, \(\angle A\) in this case, with one of its sides passing through the point \(B\). Both given angles, \(\angle A\) and \(\angle C\), share one side with each other. We do not have this shared side constructed yet. However, we do know that the side of the angle \(A\) not shared with the side of the angle \(C\) must pass through the point \(B\).
There are several ways to construct that side. One way would be to use a type of planar geometric transformation known as translation. Translation is the process of carrying every point of an object the same distance in the same direction.
In this particular case we can construct the remaining given angle, \(\angle A\), with a temporary vertex at an arbitrary point \(X\) anywhere on the side shared with the previously constructed angle \(C\). We then can slide the non-shared side of the angle \(A\) towards the point \(B\) until that side passes through that point. In the straight edge and compass constructions parlance it means constructing a straight line parallel to a given straight line passing through a given point. That line will intersect the non-shared side of the angle \(C\) at the third vertex, \(A\).
Construction Outline
Construct the given angle \(C\) with the vertex at an arbitrary point \(C\). Cut the given side \(a\) on one of the sides of the given angle \(C\) at the vertexes \(C\) and \(B\). Construct the second given angle, \(A\), on the side of the angle \(C\) opposite to \(a\) with the vertex at an arbitrary point \(X\). Construct a straight line passing through the vertex \(B\) parallel to the side of the angle \(A\) not shared with the angle \(C\). The intersection of that line with the side of the given angle \(C\) opposite to \(a\) locates \(A\)
Sample Construction
$$C, Z$$
$$Ln(C, Z)$$
$$Y \colon \quad \angle YCZ = \angle C$$
$$Ln(C, Y)$$
$$Cir(C, a)$$
$$Cir(C, a) \; \cap \; Ln(C, Z) = B \colon \quad CB = a$$
$$X \in Ln(C, Y)$$
$$W \colon \quad \angle WXC = \angle A$$
$$Ln(W, X)$$
$$U \colon \quad Ln(B, U) \; \| \; Ln(W, X)$$
$$Ln(B, U) \; \cap \; Ln(C, Y) = A$$
$$\triangle ABC$$
\(\blacksquare\)