Try constructing these figures with Sketchpad to explore some of the possibilities for building triangles from sides and angles.
Draw a figure with segments AB and BC. Then this gives two sides and the included angle that can be varied dynamically by dragging A, B or C. Then the segment AC forms a triangle in every case, except when the points are collinear (so the angle does not exist).
This time draw segment AB but also two rays AH and BG to form the two angles.
Then when the rays intersect at C, there is exactly one triangle ABC formed. Sometimes the rays do not intersect. You can explore this by measuring both angle ABG and BAH and then adding the angles to observe the angle sum when the rays do not meet.
Start with a figure with 3 segments, like this:
Label the endpoints of c as A and B. Then construct a circle with center A and radius b using the Construct Menu. Likewise construct a circle with center B and radius a. Let the two intersection points of the circles be C and D (you may have to drag something to make them intersect).
Then there are two triangles ABC and ABD formed with the given side lengths, but the two triangles are congruent.
Explore when a triangle is formed and when it is not.
Study the quadrilateral ACBD. This shape is called a kite. What relationships exist among the sides of this shape?
The easiest way of creating a figure for AAS uses a fact about parallel lines that we have not formally studied yet, but we can still do the experiment.
Construct a figure with segment AB, ray BF so that angle ABF is formed, then ray FG so that angle BFG is formed. This gives the S = AB and the two angles.
Now to experiment, construct a point H on ray BF and construct the line through H parallel to FG. Then slide H along the ray until it visually passes through A. This will happen for only one H (at most).
After this exploration, you may see how to construct the triangle exactly. Construct the line through A parallel to FG. Let C be the intersection of this line with ray BF.
This is probably the most interesting case that fails as a congruence test. There is a visual explanation at this link to a Java Sketchpad page.
Explore this page. If you wish, as a challenge, try to create a Sketchpad figure that behaves the same way.