In Geometry, we found that we could prove two triangles congruent using:

SAS  Side, Angle, Side
ASA  Angle, Side, Angle
AAS  Angle, Angle, Side
SSS  Side, Side, Side
HL  Hypotenuse Leg for Right Triangles. 

We also discovered that SSA did not
work to prove triangles congruent.
We politely called it the Donkey Theorem ;  ) 

By definition, the word
ambiguous means open to two or more
interpretations.
Such is the case for certain solutions when working with the Law of
Sines.

• If you are given two
angles and one side (ASA or AAS),
the Law of Sines will
nicely provide you with ONE solution
for a missing side. 

• Unfortunately, the
Law of Sines has a problem dealing
with SSA.
If you are given two sides and one angle (where you
must find an angle), the Law of Sines could possibly provide
you
with one or more solutions, or even no solution. 
Before we investigate this situation,
there are a few facts we need to remember.
Let's look at some cases. In each
example, decide whether the given information points to the existence of
one triangle, two triangles or no triangles.
Example 1:
In , a = 20,
c = 16, and m<A = 30º. How many distinct triangles can
be drawn given these measurements?
C = sin^{1} (0.4) = 24º
(to the nearest degree)  in Quadrant I.
Sine is also positive in Quadrant II. If we use the reference
angle 24º in Quadrant II,
the angle C is 156º.
But, with m<A = 30º and m<C = 156º the sum of the angles
would exceed 180º.
Not possible!!!!
Therefore, m<C = 24º, m<A
= 30º, and m<B = 126º and only ONE
triangle is possible.
Example 2:
In , a = 7,
c = 16, and m<A = 30º. How many distinct triangles can
be drawn given these measurements?
Since sin C must be < 1,
no angle exists for angle C.
NO triangle exists for these
measurements.
Example 3:
In , a = 10,
b = 16, and m<A = 30º. How many distinct triangles can
be drawn given these measurements?
B = sin^{1}(.8) =
53.13010 = 53º.
Angles could be 30º, 53º, and 97º : sum 180º
The angle from Quadrant II could create angles 30º, 127º, and 23º :
sum 180º
This example is the
Ambiguous Case.
The information given is the postulate SSA (or ASS, the Donkey Theorem), but the two triangles that
were created are clearly not congruent. We have two triangles with
two sides and the nonincluded angle congruent, but the triangles are not congruent to
each other.
