The ‘SSA’ triangle condition (two sides and a nonincluded angle),
does not uniquely identify a triangle.
Given two positive real numbers (two side lengths)
and a degree measure strictly between $\,0^\circ\,$ and $\,180^\circ\,$ (the angle),
there may be no triangle, exactly one triangle, or two triangles
that match the SSA information.
Let $\,a\,$ and $\,b\,$ be positive real numbers, used to denote lengths of two sides of a triangle. Let $\,\theta\,$ denote the measure of a nonincluded angle. As shown at right, $\,a\,,$ $\,b\,$ and $\,\theta\,$ form an ‘SSA’ situation—two sides and a nonincluded angle.
To significantly cut down on words, I say things like:
Note that:


In the following progression of sketches, $\,b\,$ and $\,\theta\,$ are held constant, and $\,\theta\,$ is an acute angle.
The side of length $\,a\,$ gets bigger and bigger:
$0 < a < b\sin\theta$When $\,a\,$ is less than $\,b\sin\theta\,,$ it is too short to reach the dashed line.No matter what the angle is between $\,a\,$ and $\,b\,,$ the triangle cannot be completed. For $\,0 < a < b\sin\theta\,,$ there is no triangle determined by sides $\,a\,$ and $\,b\,$ and a nonincluded angle $\,\theta\,.$ 
$0 < a < b\sin\theta$ no triangle determined by $\,a\,,$ $\,b\,$ and $\,\theta\,$ 

$a = b\sin\theta$When $\,a\,$ reaches the altitude length ($\,a = b\sin\theta\,$)then it is just long enough to hit the dashed line and form a (right) triangle. For $\,a = b\sin\theta\,,$ there is exactly one (right) triangle determined by sides $\,a\,$ and $\,b\,$ and a nonincluded angle $\,\theta\,.$ 
$a = b\sin\theta$ exactly one (right) triangle determined by $\,a\,,$ $\,b\,$ and $\,\theta\,$ 

$\,b\sin\theta < a < b\,$This is the most interesting case!This case is the reason there is no ‘SSA’ congruence theorem. For values of $\,a\,$ strictly between $\,b\sin\theta\,$ and $\,b\,,$ the circle of radius $\,a\,$ intersects the dashed line at two different points. Therefore, there are two different triangles that meet the SSA information. For $\,b\sin\theta < a < b\,,$ there are two different triangles determined by sides $\,a\,$ and $\,b\,$ and a nonincluded angle $\,\theta\,.$ 
two different triangles determined by $\,a\,,$ $\,b\,$ and $\,\theta\,$ 

$a = b$For $\,a = b\,,$ there is exactly one isosceles triangledetermined by sides $\,a\,$ and $\,b\,$ and a nonincluded angle $\,\theta\,.$ 
$a = b$ exactly one isosceles triangle 

$a > b$When $\,a\,$ is greater than $\,b\,,$ then the circle of radius $\,a\,$ intersects the dashed lineat two different points. However, only one of these points gives a triangle with interior angle $\,\theta\,.$ For $\,a > b\,,$ there is exactly one triangle determined by sides $\,a\,$ and $\,b\,$ and a nonincluded angle $\,\theta\,.$ 
$a > b$ exactly one triangle 
This graphic summarizes the ‘SSA’ situations discussed above.
Side $\,b\,$ and angle $\,\theta < 90^\circ\,$ are fixed.
The number line shows the values of $\,a\,$ corresponding to no, one, and two triangles.
The red (no $\,\triangle\,$),
blue (two $\,\triangle\,$s),
and black (one $\,\triangle\,$)
intervals change
depending upon the values of $\,b\,$ and $\,\theta\,.$
Put in your own values for $\,b\,$ and $\,\theta\,$ below—have fun!
Type in desired values for $\,b\,$ and $\,\theta\,$ below. Then, click the ‘Create Number Line Summary’ and ‘See the Triangle’ buttons. 

Note:
If $\,\theta\,$ is an obtuse angle (or a right angle), then the situation is different.
In this case, no triangle exists until $\,a > b\,,$ at which point there is a unique triangle:
Get a random SSA situation by clicking below. Sketch the given situation on a piece of paper—make it roughly to scale, but don't actually measure anything. Based on your sketch, make a conjecture as to how many triangle(s) meet this SSA configuration. Then, check your conjecture. 

You might be lamenting:
How many triangles have a $\,50^\circ\,$
angle with opposite side $\,4\,,$
and another side of length $\,7$?

How many triangles have a
$\,50^\circ\,$
angle with opposite side
$\,6\,,$
and another side of length $\,7$?

How many triangles have a
$\,50^\circ\,$
angle with opposite side $\,8\,,$
and another side of length
$\,7$?

When naming triangles, it is common to use:
When we write things like ‘$\,\sin D\,$’ or ‘$\,180^\circ  D\,$’, then $\,D\,$ refers to the measure of the angle at vertex $\,D\,.$ Precalculus triangle notation is simpler than Geometry triangle notation. Due to the nature of the work in Geometry, careful distinctions need to be made between points, angles, and numbers:
It is a nuisance to have to write things like ‘$\,\sin(m\angle P)\,$’. Cumbersome notation (without a need for such) can make simple things look hard. So—in Precalculus—when there is no possible confusion, we use simple notation and let context determine the correct interpretation. Here are some Precalculus examples:


Suppose a triangle has a $\,27^\circ\,$ angle with opposite side of length $\,23\,.$
Another side has length $\,30\,.$ If possible, solve the triangle.
Round angles to one decimal place and sides to two decimal places.
Any value used inside an arcsine should be rounded to at least four decimal places.
It's useful to make a sketch to define notation. (Try to do it approximately to scale, but certainly don't measure anything!) Note that this is an ‘SSA’ situation, and be careful! Make a conjecture about the number of triangles determined by the SSA information. Looks like a two triangle situation! 
It's also useful to use a table (or tables) to organize results:
The table(s) will be filled in as computations are completed. In this example, the tables are repeated, so you can easily see each step where information is added. When doing problems like this on your own, you won't repeat the table(s)—you'll just fill them in as you go. 











You can use
WolframAlpha to do the computations.
For example, you can type: to compute: $$ \cssId{sb156}{\frac{23\sin 116.7^\circ}{\sin 27^\circ}} $$ 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
