Remember how there was one congruency law you were never, ever, ever allowed to use in Geometry?
… it spelled the “bad word”!
Oh yeah, I remember. So?
Well, today is your lucky day! We finally, after all these years, get to use it and shout it from the rooftops!
What! You mean, SSA (i.e., the donkey theorem) is legit?
Yes!
Just because you couldn’t use SSA to prove two triangles are congruent, doesn’t mean it doesn’t hold a special place in the world of mathematics – as this lesson demonstrates. This type of triangle is called the Ambiguous Case!
Wait a minute! Why are you calling it ambiguous?
Ambiguous means that something is unclear or not exact or open to interpretation. So, if we encounter a triangle that has SSA congruency, we have an ambiguous triangle in the sense that we need to investigate more thoroughly.
Law of Sines Ambiguous Case
Why?
Because, SSA triangles can yield us one triangle, two triangles, or no triangles!
What? Why?
Remember how the sine function is positive in both the first and second quadrants? Well, that means that the sine of an acute angle (first quadrant) has the same value as the sine of an obtuse angle (second quadrant). And, this is why we could have two different triangles, as Math Bits Notebook accurately points out.
You mean, we have to solve for possibly more than one triangle?
Yep!
How do we do that? Is this going to be hard?
The process for solving Law of Sines: Ambiguous Case Triangles is really simple because all you have to do is grab some FRUIT!
Fruit?
It’s my acronym for how to solve Triangles involving the Ambiguous Case, and it’s really easy.
So, how do you find “FRUIT” and solve these types of triangles?
… watch the video, and find out!
Ambiguous Case Worksheet – [THE FRUIT METHOD]
- Ambiguous Case Resource: This handout includes the Ambiguous Case Formula, Steps for solving SSA Congruency, and 3 practice problems with solutions.
Ambiguous Case – Video
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