Remember triangle congruence in Geometry? In what ways can you prove two triangles congruent? SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), AAS (angle-angle-side), HL (hypotenuse-leg).
But why can’t you use SSA (side-side-angle)?
BECAUSE IT’S THE AMBIGUOUS CASE! (What a great SAT word - ambiguous - meaning “unclear”)
Let’s take 2 sides of a triangle and the non-included angle (SSA).
We know the lengths of the two green legs and angle A, so the dotted line is the path that line AC will take even though we don’t know how long it will be.
But we don’t know angle B, so line BC can swing either left
Notice that angle B could be either acute or obtuse. When you use the Law of Sines to find angle B, the calculator only gives you an acute angle. This will certainly be one value of angle B, OR THE ANGLE YOU CALCULATE MIGHT BE THE REFERENCE ANGLE OF OBTUSE ANGLE B!
How do you know if the obtuse angle is really possible ? This is the AMBIGUOUS part. You have to “think” about it, or rather, you need to "calculate" about it.
Take the angle you know (angle A) and add the angle B that was the result of the Law of Sines. By subtracting the sum from 180°, you will have one value of angle C.
Now use the angle calculated through the Law of Sines as the reference angle to find the other possible value of angle B. Add known angle A to it. Are there any degrees left for angle C? (Is the sum still less than 180°?) If angle A plus obtuse angle B is less than 180 degrees, you have a second possible triangle from the given information.
If you’re a fan of math, the rigorous explanation might appeal to you:
Given 2 sides and the non-included acute angle and the length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle (but less than the length of the adjacent side), this is the ambiguous case and two different triangles can be constructed.
Given: angle A, where angle A < 90°
If BC > AB(sin A)
and BC < AB,
the you have an AMBIGUOUS CASE.
Here are a couple of questions to add depth to the concepts relating to the Ambiguous Case:
1. The triangle congruence rule AAS (angle-angle-side) is really a special application of another triangle congruence rule. Which one is it? Explain why. (Hint: Think of the "no choice" rule.)
2. Why does the calculator only give you one possible value of angle B? (Hint: Think about the quadrants of inverse sine, inverse cosine, and inverse tangent.)
3. What unique characteristic of the unit circle (and inverse sine) applies when you use acute angle B as a reference angle? (Hint: Think about the sum of the degrees in a triangle and the degrees in a straight line or semicircle.)
4. The rigorous explanation emphasizes that BC must be shorter than AB. Why? (Hint: Look at the diagrams of BC swinging left and right. What would happen if BC was longer than AB?)
5. Why does the rigorous explanation stress that the known angle must be less than 90°? Why couldn't it be 90°? (Hint: Think about HL.) or obtuse? (Hint: Think about questions 1, 2, 3, and reference angles.)
When you think about it, there really isn't anything new in your math study......just more ways of using what you already know!!