1. Law of Sines The Ambiguous Case
Section 6-1

2. … this means we can solve them! Angle Side Angle - ASA
In geometry, triangles can be uniquely defined when particular combinations of sides and angles are specified
… this means we can solve them!
Angle Side Angle - ASA
Angle Angle Side – AAS
We solved these using Law of Sines
Then there are these theorems …
Side Side Side – SSS
Side Angle Side – SAS
We’ll soon solve these using the Law of Cosines
(section 6.2)

3. Side Side Angle - SSA There’s one left …
There’s a problem with solving triangles given SSA …
You could find…
No solution
One solution
Two solutions
In other words … its AMBIGUOUS … unclear
Let’s take a look at each of these possibilities.

4. Remember now … the information we’re given is two consecutive sides and the next angle …
If this side isn’t long enough, then we can’t create a triangle … no solution
So, then, what is the “right” length so we can make a triangle?
An altitude … 90 degree angle … a RIGHT triangle!
Turns out this is an important calculation … it’s
a = b sin θ
If a = b sin θ, then there is only one solution for this triangle.
The missing angle is the complement to θ
The missing side can be found using Pythagorean theorem of trigonometry. b a θ

5. BOTH! What if side a is a little too long … what would that look like?
a > b sin θ
This leg can then either swing left … or right.
So? … which one of these triangles do you solve? …
BOTH!
First, solve the acute triangle … and find angle B by Law of Sines!
Then solve for the remaining parts of the acute triangle a b θ B

6. Lastly, solve the obtuse triangle …
This next step is critical … angle B’ is ALWAYS the supplement to angle B.
B’ = 180 – m< B
Next, solve the remaining parts of the obtuse triangle. a b θ B’ B

7. What if side a is larger than side b?
Here’s the last scenario while θ is acute …
What if side a is larger than side b?
In this case, only one triangle can exist … an acute triangle which can easily be solved using law of sines.
Too long to create a triangle on this side. a θ