1. Ambiguous Case Triangles
Can given numbers make a triangle? Can a different triangle be formed with the same information?

2. Conditions for Unique Triangles
AAS
ASA
two angles must sum to less than 180º
two angles must sum to less than 180º
SSS
SAS
two shortest sides are longer than the third side
Any set of data that fits these conditions will result in one unique triangle.

3. Ambiguous Triangle Case (aka the ‘bad’ word)
SSA A a b
This diagram is deceiving -- side-side-angle data may result in two different triangles.
Side a is given but it might be possible to ‘swing’ it to either of two positions depending on the other given values.
An acute or an obtuse triangle may be possible.

4. Both values of C are possible, so 2 triangles are possible
Example (2 triangles)
Given information
Solve for sin B
Set up Law of Sines
mA = 17º
a = 5.8
b = 14.3
mB  46º
Find mB in quadrant I
Find mB in quadrant II
mB  180 – 46 = 134º
mC  (180 – 17 – 46)  117º
Find mC
mC  (180 – 17 – 134)  29º
Find mC
Both values of C are possible, so 2 triangles are possible

5. Only one value of C is possible, so only 1 triangle is possible
Example (1 triangle)
Given information
Solve for sin B
Set up Law of Sines
mA = 58º
a = 20
b = 10
mB  25º
Find mB in quadrant I
Find mB in quadrant II
mB  180 – 25 = 155º
mC  (180 – 58 – 25)  97º
Find mC
mC  (180 – 58 – 155)  -33º
Find mC
Only one value of C is possible, so only 1 triangle is possible

6. No value of B is possible, so no triangles are possible
Example (0 triangles)
Given information
Solve for sin B
Set up Law of Sines
mA = 71º
a = 12
b = 17
No value of B is possible, so no triangles are possible

7. Law of Sines Method 1) Use Law of Sines to find angle B
-If there is no value of B (for example, sin B = 2), then there are no triangles
Remember, sin x is positive in both quadrant I and II
2) Determine value of B in quadrant II
(i.e. 180 – quadrant I value)
3) Figure out the missing angle C for both values of angle B by subtracting angles A and B from 180
4) If it is possible to find angle C for
-both values of B, then there are 2 triangles
-only the quadrant I value of B, then only 1 triangle is possible

8. Solve a triangle where a = 5, b = 8 and c = 9
Draw a picture.
This is SSS 9  5
Do we know an angle and side opposite it? No, so we must use Law of Cosines. 
84.3  8
Let's use largest side to find largest angle first.
minus 2 times the product of those other sides
times the cosine of the angle between those sides
One side squared
sum of each of the other sides squared
CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction

9. How can we find one of the remaining angles?
Do we know an angle and side opposite it?  9
62.2 5 
84.3 
33.5 8
Yes, so use Law of Sines.

10. OR….. If you have SSA: (h = b Sin A) No triangles exist if:
a < b
-- One triangle exists if:
a = h
a > b
Two triangles exist if:
h < a < b