September 15, 2014
5‐7 Ambiguous Cases of the Law of Sines
When you were in Geometry class you studied triangle congruence and you were told that ASA, AAS, SAS & SSS all made two congruent triangles. Everyone in your class then snickered because they thought about the ASS combination (Angle‐Side‐Side just for the record…not the other word). Your Geometry teacher (for half of you, me) took the snickering with as much good humor as possible and told you #1 We don’t curse in Geometry class and #2 SSA (she/he cleaned it up) does not work. What do you mean it doesn’t work? It doesn’t make two congruent triangles; however, that doesn’t mean SSA doesn’t exist. It just means it didn’t work to prove two triangles congruent. Why didn’t it work? What happens when you are given a triangle with one angle and you know the opposite side and an adjacent side (aka SSA)?
So today we are going to exam the SSA combination and look at a few things. #1 Will SSA always make a triangle? (Can’t do trigonometry without a triangle)
#2 Will SSA always make just one triangle or can it sometimes make two completely different triangles with the same side measurements? If so when does it make one, when does it make two and how can we tell?
Now don’t start to panic. I’m not teaching you any new math today. We are still going to use everyone’s favorite, the Law of Sines (yay!) to solve triangles.
Today we are going to explore what happens when we are given the SSA combination. We will construct ∆ABC given ∠A, side a and side c.
Sep 23­8:49 AM
0°
211
Sep 23­8:50 AM
September 15, 2014
Construction #2: Suppose ∠A is acute & side a is bigger than side c.
0°
234
Sep 23­8:51 AM
Construction #3: Suppose ∠A is acute & side a is smaller than side c
0°
217
Sep 23­8:51 AM
September 15, 2014
height - drawn from
the vertex between
the two given sides
Sep 23­8:53 AM
Sep 15­10:28 AM
September 15, 2014
Sep 15­2:50 PM