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 - ASAAngle Angle Side – AASWe solved these using Law of SinesThen there are these theorems …Side Side Side – SSSSide Angle Side – SASWe’ll soon solve these using the Law of Cosines(section 6.2)
3Side Side Angle - SSA There’s one left … There’s a problem with solving triangles given SSA …You could find…No solutionOne solutionTwo solutionsIn other words … its AMBIGUOUS … unclearLet’s take a look at each of these possibilities.
4Remember 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 solutionSo, 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’sa = b sin θIf a = b sin θ, then there is only one solution for this triangle.The missing angle is the complement to θThe missing side can be found using Pythagorean theorem of trigonometry.baθ
5BOTH! What if side a is a little too long … what would that look like? a > b sin θThis 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 triangleabθB
6Lastly, solve the obtuse triangle … This next step is critical … angle B’ is ALWAYS the supplement to angle B.B’ = 180 – m< BNext, solve the remaining parts of the obtuse triangle.abθB’B
7What 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.aθ