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)
3 Side 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.
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 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
5 BOTH! 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
6 Lastly, 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
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.aθ