5.3 The Ambiguous Case

If we are given 2 sides and one angle opposite of one of the sides, we may not get just one answer. There are 3 possibilities … one answer, 2 answers or even no answer! The arrangement of the info is SSA Let’s think about this pictorially before trying out the math. For all these examples, we will consider being given m∠A, a, and b & we remember this was not allowed in geometry C b a A B c

❸ ❶ ❷ ❼ ❹ ❺ ❻ Ambiguous Case Multi-Flow Map This is going to help keep you organized! It can tell you how many solutions there are and it can “solve the triangle.” Ambiguous Case Multi-Flow Map Breaks down here if 0 solutions ❶ ❸   Draw a picture (use the given info) Solve for the missing angle using Law of Sines Solve for the third side using Law of Sines ❷ Find the third angle: 180 - - Super unknown ❼ Known angle ❹ ❺ Solve for the third side using Law of Sines Know opp angle Find the supplement of the 2nd angle Find the third angle: 180 - - Breaks down here if 1 solution ❻ Can this happen?

Ex 1) Solve △ABC if m∠A = 35.18°, c = 17.8 and a = 11.46 If the question had asked to determine how many solutions there are, you would have done…   B 17.8 11.46 35.18° A C And the answer is: 2 triangles Yes 

Ex 2) Solve △ABC if m∠A = 71.4°, c = 51.4 and a = 45.3 If the question had asked to determine how many solutions there are, you would have done…   B 51.4 45.3 ERROR!! 71.4° A C And the answer is: 0 triangles No Triangle Exists

Ex 3) Solve △ABC if m∠A = 132°, b = 96 and a = 105 If the question had asked to determine how many solutions there are, you would have done…   C 96 105 132° A B And the answer is: 1 triangle No!