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

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.

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.

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

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

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

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

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

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.