Chapter 13.5 Law of Sines S t T 20 R r AAS 40 49 o o Step 1 find the third angle 40 + 49 + m<T= 180 o o o m<T = 91 o Step 2 find the unknown side lengths SinR r = SinS s SinS s = SinT t Sin49 r Sin40 20 = Sin40 20 Sin91 t = rSin40 = 20Sin49 tSin40 = 20sin91 r = 20Sin49 Sin40 t = 20Sin91 Sin40 r = 23.5 t = 31.1 AAS E F D 9 23 141 e f o o Step 1 !! FIND THE OTHER ANGLE !! m<D = 180 - 141 - 23 = 16 o o o o Step 2 find the unknown side lengths Sin16 9 = Sin141 e Sin16 = Sin 23 9 f e = 9Sin141 Sin16 f = 9Sin23 Sin16 e = 20.5 f = 12.8 SinA SinB SinC a b c AAS Triangle that shows 2 angles and a side ASA Shows 2 anlges and 1 side But Wait... There's More... The Infamous... SSA AKA the ambiguous case zero, one, or two triangles are possible with SSA -one of the reasons why it is called the ambiguous case a b h A a<h No triangle Ambiguous Case

Possibel Triangles <A is Acute <A is RIght or Obtuse a=h A b a=h

One triangle A b h a a h<a<b

Two triangles h a a>b

One triangle b A a b A a<b

No triangle A a b a>b

One triangle _ _ Follow These Steps Step 1 Step 2 Step 3 use a, b, and m<A to determin the possible triangles if there is one triangle use the Law of sines to solve for the unknown if there are two triangles, use the Law of Sines to find m<B and m<B, then use the values to find the measures of the other two triangles How Does This Apply To The Real World? OK... But... Actually the Law of Sines is used extensively in Land Surveying, the land is divided into triangles then two measure one side and two angles then find the remaining sides This meathod was used to to find that Mt. Everest was the highest mountain on the planet

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