# Using the Sine Rule for Ambiguous Cases

Geometry Level 3

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Firstly, Solve $$\triangle ABC$$ where $$\hat { A } =55^{ \circ },\quad b_{ 1 }=b_{ 2 }=14.3㎝\quad and\quad b=16.3cm$$.

Find out the sum of all the angles and add it to the sum of all the sides. (Correct to 1 decimal places.)

i.e.,

$\hat { A } +\hat { B_{ 1 } } +\hat { B_{ 2 } } +\hat { C_{ 1 } } +\hat { C_{ 2 } } +AB_{ 1 }+AB_{ 2 }+B_{ 1 }C+B_{ 2 }C+AC\\ or\quad \\ \hat { A } +\hat { B_{ 1 } } +\hat { B_{ 2 } } +\hat { C_{ 1 } } +\hat { C_{ 2 } } +c_{ 1 }+c_{ 2 }+a_{ 1 }+a_{ 2 }+b\\$

(They both have the same value.)

For clarification, $$\hat { A } =55^{ \circ }\\ B_{ 1 }C=a_{ 1 }=14.3\\ B_{ 2 }C=a_{ 2 }=14.3\\ AC=b=16.3\\ AB_{ 1 }=c_{ 1 }\\ AB_{ 2 }=AB_{ 1 }+B_{ 1 }B_{ 2 }=c_{ 2 }\\$$

Details and Assumptions:

If your answer is $$\theta ^{ \circ }+\alpha ^{ \circ }+\beta ^{ \circ }+x$$ cm $$+y$$ cm, write your answer as $$\theta +\alpha +\beta +x +y$$.

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