Finding Angle C in Triangle ABC

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The discussion focuses on solving for angle C in triangle ABC, where angle B is given as 30 degrees and the relationship between the sides is defined by the equation \(\overline{BC}^2 - \overline{AB}^2 = \overline{AB} \times \overline{AC}\). The problem requires applying the Law of Cosines and algebraic manipulation to derive the value of angle C. Participants provide insights into the geometric properties and calculations necessary to arrive at the solution.

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Albert1
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$\triangle ABC,\angle B=30^o , \,\,and \,\, \overline{BC}^2 - \overline{AB}^2=\overline{AB}\times \overline{AC}\\
find \,\, \angle C=?$
 
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Albert said:
$\triangle ABC,\angle B=30^o , \,\,and \,\, \overline{BC}^2 - \overline{AB}^2=\overline{AB}\times \overline{AC}\\
find \,\, \angle C=?$
hint
prove $\angle A=2\angle C$
 
Albert said:
$\triangle ABC,\angle B=30^o , \,\,and \,\, \overline{BC}^2 - \overline{AB}^2=\overline{AB}\times \overline{AC}---(1)\\
find \,\, \angle C=?$
more hint:
in fact $\angle B=30^o$ is not important, you should prove for any triangle if $\angle A=2\angle C $ then (1) will meet
 
Albert said:
more hint:
in fact $\angle B=30^o$ is not important, you should prove for any triangle if $\angle A=2\angle C $ then (1) will meet
my solution :
View attachment 5446
 

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