MHB What is the value of Angle BMC in Triangle ABC?

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In triangle ABC, angles BAC and ACB are both 50 degrees, making angle ABC 80 degrees. Given point M inside the triangle, with angles MAC at 10 degrees and MCA at 30 degrees, the remaining angle AMC can be calculated as 180 - (10 + 30) = 140 degrees. Using the triangle angle sum property in triangle BMC, angle BMC can be determined as 180 - (80 + 140) = -40 degrees, which is not possible. Thus, the correct approach involves recalculating the angles or verifying the given conditions to find a valid angle for BMC.
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$\triangle ABC ,if \,\, \angle BAC=\angle ACB=50^o$

point $M$ is an inner point of $\triangle ABC ,$

given :$\angle MAC=10^o$ and $\angle MCA=30^o$

find the value of $\angle BMC=?$
 
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Albert said:
$\triangle ABC ,if \,\, \angle BAC=\angle ACB=50^o$

point $M$ is an inner point of $\triangle ABC ,$

given :$\angle MAC=10^o$ and $\angle MCA=30^o$

find the value of $\angle BMC=?$
my solution:
 

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