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Question 160671: The measure of one angle of a triangle is twice the measure of a second angle. The measure of the third angle is 12 less than the sum of the other two. Find the measures of the anles of the triangle.
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! had me going for a while until i remembered that the sum of the internal angles of a triangle are equal to 180 degrees.
once i remembered that the problem became a lot easier.
let a, b, and c be the angles of the triangle.
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let a = 2*b
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let c = a+b-12
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the above two statements are given.
since the sum of the internal angles of a triangle are equal to 180, then we have a third equation to work with which is
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a + b + c = 180
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we now have 3 equations for 3 unknowns and we should be able to solve for each of the unknowns.
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it looks like the easiest thing is to solve for b. once we find b, the rest should fall into place very nicely.
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since a = 2*b, we can substitute 2*b for a in the equation a + b + c = 180.
that equation then becomes 2*b + b + c = 180 which becomes 3*b + c = 180.
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looking at the equation c = a + b - 12, we solve for b as follows:
since we know that a = 2 * b, we can substitute 2*b for a and the equation becomes c = 2*b + b - 12 which becomes c = 3*b - 12.
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the equation we are working with is 3*b + c = 180 which originally started as a + b + c = 180.
we can substitute 3*b - 12 for c and the equation becomes
3*b + 3*b - 12 = 180 which becomes 6*b = 192 after adding 12 to both sides and combining like terms.
dividing both sides of the equation by 6 and we get b = 32 degrees.
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we have b = 32 degrees.
we have a = 2 * b so a = 64 degrees.
we have c = a + b - 12 which becomes c = 32 + 64 - 12 which becomes c = 96 - 12 which becomes c = 84 degrees.
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substituting in the equation a + b + c = 180, we get 32 + 64 + 84 = 180 which becomes 96 + 84 = 180 which becomes 180 = 180.
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answer is:
a = 64 degrees.
b = 32 degrees.
c = 84 degrees.
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