Question 1120157: The largest angle in a triangle is twice the degree measure of the second largest angle. one-third of the largest angle is 10 degrees larger than the difference of the other two. What is the measure in degrees of the smallest angle?
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x = the largest angle.
let y = the next largest angle.
let z = the smallest angle.
The largest angle in a triangle is twice the degree measure of the second largest angle.
x = 2y
one-third of the largest angle is 10 degrees larger than the difference of the other two.
1/3 * x = 10 + y - z
What is the measure in degrees of the smallest angle?
since x = 2y,then y = x/2.
1/3 * x = 10 + y - z becomes:
1/3 * x = 10 + x/2 - z.
since the sum of the angles in a triangle is equal to 180 degreees, then:
x + y + z = 180.
solve for z to get z = 180 - x - y
since y = x/2, this becomes z = 180 - x - x/2.
combine like terms to get z = 180 - 3x/2.
the equation of 1/3 * x = 10 + x/2 - z becomes 1/3 * x = 10 + x/2 - (180 - 3x/2).
simplify to get 1/3 * x = 10 + x/2 - 180 + 3x/2.
multiply both sides of this equation by 6 to get:
2x = 60 + 3x - 1080 + 9x
subtract 2x from both sides of this equation and subtract 60 from both sides of this equation and add 1080 to both sides of this equation and combine like terms to get:
1080 - 60 = 3x + 9x - 2x
combine like terms to get 1020 = 10x
solve for x to get x = 1020/10 = 102
since x is 102 and y is half of x, then y is 51.
since 1/3 * x = 10 + y - z, and x = 102 and y = 51, then:
1/3 * 102 = 10 + 51 - z
solve for z to get z = 10 + 51 - 1/3 * 102
this results in z = 27
your largest angle is 102.
your next largest angle is 51.
your smallest angle is 27.
your next largest angle is 1/2 of your largest angle.
102/2 = 51
1/3 * your largest angle is equal to 10 + the difference between the other two angles.
1/3 * 102 = 10 + 51 - 27
this results in 34 = 34, confirming the relationship is correct.
you were asked What is the measure in degrees of the smallest angle?.
the answer is 27 degrees.
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website! The largest angle in a triangle is twice the degree measure of the second largest angle. one-third of the largest angle is 10 degrees larger than the difference of the other two. What is the measure in degrees of the smallest angle?
Let measures of smallest, middle, and largest angles be S, M, and L, respectively
Then L = 2M ------ eq (i)
Also, 
L = 3M - 3S + 30 --------- Multiplying by LCD, 3
2M = 3M - 3S + 30 -------- Replacing L with 2M
3S - 30 = M ------- eq (ii)
In addition, S + M + L = 180______S + M + 2M = 180______S + 3M = 180 ------ eq (iii)
S + 3(3S - 30) = 180 ------- Replacing M with 3S - 30 in eq (iii)
S + 9S - 90 = 180
10S = 270
S, or measure of smallest angle = 
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