SOLUTION: In a triangle, the measure of the second angle is six times the measure of the first angle. The measure of the third angle is 18 degrees less than the sum of the measures of the s

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Hello There:
We have three unknowns (the sides of the triangle), so we need to assign three variable names for these unknowns and then come up with three equations to solve the whole system.
Let x = the first angle
Let y = the second angle
Let z = the third angle
From the problem information, we can write:
y = 6*x
and
z = y + x^2 - 18
Of course, we can also write:
x + y + z = 180
Okay, these are our three equations.
Substitute 6*x for y in the last two equations to get:
z = 6*x + x^2 - 18
and
z = 180 - x - 6*x
Setting these two equal, we get:
6*x + x^2 - 18 = 180 - x - 6*x
This is a quadratic equation. You can solve it to find that x is 9.
Since y = 6*x, that means y = 54.
180 - 9 - 54 = 117
The three sides are 9, 54, and 117.
You can check these values with the original information in the problem to see that they work.
~ Mark