One angle in a triangle measures twice the smallest angle whereas the largest angle is six times the small angle. find the measures of all three angles.

Let's say that x is the smallest angle. 
This equation could be used to solve for the smallest angle: 
x + 2x + 6x = 180 
There a 9x total in the equation equaling 180. 
Divide 180 by 9 to get 20, which is the measure of the smallest angle. 
One angle is twice that, which is 40. 
The biggest angle is 6 times that, which is 120. 
The three angles are 40º, 20º, and 120º. 
Hope this helps!