For what values of the variables must abcd be a parallelogram?

ok first things first. In all parallelograms, you now that opposite angles are numerically equal to each other and that the sum off all four angles must add up to 360.

Therefore based off the picture, angles B and D are equal to each other, and A and C are equal to each other.

(2x-10) = (y+20) B = D
y=2x-30

Now B = 2x -10 and D = 2x-30+20 = 2x-10

Now A = C

Since A = 2x+ 10, we can assume C = 2x + 10

Now we need to find the value of x

so our angle A, B, C, and D add up to 360 so the "master equation" is:

(2x - 10) + (2x - 30 + 20) + (2x + 10) + (2x + 10) = 360
2(2x -10) + 2(2x + 10) = 360
4x - 20 + 4x + 20 = 360
8x = 360
x = 45

Now that we know what x is, we can go back to the angles B and D

remember B = 2x - 10 and D = y + 20

and earlier we stated that y=2x-30
therefore y = 2(45) - 30
Y = 60

Now if you wanted to double check, all you have to do is plug in the x and y values into the equations and add up all the angles together and they should add up to 360.

I know I kinda went into too many details, but I'd rather explain it into too much detail rather than too few details and shortcuts that you wouldn't be able to follow and understand.

I hope this was helpful =D