Find the values of x and y that make the following quadrilateral a parallelogram.
(8x-8)^degrees
(6x+14)^degrees
(6y+16)^degrees
(7y+2)^degrees
assuming the angles are listed in order, recall that consecutive angles in a parallelogram are supplementary -- they add to 180°
To determine the values of x and y that make the quadrilateral a parallelogram, we need to examine the properties of parallelograms.
A parallelogram is a quadrilateral with two pairs of parallel sides. In order for the given quadrilateral to be a parallelogram, the opposite sides must be equal.
The given quadrilateral has the following sides:
Side 1: (8x-8)
Side 2: (6x+14)
Side 3: (6y+16)
Side 4: (7y+2)
To find the values of x and y that make the quadrilateral a parallelogram, we need to set the opposite sides equal to each other and solve for x and y.
Side 1 = Side 3:
8x-8 = 6y+16
Solving for x:
8x = 6y+24
x = (6y+24) / 8
x = (3y+12) / 4
Side 2 = Side 4:
6x+14 = 7y+2
Solving for x:
6x = 7y-12
x = (7y-12) / 6
x = (7y-2) / 6
Now, we have two equations for x in terms of y. By setting those equations equal to each other, we can solve for the values of x and y:
(3y+12) / 4 = (7y-2) / 6
By cross-multiplying and simplifying, we get:
18y + 72 = 28y - 8
Moving all y terms to one side and constant terms to the other side, we have:
10y = 80
Finally, solving for y:
y = 80 / 10
y = 8
Now that we have the value of y, we can find x by substituting it into one of the equations for x in terms of y:
x = (3y+12) / 4
x = (3*8+12) / 4
x = 36 / 4
x = 9
Therefore, the values of x and y that make the quadrilateral a parallelogram are x = 9 and y = 8.
To determine the values of x and y that make the given quadrilateral a parallelogram, we need to use the properties of parallelograms.
In a parallelogram, opposite sides are parallel and congruent. Therefore, the expressions for the opposite sides of the quadrilateral should be equal to each other:
8x - 8 = 6y + 16 (Equation 1)
6x + 14 = 7y + 2 (Equation 2)
We can solve this system of equations to find the values of x and y.
Let's start with Equation 1:
8x - 8 = 6y + 16
Adding 8 to both sides:
8x = 6y + 24
Dividing both sides by 8:
x = (6y + 24) / 8
x = (3y + 12) / 4 (Equation 3)
Now, let's solve Equation 2:
6x + 14 = 7y + 2
Substituting x with (3y + 12) / 4 from Equation 3:
6((3y + 12) / 4) + 14 = 7y + 2
Multiplying both sides by 4 to clear the fraction:
6(3y + 12) + 56 = 28y + 8
Expanding and simplifying:
18y + 72 + 56 = 28y + 8
Combining like terms:
18y + 128 = 28y + 8
Subtracting 18y and 8 from both sides:
120 = 10y
Dividing both sides by 10:
12 = y
Substituting the value of y back into Equation 3 to find x:
x = (3(12) + 12) / 4
x = (36 + 12) / 4
x = 48 / 4
x = 12
Therefore, the values of x and y that make the given quadrilateral a parallelogram are x = 12 and y = 12.