To determine the values of the variables \( x \) and \( y \) that will make quadrilateral ABCD a parallelogram, we need to use the property that opposite angles of a parallelogram are equal.
Let’s assign the angles as follows:
- Angle A is \( 5y^\circ \)
- Angle B is \( 5x^\circ \)
- Angle C is \( (y + 96)^\circ \)
- Angle D is \( (6x - 12)^\circ \)
Since ABCD is a parallelogram, we need to satisfy the following conditions:
- \( A + C = 180^\circ \)
- \( B + D = 180^\circ \)
Let's establish the equations based on these angles:
Equation 1:
From angles A and C: \[ 5y + (y + 96) = 180 \]
Combining like terms: \[ 6y + 96 = 180 \] Subtract 96 from both sides: \[ 6y = 84 \] Now, divide by 6: \[ y = 14 \]
Equation 2:
From angles B and D: \[ 5x + (6x - 12) = 180 \]
Combining like terms: \[ 11x - 12 = 180 \] Now, add 12 to both sides: \[ 11x = 192 \] Divide by 11: \[ x = \frac{192}{11} \approx 17.45 \]
Conclusion:
Thus, for quadrilateral ABCD to be a parallelogram, the values of the variables must be:
- \( y = 14 \)
- \( x = \frac{192}{11} \) or approximately 17.45.