To determine the values of the variables such that ABCD is a parallelogram, we need to use the properties of a parallelogram. Specifically, we rely on the properties of opposite angles in a parallelogram, which states that opposite angles are equal.
Given angles:
- \( \angle A = 5y^\circ \)
- \( \angle B = 5x^\circ \)
- \( \angle C = (y + 76)^\circ \)
- \( \angle D = (6x - 17)^\circ \)
Since angles A and C are opposite angles, we can set them equal to each other:
\[ 5y = y + 76 \]
Now, subtract \( y \) from both sides:
\[ 5y - y = 76 \ 4y = 76 \ y = \frac{76}{4} = 19 \]
Next, we use the property that angles B and D are opposite angles, so we can set them equal to each other:
\[ 5x = 6x - 17 \]
Subtract \( 5x \) from both sides:
\[ 0 = x - 17 \ x = 17 \]
Consequently, the values of the variables that would make ABCD a parallelogram are:
- \( y = 19 \)
- \( x = 17 \)
To summarize:
- \( y = 19 \)
- \( x = 17 \)