Question

For what values of the variables must ABCD be a​ parallelogram?
5 y degrees5y°
5 x degrees5x°
left parenthesis y plus 76 right parenthesis degrees(y+76)°
left parenthesis 6 x minus 17 right parenthesis degrees(6x−17)°

Answer 1

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:
- \( y = 19 \)
- \( x = 17 \)