Question

find the value of ( x ), ( y ), and ( z ) in the parallelogram below.

Explanation:

Step1: Use opposite angles property

In a parallelogram, opposite angles are equal. So set $(-7y+2)^\circ = 93^\circ$.
Solve for $y$:
$$-7y + 2 = 93$$
$$-7y = 93 - 2$$
$$-7y = 91$$
$$y = \frac{91}{-7} = -13$$

Step2: Use consecutive angles property

Consecutive angles in a parallelogram are supplementary (sum to $180^\circ$). Set $(-2x+3)^\circ + 93^\circ = 180^\circ$.
Solve for $x$:
$$-2x + 3 + 93 = 180$$
$$-2x + 96 = 180$$
$$-2x = 180 - 96$$
$$-2x = 84$$
$$x = \frac{84}{-2} = -42$$

Step3: Use opposite angles property

Opposite angles are equal, so set $(-4z-5)^\circ = (-2x+3)^\circ$. Substitute $x=-42$:
$$-4z - 5 = -2(-42) + 3$$
$$-4z - 5 = 84 + 3$$
$$-4z - 5 = 87$$
$$-4z = 87 + 5$$
$$-4z = 92$$
$$z = \frac{92}{-4} = -23$$

Answer:

$x=-42$, $y=-13$, $z=-23$