Question

find the value of x, y, and z in the parallelogram below.
(3x-6)°
111°
(8y-1)°
(-4z-3)°
answer attempt 1 out of 2
x =
y =
z =

Explanation:

Step1: Find x using opposite angles in parallelogram

In a parallelogram, opposite angles are equal. So, \(3x - 6 = 111\)
\(3x = 111 + 6\)
\(3x = 117\)
\(x = \frac{117}{3}\)
\(x = 39\)

Step2: Find y using consecutive angles in parallelogram

Consecutive angles in a parallelogram are supplementary (sum to \(180^\circ\)). So, \(8y - 1 + 111 = 180\)
\(8y + 110 = 180\)
\(8y = 180 - 110\)
\(8y = 70\)
\(y = \frac{70}{8} = 8.75\) (Wait, correction: Wait, actually, the angle \(8y - 1\) should be equal to the angle opposite to it? Wait no, in the parallelogram, let's recheck. Wait the angles: the angle \(3x - 6\) and \(111^\circ\) are opposite? Wait no, looking at the diagram, the angles: let's see, the angle \(3x - 6\) and \(-4z - 3\) – no, wait, in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. Wait the angle \(111^\circ\) and \(8y - 1\): are they consecutive? Wait no, let's re-express. Wait the angle \(3x - 6\) and \(111^\circ\) – maybe \(3x - 6\) is opposite to \(111^\circ\), so \(3x - 6 = 111\) (correct as step1). Then the angle \(8y - 1\) and \(-4z - 3\) – wait, no, consecutive angles: \(111^\circ\) and \(8y - 1\) should be supplementary? Wait no, maybe I made a mistake. Wait, in a parallelogram, opposite angles are equal, so angle \(A =\) angle \(C\), angle \(B =\) angle \(D\). Also, angle \(A +\) angle \(B = 180^\circ\). So let's label the parallelogram: let's say the top angle is \(3x - 6\), right angle is \(111^\circ\), bottom angle is \(-4z - 3\), left angle is \(8y - 1\). So top and bottom are opposite, right and left are opposite? Wait no, top and bottom: no, in a parallelogram, opposite angles are (top, bottom) and (left, right). Wait, no, adjacent angles are consecutive. So top angle (\(3x - 6\)) and right angle (\(111^\circ\)) are consecutive, so they should be supplementary? Wait no, that can't be, because \(3x - 6 + 111 = 180\) would give \(3x = 75\), \(x = 25\), but that contradicts. Wait, maybe the angle \(3x - 6\) is opposite to \(-4z - 3\), and \(111^\circ\) is opposite to \(8y - 1\). Wait, that makes sense. So opposite angles: \(3x - 6 = -4z - 3\) (no, that might not be). Wait, no, let's check the sum of angles in a parallelogram: sum of all angles is \(360^\circ\), and opposite angles are equal. So let's denote:

Let angle 1: \(3x - 6\)

Angle 2: \(111^\circ\)

Angle 3: \(-4z - 3\)

Angle 4: \(8y - 1\)

So angle 1 = angle 3, angle 2 = angle 4.

Also, angle 1 + angle 2 + angle 3 + angle 4 = 360

Since angle 1 = angle 3, angle 2 = angle 4, so 2(angle 1) + 2(angle 2) = 360 => angle 1 + angle 2 = 180

So angle 1 + angle 2 = 180 => \(3x - 6 + 111 = 180\)

Wait, that's the correct approach. So \(3x - 6 + 111 = 180\)

\(3x + 105 = 180\)

\(3x = 75\)

\(x = 25\) (Oh! I made a mistake earlier. So step1 correction: \(3x - 6 + 111 = 180\) (consecutive angles are supplementary))

So step1 corrected:

Step1: Find x (consecutive angles supplementary)

\(3x - 6 + 111 = 180\)

\(3x + 105 = 180\)

\(3x = 75\)

\(x = 25\)

Step2: Find y (opposite angles equal)

Since angle 2 (111°) and angle 4 (8y - 1) are opposite, so \(8y - 1 = 111\)

\(8y = 112\)

\(y = 14\)

Step3: Find z (opposite angles equal, angle 1 and angle 3)

Angle 1: \(3x - 6 = 3*25 - 6 = 75 - 6 = 69°\)

Angle 3: \(-4z - 3\), so \( -4z - 3 = 69\)

\(-4z = 72\)

\(z = -18\)

Wait, let's verify:

Angle 1: 69°, Angle 2: 111°, Angle 3: 69°, Angle 4: 111°

Sum: 69 + 111 + 69 + 111 = 360°, which is correct.

So correcting step1: I initially thought opposite angles, but actually, consecutive angles are supplementary, so \(3x - 6\) and \(111°\) are consecutive, so they sum to 180. Then angle 4 (8y - 1) is opposite to angle 2 (111°), so equal. Angle 3 (-4z - 3) is opposite to angle 1 (3x - 6), so equal.

So step1: \(3x - 6 + 111 = 180\) => \(3x = 75\) => \(x = 25\)

Step2: \(8y - 1 = 111\) => \(8y = 112\) => \(y = 14\)

Step3: \(3x - 6 = -4z - 3\) => \(3*25 - 6 = -4z - 3\) => \(75 - 6 = -4z - 3\) => \(69 = -4z - 3\) => \(-4z = 72\) => \(z = -18\)

Let's check all angles:

Angle 1: \(3x - 6 = 75 - 6 = 69°\)

Angle 2: 111°

Angle 3: \(-4z - 3 = -4*(-18) - 3 = 72 - 3 = 69°\)

Angle 4: \(8y - 1 = 112 - 1 = 111°\)

Sum: 69 + 111 + 69 + 111 = 360°, correct. Consecutive angles: 69 + 111 = 180°, correct. Opposite angles equal: 69=69, 111=111, correct.

So the correct steps:

Step1: Solve for x (consecutive angles supplementary)

\(3x - 6 + 111 = 180\)
\(3x + 105 = 180\)
\(3x = 75\)
\(x = 25\)

Step2: Solve for y (opposite angles equal)

\(8y - 1 = 111\)
\(8y = 112\)
\(y = 14\)

Step3: Solve for z (opposite angles equal)

\(3x - 6 = -4z - 3\)
Substitute \(x = 25\):
\(3(25) - 6 = -4z - 3\)
\(75 - 6 = -4z - 3\)
\(69 = -4z - 3\)
\(-4z = 72\)
\(z = -18\)

Answer:

\(x = 25\), \(y = 14\), \(z = -18\)