Question

question
find the value of x, y, and z in the parallelogram below.
(5z+1)°
69°
(8y+5)°
(-8x+7)°
answer attempt 1 out of 2
x =
y =
z =
submit answer

Explanation:

Step1: Recall parallelogram angle properties

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to \(180^\circ\)).

Step2: Solve for \(x\)

The angle \(69^\circ\) and \((-8x + 7)^\circ\) – wait, no, opposite angles: \(69^\circ\) should equal \((8y + 5)^\circ\)? Wait, no, let's check again. Wait, the angle \(69^\circ\) and \((8y + 5)^\circ\)? Wait, no, in a parallelogram, opposite angles are equal. So the angle \(69^\circ\) and \((8y + 5)^\circ\)? Wait, no, the angle \(69^\circ\) and \((-8x + 7)^\circ\)? Wait, no, let's list the angles: one angle is \(69^\circ\), another is \((-8x + 7)^\circ\), another is \((5z + 1)^\circ\), and \((8y + 5)^\circ\). Wait, actually, in a parallelogram, opposite angles are equal. So \(69^\circ = (8y + 5)^\circ\)? Wait, no, maybe \(69^\circ\) and \((-8x + 7)^\circ\) are not opposite. Wait, let's correct: in a parallelogram, opposite angles are equal. So the angle \(69^\circ\) and \((8y + 5)^\circ\) – no, wait, the angle \(69^\circ\) and \((5z + 1)^\circ\) are consecutive? Wait, no, let's think again. Wait, the angle \(69^\circ\) and \((-8x + 7)^\circ\): no, maybe \(69^\circ\) and \((8y + 5)^\circ\) are opposite? Wait, no, let's check the other pair. Wait, the angle \(69^\circ\) and \((-8x + 7)^\circ\) – no, consecutive angles. Wait, consecutive angles in a parallelogram are supplementary. So \(69^\circ + (5z + 1)^\circ = 180^\circ\)? Wait, no, maybe \(69^\circ\) and \((-8x + 7)^\circ\) are opposite? Wait, no, I think I made a mistake. Let's start over.

In a parallelogram, opposite angles are equal. So:

1. \(69^\circ = (8y + 5)^\circ\) (opposite angles)
2. \(69^\circ + (-8x + 7)^\circ = 180^\circ\) (consecutive angles, supplementary)
3. \(69^\circ + (5z + 1)^\circ = 180^\circ\) (consecutive angles, supplementary)

Wait, no, consecutive angles are supplementary. So let's solve for \(x\) first. Consecutive angles: \(69 + (-8x + 7) = 180\)? Wait, no, \(69 + (5z + 1) = 180\)? Wait, no, let's take the angle \(69^\circ\) and \((-8x + 7)^\circ\) as consecutive? Wait, no, maybe \(69^\circ\) and \((5z + 1)^\circ\) are consecutive. Wait, let's check the equations:

First, for \(x\):

Consecutive angles: \(69 + (-8x + 7) = 180\)? Wait, no, that would be if they are consecutive. Wait, no, maybe \(69^\circ\) and \((-8x + 7)^\circ\) are opposite? No, opposite angles are equal. So if \(69^\circ\) and \((-8x + 7)^\circ\) are opposite, then \(69 = -8x + 7\). Let's solve that:

\(69 = -8x + 7\)

Subtract 7: \(62 = -8x\)

\(x = 62 / (-8) = -7.75\)? That can't be. So maybe that's wrong.

Wait, maybe \(69^\circ\) and \((8y + 5)^\circ\) are opposite. So \(69 = 8y + 5\)

Subtract 5: \(64 = 8y\)

\(y = 8\)

Then, for \(x\): the angle \(69^\circ\) and \((-8x + 7)^\circ\) are consecutive, so they sum to \(180^\circ\):

\(69 + (-8x + 7) = 180\)

\(76 - 8x = 180\)

\(-8x = 104\)

\(x = -13\)? No, that's not right. Wait, maybe I mixed up the angles.

Wait, another approach: in a parallelogram, opposite angles are equal. So the angle \(69^\circ\) and \((5z + 1)^\circ\) are not opposite. Wait, maybe the angle \(69^\circ\) and \((-8x + 7)^\circ\) are opposite. Wait, let's check the other pair: \((5z + 1)^\circ\) and \((8y + 5)^\circ\) are opposite. So:

\(5z + 1 = 8y + 5\)

And \(69 + (5z + 1) = 180\) (consecutive angles)

Let's try that.

Step3: Solve for \(z\) first (using consecutive angles)

Consecutive angles in a parallelogram sum to \(180^\circ\). So \(69^\circ + (5z + 1)^\circ = 180^\circ\)

\(69 + 5z + 1 = 180\)

\(70 + 5z = 180\)

\(5z = 110\)

\(z = 22\)

Step4: Solve for \(x\) (using opposite angles? Wait, no, consecutive angles)

Wait, the angle \(69^\circ\) and \((-8x + 7)^\circ\) – are they consecutive? Wait, if \(z = 22\), then \((5z + 1) = 111^\circ\). Then the other angle \((-8x + 7)^\circ\) should be equal to \((8y + 5)^\circ\) (opposite angles). And \(69^\circ + 111^\circ = 180^\circ\), which works (consecutive angles). Now, the angle \(69^\circ\) and \((-8x + 7)^\circ\) – wait, no, the angle \(69^\circ\) and \((-8x + 7)^\circ\) should be equal to \((8y + 5)^\circ\)? Wait, no, let's correct.

Wait, in a parallelogram, opposite angles are equal. So:

- Angle 1: \(69^\circ\)
- Angle 2: \((-8x + 7)^\circ\)
- Angle 3: \((5z + 1)^\circ\)
- Angle 4: \((8y + 5)^\circ\)

So Angle 1 = Angle 4 (opposite), so \(69 = 8y + 5\)

Angle 2 = Angle 3 (opposite), so \(-8x + 7 = 5z + 1\)

And Angle 1 + Angle 3 = 180 (consecutive), so \(69 + (5z + 1) = 180\)

Let's solve \(69 + 5z + 1 = 180\)

\(70 + 5z = 180\)

\(5z = 110\)

\(z = 22\) (as before)

Now, solve for \(y\): \(69 = 8y + 5\)

\(8y = 69 - 5 = 64\)

\(y = 8\)

Now, solve for \(x\): \(-8x + 7 = 5z + 1\) (since Angle 2 = Angle 3)

We know \(z = 22\), so \(5z + 1 = 111\)

Thus, \(-8x + 7 = 111\)

\(-8x = 111 - 7 = 104\)

\(x = 104 / (-8) = -13\)? Wait, that can't be. Wait, maybe Angle 2 and Angle 1 are consecutive. So Angle 1 + Angle 2 = 180.

So \(69 + (-8x + 7) = 180\)

\(76 - 8x = 180\)

\(-8x = 104\)

\(x = -13\). Hmm, but angles can't be negative? Wait, \(-8x + 7\): if \(x = -13\), then \(-8*(-13) + 7 = 104 + 7 = 111^\circ\), which is positive. Okay, that works.

So:

- \(x = -13\)
- \(y = 8\)
- \(z = 22\)

Wait, let's verify:

- Angles: \(69^\circ\), \(111^\circ\) (since \(-8*(-13)+7=111\)), \(111^\circ\) (since \(5*22 +1=111\)), \(69^\circ\) (since \(8*8 +5=69\)). Yes, opposite angles are equal (\(69=69\), \(111=111\)) and consecutive angles sum to \(180\) (\(69+111=180\)). Perfect.

Answer:

\(x = -13\), \(y = 8\), \(z = 22\)