Question

question
find the value of x, y, and z in the parallelogram below.
(-5y-6)°
(-x-1)°
(2z+5)°
69°
answer attempt 1 out of 2
x =
y =
z =

Explanation:

Step1: Recall properties of parallelograms

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

Step2: Find \(x\)

The angle \(-x - 1\) and \(69^\circ\) are opposite angles? Wait, no, wait. Wait, in a parallelogram, opposite angles are equal. Wait, the angle \(-x - 1\) and the angle \(69^\circ\)? Wait, no, let's look at the diagram. Wait, the angle labeled \(69^\circ\) and the angle labeled \((-x - 1)^\circ\) – wait, no, maybe consecutive angles? Wait, no, let's correct. In a parallelogram, opposite angles are equal. So the angle \((-x - 1)^\circ\) and \(69^\circ\) – wait, no, maybe the angle \((-x - 1)^\circ\) and the angle \((-5y - 6)^\circ\)? Wait, no, let's re-examine. Wait, the angle \(69^\circ\) and the angle \((-x - 1)^\circ\) – wait, no, consecutive angles in a parallelogram are supplementary. Wait, the angle \(69^\circ\) and the angle \((-x - 1)^\circ\) – wait, maybe the angle \(69^\circ\) and the angle \((-x - 1)^\circ\) are supplementary? Wait, no, let's check the other angle. Wait, the angle \((2z + 5)^\circ\) and the angle \((-5y - 6)^\circ\) – no, opposite angles. Wait, in a parallelogram, opposite angles are equal. So the angle \(69^\circ\) should be equal to the angle opposite to it, which is \((-x - 1)^\circ\)? Wait, no, that would give \(-x - 1 = 69\), but that would make \(x\) negative, which is odd. Wait, maybe I got the angles wrong. Wait, maybe the angle \(69^\circ\) and the angle \((-5y - 6)^\circ\) are opposite? Wait, no, the diagram: let's assume the parallelogram has angles: top left \((-5y - 6)^\circ\), top right \((-x - 1)^\circ\), bottom right \(69^\circ\), bottom left \((2z + 5)^\circ\). So in a parallelogram, opposite angles are equal: top left = bottom right? No, top left and bottom right are opposite? Wait, no, top left and bottom right: no, top left and bottom right are not opposite. Wait, top left and bottom right: no, in a parallelogram, opposite angles are (top left, bottom right) and (top right, bottom left)? Wait, no, standard parallelogram: angle A (top left), angle B (top right), angle C (bottom right), angle D (bottom left). Then angle A = angle C, angle B = angle D. And angle A + angle B = 180 degrees (consecutive angles are supplementary).

So in this case, angle C is \(69^\circ\), so angle A (top left) should be equal to angle C? Wait, no, angle A is \((-5y - 6)^\circ\), angle C is \(69^\circ\). Then angle B (top right) is \((-x - 1)^\circ\), angle D (bottom left) is \((2z + 5)^\circ\). So angle B = angle D, and angle A = angle C.

Wait, so angle A = angle C: \(-5y - 6 = 69\)? Wait, that would be \(-5y = 75\), so \(y = -15\), which is possible? Wait, but let's check angle B and angle D: \(-x - 1 = 2z + 5\). Also, consecutive angles: angle A + angle B = 180, so \((-5y - 6) + (-x - 1) = 180\).

Wait, let's start over.

Property 1: Opposite angles in a parallelogram are equal.

So:

1. \((-5y - 6)^\circ = 69^\circ\) (opposite angles)
2. \((-x - 1)^\circ = (2z + 5)^\circ\) (opposite angles)
3. Consecutive angles are supplementary: \((-5y - 6) + (-x - 1) = 180\) (or \((2z + 5) + 69 = 180\))

Let's solve equation 1 first: \(-5y - 6 = 69\)

Step3: Solve for \(y\)

\(-5y - 6 = 69\)

Add 6 to both sides: \(-5y = 69 + 6 = 75\)

Divide both sides by -5: \(y = 75 / (-5) = -15\)

Step4: Solve for \(x\) using consecutive angles

Consecutive angles: \((-5y - 6) + (-x - 1) = 180\)

We know \(y = -15\), so substitute:

\(-5(-15) - 6 + (-x - 1) = 180\)

Calculate \(-5(-15) = 75\), so:

\(75 - 6 - x - 1 = 180\)

Simplify: \(68 - x = 180\)

Subtract 68: \(-x = 180 - 68 = 112\)

Multiply by -1: \(x = -112\)

Wait, but let's check with the other angle. The angle \(69^\circ\) and the angle \((-x - 1)^\circ\) – are they supplementary? Let's check: \(69 + (-x - 1) = 68 - x\). If \(x = -112\), then \(68 - (-112) = 68 + 112 = 180\), which works. Good.

Step5: Solve for \(z\)

Using opposite angles: \((-x - 1) = (2z + 5)\)

We know \(x = -112\), so substitute:

\(-(-112) - 1 = 2z + 5\)

Simplify: \(112 - 1 = 2z + 5\)

\(111 = 2z + 5\)

Subtract 5: \(106 = 2z\)

Divide by 2: \(z = 53\)

Wait, let's verify all angles:

- \(y = -15\), so \(-5y - 6 = -5(-15) - 6 = 75 - 6 = 69^\circ\), which matches the opposite angle \(69^\circ\). Good.

- \(x = -112\), so \(-x - 1 = -(-112) - 1 = 112 - 1 = 111^\circ\)

- \(z = 53\), so \(2z + 5 = 2(53) + 5 = 106 + 5 = 111^\circ\), which matches \(-x - 1 = 111^\circ\). Good.

- Consecutive angles: \(69^\circ + 111^\circ = 180^\circ\), which is correct for a parallelogram. Perfect.

So:

\(x = -112\)

\(y = -15\)

\(z = 53\)

Answer:

\(x = -112\), \(y = -15\), \(z = 53\)