Question

find the value of x, y, and z in the parallelogram below.
answer
x =
y =
z =

Explanation:

Step1: Recall properties of parallelograms

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to \(180^\circ\)). Also, opposite angles are equal. Let's identify the angles:

• The angle \(111^\circ\) and the angle \((6x + 9)^\circ\) are opposite? Wait, no, in a parallelogram, opposite angles are equal. Wait, let's check the angles: the angles given are \((6x + 9)^\circ\), \(111^\circ\), \((8y - 1)^\circ\), and \((4z - 3)^\circ\). Wait, maybe the \(111^\circ\) and \((6x + 9)^\circ\) are consecutive? Wait, no, let's correct: in a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)), and opposite angles are equal.

Wait, maybe the \(111^\circ\) and \((6x + 9)^\circ\) are opposite? No, wait, let's look at the diagram. Wait, the angles: let's assume that the angle \(111^\circ\) and \((8y - 1)^\circ\) are opposite? No, maybe the angle \(111^\circ\) and \((6x + 9)^\circ\) are consecutive. Wait, let's re-express:

In a parallelogram, consecutive angles are supplementary. So if one angle is \(111^\circ\), the consecutive angle should be \(180 - 111 = 69^\circ\). So let's set \(6x + 9 = 69\) (since \((6x + 9)^\circ\) is consecutive to \(111^\circ\))? Wait, no, maybe the angle \((8y - 1)^\circ\) is equal to \(111^\circ\) (opposite angles). Wait, no, let's start with \(x\):

Wait, let's assume that \((6x + 9)^\circ\) and \(111^\circ\) are supplementary (consecutive angles). So:

\(6x + 9 + 111 = 180\)

Step2: Solve for \(x\)

Simplify the equation:

\(6x + 120 = 180\)

Subtract 120 from both sides:

\(6x = 180 - 120 = 60\)

Divide by 6:

\(x = \frac{60}{6} = 10\)

Now for \(y\): the angle \((8y - 1)^\circ\) should be equal to \(111^\circ\) (opposite angles in parallelogram). So:

\(8y - 1 = 111\)

Step3: Solve for \(y\)

Add 1 to both sides:

\(8y = 111 + 1 = 112\)

Divide by 8:

\(y = \frac{112}{8} = 14\)

Now for \(z\): the angle \((4z - 3)^\circ\) should be equal to \(111^\circ\)? Wait, no, wait, the angle \((4z - 3)^\circ\) and \((8y - 1)^\circ\) are opposite? Wait, no, wait, the angle \(111^\circ\) and \((4z - 3)^\circ\) are opposite? Wait, no, let's check: in a parallelogram, opposite angles are equal. So if \((8y - 1)^\circ = 111^\circ\) (we found \(y = 14\), so \(8*14 -1 = 112 -1 = 111\), correct). Then the other pair of opposite angles: \((6x + 9)^\circ\) and \((4z - 3)^\circ\) should be equal. We found \(x = 10\), so \(6*10 +9 = 69\). So:

\(4z - 3 = 69\)

Step4: Solve for \(z\)

Add 3 to both sides:

\(4z = 69 + 3 = 72\)

Divide by 4:

\(z = \frac{72}{4} = 18\)

Wait, let's verify:

• \(x = 10\): \(6x +9 = 6*10 +9 = 69\). Then \(69 + 111 = 180\) (supplementary, correct, since consecutive angles in parallelogram sum to 180).
• \(y = 14\): \(8y -1 = 8*14 -1 = 112 -1 = 111\), which is equal to the given \(111^\circ\) (opposite angles, correct).
• \(z = 18\): \(4z -3 = 4*18 -3 = 72 -3 = 69\), which is equal to \(6x +9 = 69\) (opposite angles, correct).

So that works.

Answer:

\(x = 10\), \(y = 14\), \(z = 18\)