Question

m∠rbm = 8z - 10
m∠bmu = 12z + 10
m∠urb = 59k
given the parallelogram in the image, solve for k.
k = 5
k = 2
k = 9
k = 1.5

Explanation:

Answer:

We first need to use the property of parallelograms that adjacent - angles are supplementary. In parallelogram \(BMUR\), \(\angle RBM\) and \(\angle BMU\) are adjacent angles, so \(m\angle RBM + m\angle BMU=180^{\circ}\).

Step1: Set up the equation for adjacent - angles

\((8z - 10)+(12z + 10)=180\)
Combining like - terms: \(8z+12z-10 + 10=180\), which simplifies to \(20z=180\).
Dividing both sides by 20, we get \(z = 9\).

Step2: Use the property of opposite - angles

In a parallelogram, opposite angles are equal. \(\angle URB\) and \(\angle BMU\) are opposite angles, so \(m\angle URB=m\angle BMU\).
We know \(m\angle BMU=12z + 10\), substituting \(z = 9\), we have \(m\angle BMU=12\times9+10=108 + 10=118^{\circ}\).
And \(m\angle URB = 59k\), so \(59k=118\).

Step3: Solve for \(k\)

Dividing both sides of the equation \(59k=118\) by 59, we get \(k = 2\).
So the answer is \(k = 2\).