Question

quadrilateral wxyz is a rhombus and ( mangle wzx = 2z + 61^{circ}). what is the value of ( z)?

Explanation:

Step1: Properties of a rhombus

In a rhombus, the diagonals bisect the angles. Also, adjacent angles of a rhombus are supplementary. But here, we use the property that in a rhombus \(WXYZ\), \(\angle Y + 2\angle WZX=180^{\circ}\) (since \(\angle Y\) and \(\angle W\) are adjacent angles and the diagonal \(XZ\) bisects \(\angle W\)). Given \(\angle Y = 54^{\circ}\) and \(\angle WZX=2z + 61^{\circ}\).

Step2: Set up the equation

Substitute the values into the equation \(\angle Y+2\angle WZX = 180^{\circ}\). So, \(54+2(2z + 61)=180\).
First, expand the left - hand side: \(54 + 4z+122 = 180\).
Combine like terms: \(4z+176 = 180\).
Subtract 176 from both sides: \(4z=180 - 176\).
\(4z = 4\).
Divide both sides by 4: \(z=\frac{4}{4}\).

Answer:

\(z = 1\)