Question

1. find w, x, y, and z:
2. find w, x, y, and z:

Explanation:

1.

Step1: Identify the type of quadrilateral

Since \(AB = BC=CD = DA = 7\), quadrilateral \(ABCD\) is a rhombus. In a rhombus, \(AD\parallel BC\), \(AB\parallel CD\), and the diagonals bisect the vertex angles.

Step2: Find \(w\)

Because \(AB\parallel CD\), \(\angle CAD=\angle ACD\) (alternate - interior angles). Also, in \(\triangle ABD\), \(AD = AB = 7\), so \(\triangle ABD\) is isosceles. \(\angle ADB=\angle ABD\). Since \(AC\) bisects \(\angle DAB\), \(\angle DAB = 2\times50^{\circ}=100^{\circ}\). In \(\triangle ABD\), \(w = 50^{\circ}\) (diagonal \(AC\) bisects \(\angle DAB\)).

Step3: Find \(x\)

Since \(AD\parallel BC\), \(\angle ADB=\angle CBD\) (alternate - interior angles). In \(\triangle ABD\), \(x = 50^{\circ}\) (because \(AD = AB\), \(\angle ADB=\angle ABD\)).

Step4: Find \(y\)

\(y = 50^{\circ}\) (because \(AD\parallel BC\), \(\angle CAD=\angle ACB\) (alternate - interior angles) and \(AC\) bisects \(\angle DAB\))

Step5: Find \(z\)

In a rhombus, the diagonals are perpendicular bisectors of each other. \(\angle BEC = 90^{\circ}\). In \(\triangle BEC\), \(z=40^{\circ}\) (\(180-(90 + 50)=40\))

Step1: Identify the type of quadrilateral

Since \(AB = BC=CD = DA = 4\), quadrilateral \(ABCD\) is a rhombus. In a rhombus, \(AD\parallel BC\), \(AB\parallel CD\).

Step2: Find \(w\)

In \(\triangle ABD\), \(AD = AB = 4\), \(\angle DAB\) has an angle of \(60^{\circ}\). \(\triangle ABD\) is an equilateral triangle. So \(w = 60^{\circ}\)

Step3: Find \(x\)

Since \(AD\parallel BC\), \(\angle ADB=\angle CBD\) (alternate - interior angles). \(\triangle ABD\) is equilateral, \(\angle ADB = 60^{\circ}\). In \(\triangle BCD\), \(BC = CD\), \(\angle CBD=\angle CDB\). \(\angle BCD=120^{\circ}\) (adjacent angles in a rhombus are supplementary to \(\angle DAB\)). Using the angle - sum property of a triangle in \(\triangle BCD\), if \(\angle CBD=\angle CDB = 30^{\circ}\), and since \(AB\parallel CD\), \(x = 60^{\circ}\) (alternate - interior angles with \(\angle DAB\) considering the properties of the rhombus and parallel lines)

Step4: Find \(y\)

\(y = 30^{\circ}\) (because in \(\triangle BCD\), \(BC = CD\), \(\angle CBD=\angle CDB\) and \(\angle BCD = 120^{\circ}\), \(y=\frac{180 - 120}{2}=30^{\circ}\))

Step5: Find \(z\)

\(z = 30^{\circ}\) (because \(AD\parallel BC\), \(\angle ADB=\angle CBD\) (alternate - interior angles) and \(\angle ADB = 30^{\circ}\) from \(\triangle ABD\) properties)

Answer:

\(w = 50^{\circ}\), \(x = 50^{\circ}\), \(y = 50^{\circ}\), \(z = 40^{\circ}\)

2.