Question

if \\(overline{xy}\\) is not parallel to \\(overline{wz}\\), what is \\(m\angle z\\)?
\\(m\angle z = \square^\circ\\)

Explanation:

Step1: Identify the polygon type

The figure has two pairs of adjacent sides equal (marked with ticks), so it's a kite. In a kite, one pair of opposite angles (between the unequal sides) are equal, and the sum of adjacent angles between the equal sides is \(180^\circ\)? Wait, no, in a kite, one pair of opposite angles (the ones between the unequal sides) are equal, and the other two angles (between the equal sides) are supplementary? Wait, actually, in a kite, one pair of opposite angles are equal, and the sum of the other two angles (the ones adjacent to the unequal sides) is \(180^\circ\)? Wait, no, let's recall: in a kite, two distinct pairs of adjacent sides are equal. The angles between the unequal sides are equal, and the sum of the angles between the equal sides: actually, in a kite, one pair of opposite angles (the ones that are between the unequal sides) are equal, and the other two angles (between the equal sides) are supplementary? Wait, no, let's look at the diagram. The sides: WZ and XY are marked with one tick, WX and ZY are marked with another tick? Wait, no, the diagram: WZ has one tick, ZY has one tick (arrow), WX has one tick (arrow), XY has one tick. Wait, maybe it's a kite with two pairs of adjacent equal sides: WZ = XY and WX = ZY? Wait, no, the markings: WZ has a red tick, ZY has a red arrow, WX has a red arrow, XY has a red tick. So WZ = XY (tick) and WX = ZY (arrow). So it's a kite with WZ = XY and WX = ZY. In a kite, one pair of opposite angles are equal, and the sum of the other two angles: actually, in a kite, the sum of the angles between the unequal sides? Wait, no, let's recall the properties of a kite. A kite has two distinct pairs of adjacent sides equal. One pair of opposite angles (the ones that are between the unequal sides) are equal. The diagonals are perpendicular, and one diagonal bisects the other. Also, in a kite, the sum of the measures of two angles that are between the unequal sides: wait, no, let's look at the given angle. Angle Y is \(103^\circ\). We need to find angle Z. In a kite, the angles between the equal sides: wait, WZ and ZY: WZ is equal to XY, ZY is equal to WX. So angle at Z and angle at X: no, wait, angle at Y is \(103^\circ\), angle at Z: in a kite, if two adjacent sides are equal (WZ and ZY? No, WZ and XY are equal, WX and ZY are equal. So the vertices are W, Z, Y, X. So sides: WZ, ZY, YX, XW. WZ = YX (tick), ZY = XW (arrow). So it's a kite with WZ = YX and ZY = XW. So the angles at Z and at X: no, angles at W and at Y? Wait, angle at Y is \(103^\circ\), angle at Z: in a kite, the sum of angle Y and angle Z: wait, no, in a kite, one pair of opposite angles are equal, and the other two angles are supplementary? Wait, no, let's think again. In a kite, the sum of the measures of two angles that are between the unequal sides: actually, in a kite, if you have two pairs of adjacent equal sides, then the angles between the equal sides (the ones that are not the equal angles) are supplementary. Wait, maybe I'm overcomplicating. Let's see: in a kite, one pair of opposite angles are equal, and the other two angles are supplementary. Wait, no, let's take a kite: two adjacent sides equal (AB = AD) and two adjacent sides equal (BC = DC). Then angle B and angle D are equal, and angle A + angle C = 180°. Wait, no, that's not right. Wait, in a kite with AB = AD and BC = DC, angle at B and angle at D: no, angle at A and angle at C: angle at A is between the two equal sides (AB and AD), angle at C is between the two equal sides (BC and DC). Then angle B and angle D are equal. And angle A + angle C = 180°? No, that's not a standard property. Wait, maybe this is a kite where the two angles between the equal sides (the ones that are not the equal angles) are supplementary. Wait, the given angle is angle Y (103°), and we need angle Z. If angle Y and angle Z are adjacent angles in the kite, and in a kite, adjacent angles between the equal sides: wait, maybe it's a kite where WZ || XY? But the problem says XY is not parallel to WZ. Wait, no, the problem says "If XY is not parallel to WZ", so we can't use parallel lines. But in a kite, even if the sides are not parallel, the property of the angles: in a kite, one pair of opposite angles are equal, and the sum of the other two angles is 180°? Wait, no, let's check the sum of interior angles of a quadrilateral. The sum of interior angles of a quadrilateral is \(360^\circ\). In a kite, one pair of opposite angles are equal. Let's denote angle W = angle X (if that's the equal pair) or angle Z = angle Y? Wait, no, the markings: WZ = XY (tick), ZY = XW (arrow). So the equal sides are WZ = XY and ZY = XW. So the angles opposite each other: angle W and angle Y, angle Z and angle X? No, angle at W: between WZ and WX, angle at Y: between YX and YZ. Angle at Z: between WZ and ZY, angle at X: between XW and XY. So if WZ = XY and ZY = XW, then triangle WZ X and triangle YX Z? No, maybe it's a kite with two pairs of adjacent equal sides: WZ = ZY and WX = XY? No, the markings are different. Wait, the diagram: WZ has a tick, ZY has an arrow, WX has an arrow, XY has a tick. So WZ = XY (tick) and ZY = WX (arrow). So it's a kite with WZ = XY and ZY = WX. So the quadrilateral is W-Z-Y-X-W. So sides: WZ, ZY, YX, XW. WZ = YX (tick), ZY = XW (arrow). So it's a kite. In a kite, one pair of opposite angles are equal. Let's see: angle at Z and angle at X? No, angle at W and angle at Y? Wait, angle at Y is \(103^\circ\). Let's assume that angle at Z and angle at X are equal, and angle at W and angle at Y are equal? No, that can't be. Wait, no, in a kite, the angles between the equal sides: WZ and ZY: WZ is equal to XY, ZY is equal to WX. So angle at Z is between WZ and ZY, angle at X is between XW and XY. Since WZ = XY and ZY = XW, triangles WZ Z and X Y Y? No, maybe using the sum of interior angles. The sum of interior angles of a quadrilateral is \(360^\circ\). In a kite, one pair of opposite angles are equal. Let's say angle W = angle X, and angle Z = angle Y? No, angle Y is \(103^\circ\), that would make angle Z \(103^\circ\), but that might not be right. Wait, no, maybe I made a mistake. Wait, in a kite, the two angles that are between the unequal sides are equal, and the other two angles are supplementary. Wait, no, let's look for similar problems. In a kite with two pairs of adjacent equal sides, if one angle is given, the opposite angle (if it's the equal pair) is equal, and the other two angles are supplementary. Wait, but in this case, the angle given is \(103^\circ\), and we need to find angle Z. If angle Y and angle Z are adjacent, and in a kite, adjacent angles between the equal sides: wait, maybe it's a kite where the sum of angle Y and angle Z is \(180^\circ\)? No, \(180 - 103 = 77\), but that doesn't seem right. Wait, no, let's recall the properties of a kite. A kite has two distinct pairs of adjacent sides equal. One pair of opposite angles (the ones that are between the unequal sides) are equal. The diagonals are perpendicular, and one diagonal bisects the other. Also, in a kite, the sum of the measures of two angles that are between the unequal sides: wait, no, let's calculate the sum of interior angles. Quadrilateral: sum is \(360^\circ\). In a kite, one pair of opposite angles are equal. Let's denote angle W = angle X = a, and angle Z = angle Y = b. Then \(2a + 2b = 360\), so \(a + b = 180\). But angle Y is \(103^\circ\), so angle Z would be \(103^\circ\)? No, that can't be, because then \(a = 77^\circ\), but that would mean angle W and angle X are \(77^\circ\). But maybe the equal angles are angle Z and angle X, and angle W and angle Y? No, the diagram shows angle Y is \(103^\circ\), and we need angle Z. Wait, maybe the kite is such that WZ and XY are parallel? But the problem says they are not. Wait, no, the markings: WZ has a tick, ZY has an arrow, WX has an arrow, XY has a tick. So WZ = XY (tick) and ZY = WX (arrow). So it's a kite with WZ = XY and ZY = WX. So the sides: WZ, ZY, YX, XW. So WZ = YX, ZY = XW. So it's a kite, so the angles at Z and at X are equal, and angles at W and at Y are equal? Wait, angle at Y is \(103^\circ\), so angle at W is \(103^\circ\), and angle at Z and angle at X are equal. Then sum of angles: \(103 + 103 + 2\angle Z = 360\). So \(206 + 2\angle Z = 360\), so \(2\angle Z = 154\), so \(\angle Z = 77^\circ\). Ah, that makes sense. So step by step:

Step1: Recall the sum of interior angles of a quadrilateral

The sum of the interior angles of any quadrilateral is \(360^\circ\). So for quadrilateral WZYX, we have:
\(m\angle W + m\angle Z + m\angle Y + m\angle X = 360^\circ\)

Step2: Identify the equal angles in the kite

In a kite with two pairs of adjacent equal sides (WZ = XY and ZY = XW), the angles opposite each other (angle W and angle Y, angle Z and angle X) are equal? Wait, no, in a kite, one pair of opposite angles are equal. Wait, actually, in a kite, the two angles that are between the unequal sides are equal. Wait, maybe I mixed up. Let's correct: in a kite, the two angles that are not between the equal sides (i.e., the angles that are between the unequal sides) are equal. Wait, no, let's look at the sides: WZ = XY (equal) and ZY = XW (equal). So the sides WZ and XY are equal, ZY and XW are equal. So the angles at W and at Y: angle at W is between WZ and WX, angle at Y is between YX and YZ. Since WZ = YX and WX = YZ, triangles WZW and YXY? No, maybe using the SAS congruence. Triangles WZ X and YX Z? No, maybe it's better to use the property that in a kite, one pair of opposite angles are equal. Let's assume that angle W = angle Y and angle Z = angle X. Wait, angle Y is \(103^\circ\), so angle W is \(103^\circ\). Then the sum of angle W and angle Y is \(103 + 103 = 206^\circ\). Then the sum of angle Z and angle X is \(360 - 206 = 154^\circ\). Since angle Z = angle X (because the kite has WZ = XY and ZY = XW, so the angles at Z and X are equal), then \(m\angle Z = \frac{154}{2} = 77^\circ\).

Answer:

\(77\)