Question

question 13 of 27
what is the measure of ∠z in the parallelogram shown?
image of parallelogram with vertices x, y, w, z; ∠x is 30°
a. 90°
b. 190°
c. 150°
d. 30°

Explanation:

Step1: Recall parallelogram angle properties

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to \(180^\circ\)).

Step2: Identify angle relationship

\(\angle X\) and \(\angle Z\) are not opposite or consecutive? Wait, no, in parallelogram \(XYWZ\), sides \(XY \parallel WZ\) and \(XW \parallel YZ\). So \(\angle X\) and \(\angle W\) are consecutive, \(\angle X\) and \(\angle Z\)? Wait, no, let's label the parallelogram: vertices \(X, Y, Z, W\) in order, so \(XY \parallel WZ\) and \(XW \parallel YZ\). Then \(\angle X\) and \(\angle Z\)? Wait, no, \(\angle X\) and \(\angle Y\) are consecutive? Wait, no, in a parallelogram, consecutive angles (adjacent) are supplementary. Wait, \(\angle X\) is \(30^\circ\), and \(\angle Z\) – wait, no, actually, in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. Wait, \(\angle X\) and \(\angle Z\) – no, \(\angle X\) and \(\angle W\) are consecutive, \(\angle X\) and \(\angle Y\) are consecutive? Wait, maybe I mixed up. Let's correct: in parallelogram \(ABCD\), \(\angle A = \angle C\), \(\angle B = \angle D\), and \(\angle A + \angle B = 180^\circ\), \(\angle B + \angle C = 180^\circ\), etc. So in this parallelogram \(XYWZ\), \(\angle X\) and \(\angle Z\) – wait, no, \(\angle X\) and \(\angle W\) are consecutive, \(\angle X\) and \(\angle Y\) are consecutive? Wait, the diagram: \(X\) connected to \(Y\) and \(W\), \(Y\) connected to \(X\) and \(Z\), \(Z\) connected to \(Y\) and \(W\), \(W\) connected to \(Z\) and \(X\). So sides: \(XY \parallel WZ\), \(XW \parallel YZ\). So \(\angle X\) (at vertex \(X\)) and \(\angle Z\) (at vertex \(Z\)) – are they opposite? Wait, \(\angle X\) and \(\angle Z\): no, \(\angle X\) and \(\angle Y\) are adjacent, \(\angle X\) and \(\angle W\) are adjacent. Wait, maybe the problem is that \(\angle X\) and \(\angle Z\) – no, wait, in a parallelogram, alternate interior angles? Wait, no, the key property: consecutive angles in a parallelogram are supplementary (sum to \(180^\circ\)), and opposite angles are equal. Wait, \(\angle X\) is \(30^\circ\), and \(\angle Z\) – wait, maybe \(\angle X\) and \(\angle Z\) are not opposite. Wait, no, let's look at the vertices: \(X, Y, Z, W\) in order, so \(X\) and \(Z\) are opposite? No, \(X\) and \(Z\) would be connected by a diagonal, \(Y\) and \(W\) by the other diagonal. So opposite angles: \(\angle X = \angle Z\)? No, that can't be, because if \(\angle X\) is \(30^\circ\), and if consecutive angles are supplementary, then \(\angle X + \angle W = 180^\circ\), \(\angle W + \angle Z = 180^\circ\), so \(\angle X = \angle Z\)? Wait, no, that would mean \(\angle X = \angle Z\), but that would be if they are opposite. Wait, maybe I made a mistake. Wait, in a parallelogram, opposite angles are equal, so \(\angle X = \angle Z\)? But that would be \(30^\circ\), but option D is \(30^\circ\), but let's check again. Wait, no, maybe the diagram is different. Wait, the angle at \(X\) is \(30^\circ\), and the angle at \(Z\) – wait, maybe the sides \(XW\) and \(YZ\) are parallel, and \(XY\) and \(WZ\) are parallel. So \(\angle X\) and \(\angle Z\) – are they same-side interior angles? Wait, no, when two parallel lines are cut by a transversal, same-side interior angles are supplementary. Wait, \(XW \parallel YZ\), and transversal \(XZ\)? No, \(XZ\) is a diagonal. Wait, maybe the correct approach: in a parallelogram, consecutive angles are supplementary. So \(\angle X\) and \(\angle W\) are consecutive, so \(\angle X + \angle W = 180^\circ\). Then \(\angle W\) and \(\angle Z\) are consecutive, so \(\angle W + \angle Z = 180^\circ\). Therefore, \(\angle X = \angle Z\)? No, that would mean \(\angle X = \angle Z\), but that would be if they are opposite. Wait, no, opposite angles are equal, so \(\angle X = \angle Z\), and \(\angle Y = \angle W\). But then if \(\angle X\) is \(30^\circ\), \(\angle Z\) is \(30^\circ\), but that's option D. But wait, maybe I got the consecutive angles wrong. Wait, let's take a standard parallelogram: \(ABCD\), with \(AB \parallel CD\) and \(AD \parallel BC\). Then \(\angle A + \angle B = 180^\circ\), \(\angle B + \angle C = 180^\circ\), so \(\angle A = \angle C\), \(\angle B = \angle D\). So in this problem, the parallelogram is \(XYWZ\), so \(XY \parallel WZ\) and \(XW \parallel YZ\). So \(\angle X\) is at \(X\), between \(XY\) and \(XW\). \(\angle Z\) is at \(Z\), between \(WZ\) and \(YZ\). Since \(XW \parallel YZ\) and \(XY \parallel WZ\), the angle at \(X\) and angle at \(Z\) – are they equal? Wait, no, maybe the angle at \(X\) and angle at \(W\) are supplementary. Let's say \(\angle X = 30^\circ\), then \(\angle W = 180^\circ - 30^\circ = 150^\circ\). Then \(\angle Z\) is equal to \(\angle X\)? No, that can't be. Wait, no, opposite angles: \(\angle X\) and \(\angle Z\) are opposite? Wait, \(X\) and \(Z\) are opposite vertices, so \(\angle X = \angle Z\), and \(\angle Y = \angle W\). But then if \(\angle X\) is \(30^\circ\), \(\angle Z\) is \(30^\circ\), but that's option D. But wait, maybe the diagram is such that \(\angle X\) and \(\angle Z\) are consecutive? No, that doesn't make sense. Wait, maybe I made a mistake in the property. Wait, no, in a parallelogram, opposite angles are equal, consecutive angles are supplementary. So if \(\angle X\) is \(30^\circ\), then the angle opposite to it (which is \(\angle Z\)) is also \(30^\circ\), and the consecutive angles (like \(\angle Y\) and \(\angle W\)) are \(150^\circ\). But wait, the options include \(30^\circ\) (D) and \(150^\circ\) (C). Wait, maybe the diagram is labeled differently. Wait, the vertices are \(X, Y, Z, W\) in order, so the sides are \(XY\), \(YZ\), \(ZW\), \(WX\)? No, the diagram shows \(X\) connected to \(Y\) and \(W\), \(Y\) connected to \(X\) and \(Z\), \(Z\) connected to \(Y\) and \(W\), \(W\) connected to \(Z\) and \(X\). So it's a parallelogram with sides \(XY\), \(YZ\), \(ZW\), \(WX\)? No, \(XY\) is parallel to \(WZ\), and \(XW\) is parallel to \(YZ\). So the angles: at \(X\), between \(XY\) and \(XW\) (30°), at \(Z\), between \(WZ\) and \(YZ\). Since \(XY \parallel WZ\) and \(XW\) is a transversal, \(\angle X\) and \(\angle W\) are consecutive (supplementary). Then \(XW \parallel YZ\) and \(WZ\) is a transversal, so \(\angle W\) and \(\angle Z\) are consecutive (supplementary). Therefore, \(\angle X = \angle Z\)? Wait, no, \(\angle X + \angle W = 180^\circ\), \(\angle W + \angle Z = 180^\circ\), so \(\angle X = \angle Z\). So \(\angle Z = 30^\circ\)? But that's option D. But wait, maybe the angle at \(X\) and angle at \(Z\) are not opposite. Wait, maybe the diagram is a rhombus or something, but no, it's a parallelogram. Wait, maybe I misread the angle. Wait, the angle at \(X\) is \(30^\circ\), and we need to find \(\angle Z\). Wait, maybe the correct property is that in a parallelogram, consecutive angles are supplementary, so \(\angle X + \angle Z = 180^\circ\)? No, that would be if they are consecutive, but they are not. Wait, no, let's draw it: \(X\)---\(Y\)---\(Z\), \(X\)---\(W\)---\(Z\), so \(XY\) and \(WZ\) are top and bottom, \(XW\) and \(YZ\) are left and right. So angle at \(X\) is between top (XY) and left (XW), angle at \(Z\) is between bottom (WZ) and right (YZ). Since left (XW) and right (YZ) are parallel, and top (XY) and bottom (WZ) are parallel, the angle at \(X\) and angle at \(Z\) – are they equal? Yes, because they are opposite angles. So \(\angle Z = \angle X = 30^\circ\)? But that's option D. Wait, but maybe the question is about consecutive angles. Wait, no, the options: A is 90, B is 190 (invalid, since angles in a parallelogram are less than 180), C is 150, D is 30. Wait, maybe I made a mistake. Wait, no, in a parallelogram, opposite angles are equal, so if \(\angle X\) is 30°, \(\angle Z\) is 30°, so answer is D? But wait, maybe the diagram is different. Wait, maybe the angle at \(X\) is 30°, and \(\angle Z\) is consecutive to \(\angle W\), which is consecutive to \(\angle X\). Wait, \(\angle X + \angle W = 180°\), so \(\angle W = 150°\), then \(\angle Z = \angle X = 30°\)? No, that's opposite. Wait, I think I was wrong earlier. Let's check the properties again: In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to 180°). So if \(\angle X\) is 30°, then the angle opposite to it (let's say \(\angle Z\)) is also 30°, and the angles adjacent to \(\angle X\) (like \(\angle Y\) and \(\angle W\)) are 150° each. So the measure of \(\angle Z\) is 30°, which is option D. But wait, maybe the diagram is labeled such that \(\angle X\) and \(\angle Z\) are adjacent? No, in the diagram, \(X\) and \(Z\) are connected by a diagonal, so they are opposite vertices. Therefore, \(\angle Z = \angle X = 30°\).

Answer:

D. 30°