Question

question 2
given wxyz is a parallelogram, what is the measure of ∠z?
image of parallelogram wxyz with ∠x = 35°
options: 25°, 35°, 55°, 145°

Explanation:

Step1: Recall parallelogram angle property

In a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)), and opposite angles are equal. Also, \(XY \parallel WZ\) and \(XW \parallel YZ\), so \(\angle X\) and \(\angle Z\) are consecutive angles? Wait, no—wait, in parallelogram \(WXYZ\), sides \(XW\) and \(YZ\) are parallel, and \(XY\) and \(WZ\) are parallel. So \(\angle X\) and \(\angle Z\): Wait, actually, in a parallelogram, consecutive angles (adjacent angles) are supplementary, and opposite angles are equal. Wait, let's label the parallelogram: vertices \(W, X, Y, Z\) in order, so sides \(WX\) and \(YZ\) are parallel, \(XY\) and \(WZ\) are parallel. So \(\angle X\) (at vertex \(X\)) and \(\angle Z\) (at vertex \(Z\)): Wait, no, \(\angle X\) and \(\angle W\) are consecutive, \(\angle X\) and \(\angle Y\) are consecutive? Wait, maybe I mixed up. Wait, in a parallelogram, opposite angles are equal. So \(\angle X = \angle Z\)? Wait, no, wait: let's check the sides. \(X\) is connected to \(Y\) and \(W\), \(Z\) is connected to \(Y\) and \(W\). So \(XY \parallel WZ\), and \(XW \parallel YZ\). So the angle at \(X\) (\(\angle X\)) and angle at \(Z\) (\(\angle Z\)): are they consecutive or opposite? Wait, in parallelogram \(WXYZ\), the vertices are in order, so the angles are \(\angle W\), \(\angle X\), \(\angle Y\), \(\angle Z\), with \(\angle W\) opposite \(\angle Y\), and \(\angle X\) opposite \(\angle Z\)? Wait, no, that's not right. Wait, in a parallelogram, opposite angles are equal. So if \(WXYZ\) is a parallelogram, then \(WX \parallel YZ\) and \(XY \parallel WZ\). So \(\angle X\) and \(\angle Z\): let's see, \(X\) is between \(W\) and \(Y\), \(Z\) is between \(Y\) and \(W\). So the sides \(XW\) and \(YZ\) are parallel, and \(XY\) and \(WZ\) are parallel. So the angle at \(X\) (between \(XW\) and \(XY\)) and the angle at \(Z\) (between \(YZ\) and \(WZ\)): since \(XW \parallel YZ\) and \(XY \parallel WZ\), the consecutive angles (like \(\angle X\) and \(\angle W\)) are supplementary, but \(\angle X\) and \(\angle Z\): wait, no, actually, in a parallelogram, consecutive angles (adjacent) are supplementary, and opposite angles are equal. Wait, maybe I made a mistake. Wait, let's take a standard parallelogram: \(ABCD\), with \(AB \parallel CD\) and \(AD \parallel BC\). Then \(\angle A = \angle C\), \(\angle B = \angle D\), and \(\angle A + \angle B = 180^\circ\), etc. So in our case, \(WXYZ\), so let's map to \(ABCD\): \(W \to A\), \(X \to B\), \(Y \to C\), \(Z \to D\). So \(\angle X\) (at \(X\)) is like \(\angle B\), \(\angle Z\) (at \(Z\)) is like \(\angle D\). Wait, no, \(Z\) would be \(D\), so \(\angle D\) is opposite \(\angle B\)? No, \(\angle A\) ( \(W\)) and \(\angle C\) ( \(Y\)) are opposite, \(\angle B\) ( \(X\)) and \(\angle D\) ( \(Z\)) are opposite? Wait, no, in \(ABCD\), \(\angle A\) (at \(A\)) and \(\angle C\) (at \(C\)) are opposite, \(\angle B\) (at \(B\)) and \(\angle D\) (at \(D\)) are opposite. So in \(WXYZ\), \(\angle W\) (at \(W\)) and \(\angle Y\) (at \(Y\)) are opposite, \(\angle X\) (at \(X\)) and \(\angle Z\) (at \(Z\)) are opposite? Wait, that can't be, because in the diagram, \(\angle X\) is \(35^\circ\), and the options include \(35^\circ\) and \(145^\circ\). Wait, maybe I got the consecutive angles wrong. Wait, in the diagram, the parallelogram is drawn with \(X\) connected to \(Y\) and \(W\), \(Y\) connected to \(X\) and \(Z\), \(Z\) connected to \(Y\) and \(W\), \(W\) connected to \(Z\) and \(X\). So the sides \(XY\) and \(WZ\) are parallel, \(XW\) and \(YZ\) are parallel. So \(\angle X\) (at \(X\), between \(XW\) and \(XY\)) and \(\angle Z\) (at \(Z\), between \(YZ\) and \(WZ\)): since \(XW \parallel YZ\) and \(XY \parallel WZ\), the angle at \(X\) and angle at \(Z\): are they same - side interior angles? Wait, no, \(\angle X\) and \(\angle Z\): let's see, \(XW\) is parallel to \(YZ\), and \(WZ\) is a transversal. So \(\angle X\) (at \(X\), between \(XW\) and \(XY\)) and \(\angle Z\) (at \(Z\), between \(YZ\) and \(WZ\)): actually, in a parallelogram, consecutive angles (adjacent) are supplementary. Wait, maybe \(\angle X\) and \(\angle Z\) are consecutive? No, that doesn't make sense. Wait, let's look at the diagram again: the parallelogram has vertices \(X, Y, Z, W\) (maybe the order is \(X - Y - Z - W - X\)). So \(XY\) is parallel to \(WZ\), and \(XW\) is parallel to \(YZ\). So \(\angle X\) is between \(XW\) and \(XY\), \(\angle Z\) is between \(YZ\) and \(WZ\). Since \(XW \parallel YZ\) and \(XY \parallel WZ\), then \(\angle X\) and \(\angle Z\): wait, \(\angle X\) and \(\angle W\) are consecutive (between \(XW\) and \(XY\), and \(XW\) and \(WZ\)), so they should be supplementary. But \(\angle X\) is \(35^\circ\), so \(\angle W\) would be \(180 - 35 = 145^\circ\). Then \(\angle Z\) is opposite to \(\angle X\)? No, wait, \(\angle X\) and \(\angle Z\): if \(X - Y - Z - W - X\), then \(\angle X\) (at \(X\)) and \(\angle Z\) (at \(Z\)): are they opposite? Wait, no, \(\angle X\) is adjacent to \(\angle Y\) and \(\angle W\), \(\angle Z\) is adjacent to \(\angle Y\) and \(\angle W\). Wait, maybe the correct property is that in a parallelogram, consecutive angles (adjacent) are supplementary, and opposite angles are equal. So \(\angle X\) and \(\angle Z\): are they opposite or adjacent? Wait, in the diagram, \(\angle X\) is at vertex \(X\), between \(XW\) and \(XY\), \(\angle Z\) is at vertex \(Z\), between \(YZ\) and \(WZ\). Since \(XY \parallel WZ\) and \(XW \parallel YZ\), then \(\angle X\) and \(\angle Z\) are same - side interior angles? No, that's not right. Wait, maybe I made a mistake in the property. Let's recall: in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. So if \(WXYZ\) is a parallelogram, then \(\angle X = \angle Z\) if they are opposite, or \(\angle X + \angle Z = 180^\circ\) if they are consecutive. Wait, the options are \(25^\circ\), \(35^\circ\), \(55^\circ\), \(145^\circ\). Since \(\angle X\) is \(35^\circ\), if \(\angle Z\) is supplementary, it would be \(145^\circ\), but if they are equal, it's \(35^\circ\). Wait, let's check the sides: \(X\) to \(Y\) to \(Z\) to \(W\) to \(X\). So \(XY\) is parallel to \(WZ\), \(XW\) is parallel to \(YZ\). So the angle at \(X\) (\(\angle X\)) and angle at \(Z\) (\(\angle Z\)): when you have a parallelogram, the angles adjacent to a side are supplementary. Wait, maybe \(\angle X\) and \(\angle Z\) are consecutive angles (adjacent), so they should be supplementary. Wait, no, that can't be, because in a parallelogram, opposite angles are equal. Wait, let's take a concrete example: a parallelogram with angle \(35^\circ\) at \(X\), then the angle opposite to \(X\) (say \(Z\)) should be \(35^\circ\), and the adjacent angles ( \(Y\) and \(W\)) should be \(145^\circ\). Wait, but in the diagram, the parallelogram is drawn such that \(X\) and \(Z\) are at the "ends" of the slanted sides. Wait, maybe the correct property is that in a parallelogram, opposite angles are equal. So \(\angle X = \angle Z\). So if \(\angle X = 35^\circ\), then \(\angle Z = 35^\circ\)? But that seems conflicting with the supplementary idea. Wait, no, I think I messed up the consecutive angles. Let's re - state the properties of a parallelogram:

1. Opposite sides are parallel and equal.
2. Opposite angles are equal.
3. Consecutive (adjacent) angles are supplementary (sum to \(180^\circ\)).

So in parallelogram \(WXYZ\), let's define the vertices in order \(W, X, Y, Z\), so:

- \(WX \parallel YZ\) and \(WX = YZ\)
- \(XY \parallel WZ\) and \(XY = WZ\)
- \(\angle W=\angle Y\) and \(\angle X = \angle Z\)
- \(\angle W+\angle X = 180^\circ\), \(\angle X+\angle Y = 180^\circ\), etc.

Wait, now I see my mistake earlier. So \(\angle X\) and \(\angle Z\) are opposite angles, so they are equal. Therefore, if \(\angle X = 35^\circ\), then \(\angle Z = 35^\circ\). But wait, the options include \(35^\circ\) and \(145^\circ\). Wait, maybe the order of the vertices is different. Maybe the parallelogram is \(X - Y - Z - W - X\), so \(X\) is connected to \(Y\) and \(W\), \(Y\) to \(X\) and \(Z\), \(Z\) to \(Y\) and \(W\), \(W\) to \(Z\) and \(X\). So \(XY\) is parallel to \(WZ\), \(XW\) is parallel to \(YZ\). Then \(\angle X\) is between \(XW\) and \(XY\), \(\angle Z\) is between \(YZ\) and \(WZ\). Now, \(XW\) is parallel to \(YZ\), and \(WZ\) is a transversal. So the angle at \(X\) (\(\angle X\)) and angle at \(Z\) (\(\angle Z\)): are they same - side interior angles? Wait, no, \(\angle X\) and \(\angle W\) are same - side interior angles with respect to transversal \(XW\) and parallel lines \(XY\) and \(WZ\), so they should be supplementary. But \(\angle X\) and \(\angle Z\): if \(XW \parallel YZ\) and \(XY \parallel WZ\), then \(\angle X\) and \(\angle Z\) are actually equal, because they are opposite angles. Wait, I think the key is to look at the diagram: the angle at \(X\) is \(35^\circ\), and in the parallelogram, \(\angle X\) and \(\angle Z\) are opposite angles, so they are equal. Therefore, \(\angle Z = 35^\circ\). Wait, but let's check the supplementary angles. If \(\angle X = 35^\circ\), then the angle adjacent to \(\angle X\) (say \(\angle W\)) would be \(180 - 35=145^\circ\), and \(\angle Z\) is opposite to \(\angle X\), so \(35^\circ\). So the measure of \(\angle Z\) is \(35^\circ\).

Step2: Determine the measure of \(\angle Z\)

Since \(WXYZ\) is a parallelogram, opposite angles are equal. \(\angle X\) and \(\angle Z\) are opposite angles. Given \(\angle X = 35^\circ\), so \(\angle Z=\angle X = 35^\circ\).

Answer:

\(35^\circ\) (the option with \(35^\circ\))