Question

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

Explanation:

Step1: Recall parallelogram angle property

In a parallelogram, consecutive angles are supplementary, and opposite angles are equal? Wait, no, wait. Wait, in parallelogram WXYZ, sides WX and YZ are parallel, and sides XY and WZ are parallel. So angle X and angle Z: Wait, no, angle X and angle W are consecutive, angle X and angle Z: Wait, let's label the parallelogram. The vertices are W, X, Y, Z in order? Wait, the figure is X connected to Y, Y to Z, Z to W, W to X? Wait, the labels are X, Y, Z, W. So sides XY and WZ are parallel, sides XW and YZ are parallel. So angle at X (∠X) and angle at Z (∠Z): Wait, no, consecutive angles? Wait, in a parallelogram, consecutive angles are supplementary (sum to 180°), and opposite angles are equal. Wait, let's check the sides. If WXYZ is a parallelogram, then WX || YZ and XY || WZ. So angle X and angle W are consecutive (between WX and XY), angle X and angle Z: Wait, angle X is at vertex X, between sides XW and XY. Angle Z is at vertex Z, between sides YZ and WZ. Since XW || YZ and XY || WZ, then angle X and angle Z: Wait, no, angle X and angle W are consecutive, angle X and angle Y are... Wait, maybe I got the order wrong. Let's look at the diagram: X is connected to Y (top side), Y to Z (right side), Z to W (bottom side), W to X (left side). So the sides: XY (top), YZ (right), ZW (bottom), WX (left). So XY || ZW, and WX || YZ. So angle at X (∠X) is between WX (left) and XY (top), angle at Z (∠Z) is between YZ (right) and ZW (bottom). Since WX || YZ and XY || ZW, then angle X and angle Z: Wait, angle X and angle W are consecutive (WX and XY), angle X and angle Z: Wait, no, angle X and angle Z are... Wait, in a parallelogram, consecutive angles are supplementary. Wait, angle X (35°) and angle W: are they consecutive? Then angle X + angle W = 180°, angle W + angle Z = 180°, so angle X = angle Z? Wait, no, that would be if they are opposite. Wait, maybe I mixed up opposite and consecutive. Let's recall: In a parallelogram, opposite angles are equal, and consecutive angles are supplementary. So opposite angles: ∠X and ∠Z? Wait, no, ∠X and ∠Y? Wait, no, let's label the vertices in order: W, X, Y, Z. So the angles: ∠W, ∠X, ∠Y, ∠Z. Then opposite angles: ∠W and ∠Y, ∠X and ∠Z. Wait, no, in a parallelogram, the vertices are in order, so W connected to X, X to Y, Y to Z, Z to W. So the angles are at each vertex: ∠W (at W, between ZW and WX), ∠X (at X, between WX and XY), ∠Y (at Y, between XY and YZ), ∠Z (at Z, between YZ and ZW). Then opposite angles: ∠W and ∠Y, ∠X and ∠Z. Wait, no, that's not right. Wait, in a parallelogram, opposite sides are parallel, so WX || YZ and XY || ZW. So the angle at X (∠X) is between WX (left) and XY (top), angle at Z (∠Z) is between YZ (right) and ZW (bottom). Since WX || YZ, and XY || ZW, then the transversal XZ? No, the transversal is XY and ZW. Wait, maybe the correct property is that consecutive angles (adjacent angles) are supplementary. So angle X (35°) and angle Y: are they consecutive? No, angle X and angle W are consecutive (WX and XY), angle X and angle Y are... Wait, maybe the diagram is labeled as X, Y, Z, W in order, so the angles: ∠X (at X), ∠Y (at Y), ∠Z (at Z), ∠W (at W). Then opposite angles: ∠X and ∠Z, ∠Y and ∠W. Wait, no, that would be if it's a parallelogram with vertices in order, so WXYZ: W to X, X to Y, Y to Z, Z to W. So sides: WX, XY, YZ, ZW. Then opposite sides: WX || YZ, XY || ZW. Opposite angles: ∠W and ∠Y, ∠X and ∠Z. Wait, no, that's not correct. Wait, let's take a standard parallelogram ABCD, with AB || CD and AD || BC. Then opposite angles: ∠A = ∠C, ∠B = ∠D. Consecutive angles: ∠A + ∠B = 180°, ∠B + ∠C = 180°, etc. So applying that to WXYZ: Let's map A to W, B to X, C to Y, D to Z. Then AB is WX, BC is XY, CD is YZ, DA is ZW. Then opposite angles: ∠W (A) = ∠Y (C), ∠X (B) = ∠Z (D). Wait, that makes sense. So ∠X (at X, which is B) is equal to ∠Z (at Z, which is D). Wait, but in the diagram, ∠X is 35°, so ∠Z would be 35°? But that contradicts the consecutive angles. Wait, no, maybe I mapped it wrong. Alternatively, maybe the sides are XY and WZ (top and bottom), and XW and YZ (left and right). So XY || WZ, XW || YZ. Then angle at X (∠X) is between XW (left) and XY (top), angle at Z (∠Z) is between YZ (right) and WZ (bottom). Since XW || YZ and XY || WZ, then angle X and angle Z: are they consecutive? Wait, angle X and angle W are consecutive (XW and XY), so ∠X + ∠W = 180°. Angle W and angle Z are consecutive (WZ and YZ), so ∠W + ∠Z = 180°. Therefore, ∠X = ∠Z. Wait, because ∠X + ∠W = 180° and ∠W + ∠Z = 180°, so ∠X = ∠Z. So if ∠X is 35°, then ∠Z is also 35°? But wait, the options include 35°, so that would be the answer. Wait, but let's check again. If WXYZ is a parallelogram, then opposite angles are equal. Wait, maybe I had the opposite angles wrong. Let's look at the diagram: the vertices are X, Y, Z, W. So the angles: ∠X (at X), ∠Y (at Y), ∠Z (at Z), ∠W (at W). So sides: XY (X to Y), YZ (Y to Z), ZW (Z to W), WX (W to X). So XY || ZW, XW || YZ. Therefore, ∠X and ∠Z: are they opposite? No, ∠X is between XW and XY, ∠Z is between YZ and ZW. Since XW || YZ and XY || ZW, then ∠X and ∠Z are... Wait, ∠X and ∠Y are consecutive (XY and YZ), ∠Y and ∠Z are consecutive (YZ and ZW), ∠Z and ∠W are consecutive (ZW and WX), ∠W and ∠X are consecutive (WX and XY). So consecutive angles: ∠X + ∠Y = 180°, ∠Y + ∠Z = 180°, ∠Z + ∠W = 180°, ∠W + ∠X = 180°. Therefore, ∠X = ∠Z (because ∠X + ∠Y = 180° and ∠Y + ∠Z = 180°, so ∠X = ∠Z) and ∠Y = ∠W. So if ∠X is 35°, then ∠Z is also 35°? But wait, that would mean they are equal, but consecutive angles are supplementary. Wait, no, ∠X and ∠Z are not consecutive, they are... Wait, maybe I made a mistake. Let's take a concrete example: a parallelogram with ∠X = 35°, then ∠W (consecutive to ∠X) is 180° - 35° = 145°, then ∠Z (consecutive to ∠W) is 180° - 145° = 35°, so ∠Z = 35°. Yes, that makes sense. So ∠Z is equal to ∠X, which is 35°.

Step2: Determine the measure of ∠Z

Since WXYZ is a parallelogram, opposite angles are equal? Wait, no, in the consecutive angles, we saw that ∠X = ∠Z. Wait, actually, in a parallelogram, opposite angles are equal. Wait, ∠X and ∠Z: are they opposite? Let's see the vertices: X, Y, Z, W. So opposite vertices: X and Z, Y and W. Therefore, opposite angles: ∠X (at X) and ∠Z (at Z) are opposite, so they are equal. Therefore, ∠Z = ∠X = 35°.

Answer:

35° (the option with 35°)