Question

find the measure of the three missing angles in the parallelogram below.

answer attempt 1 out of 3
x =
y = 112
z =

Explanation:

Step1: Recall parallelogram angle properties

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

Step2: Find \(x\) and \(z\) (opposite angles)

The given angle is \(112^\circ\). The angle opposite to \(112^\circ\) (let's say \(x\) or \(z\)? Wait, no: the angle \(112^\circ\) and angle \(x\) – wait, no, in the parallelogram, consecutive angles are supplementary. Wait, the angle \(112^\circ\) and angle \(x\): wait, no, let's label the parallelogram. Let's say the angles are: top angle \(x\), right angle \(112^\circ\), bottom angle \(z\), left angle \(y = 112^\circ\) (since opposite angles are equal). Then consecutive angles: \(x\) and \(112^\circ\) are consecutive? Wait, no, in a parallelogram, adjacent angles are supplementary. So if one angle is \(112^\circ\), the adjacent angle is \(180 - 112 = 68^\circ\). So \(x\) and \(z\) should be \(68^\circ\), and \(y\) is \(112^\circ\) (opposite to \(112^\circ\)). Wait, let's correct: in a parallelogram, opposite angles are equal. So angle opposite to \(112^\circ\) is \(y\)? Wait, no, the diagram: the right angle is \(112^\circ\), left angle is \(y\), top is \(x\), bottom is \(z\). So opposite angles: \(x\) and \(z\) are opposite? No, wait, in a parallelogram, opposite angles are equal. So if right angle is \(112^\circ\), then left angle \(y\) is equal to right angle? No, wait, no: adjacent angles are supplementary. Wait, maybe I mixed up. Let's recall: in a parallelogram, \(AB \parallel CD\) and \(AD \parallel BC\). So angle at \(A\) and angle at \(B\) are supplementary, angle at \(B\) and angle at \(C\) are supplementary, etc. So if one angle is \(112^\circ\), the angle adjacent to it is \(180 - 112 = 68^\circ\). So the angle opposite to \(112^\circ\) is also \(112^\circ\), and the angle opposite to \(68^\circ\) is \(68^\circ\). So in the diagram, the right angle is \(112^\circ\), so the left angle \(y\) is equal to right angle? Wait, no, the user already filled \(y = 112\), so that's correct (opposite angles). Then the top angle \(x\) and bottom angle \(z\) are opposite to each other, and adjacent to \(112^\circ\), so they should be \(180 - 112 = 68^\circ\). So \(x = 68\), \(z = 68\), \(y = 112\).

Step3: Calculate \(x\) and \(z\)

Since consecutive angles in a parallelogram sum to \(180^\circ\), \(x + 112^\circ = 180^\circ\), so \(x = 180 - 112 = 68^\circ\). Similarly, \(z\) is opposite to \(x\), so \(z = x = 68^\circ\) (opposite angles in parallelogram are equal).

Answer:

\(x = 68\), \(y = 112\), \(z = 68\)