Question

find the measure of the three missing angles in the parallelogram below.
(parallelogram image with angles x°, y°, z°, 111°)
answer attempt 1 out of 3
x =
y =
z =

Explanation:

Step1: Recall properties of parallelograms

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

Step2: Find \(x\)

The angle \(x\) is opposite to \(z\)? Wait, no, the angle \(111^\circ\) and \(x\)? Wait, no, let's look at the diagram. The angle \(111^\circ\) and \(y\) are consecutive? Wait, no, in a parallelogram, opposite angles are equal. So the angle \(111^\circ\) and \(y\) – wait, no, the angle labeled \(111^\circ\) and the angle \(x\): wait, no, let's correct. In a parallelogram, opposite angles are equal. So the angle \(111^\circ\) and \(y\) – no, wait, the angle \(111^\circ\) and \(x\)? Wait, no, the angle \(111^\circ\) and \(z\)? Wait, no, let's see: in a parallelogram, consecutive angles are supplementary. So the angle \(111^\circ\) and \(x\) are consecutive? Wait, no, the angle \(111^\circ\) and \(x\) – wait, the angle \(111^\circ\) and \(x\): no, the angle \(111^\circ\) and \(y\) are consecutive? Wait, maybe I mixed up. Let's recall: in a parallelogram, opposite angles are equal. So the angle \(111^\circ\) and \(y\) – no, the angle \(111^\circ\) and \(x\) are opposite? Wait, no, the angle \(111^\circ\) and \(y\) are opposite? Wait, no, the diagram: the parallelogram has angles \(y\), \(x\), \(111^\circ\), \(z\). So \(y\) is opposite to \(111^\circ\)? No, \(y\) and \(111^\circ\) – wait, no, in a parallelogram, opposite angles are equal. So \(y = 111^\circ\)? Wait, no, consecutive angles are supplementary. So if one angle is \(111^\circ\), the consecutive angle is \(180 - 111 = 69^\circ\). So \(x\) and \(z\) are equal to \(69^\circ\), and \(y\) is equal to \(111^\circ\). Wait, let's verify.

So, step 1: Opposite angles in a parallelogram are equal. So angle \(y\) is opposite to \(111^\circ\)? Wait, no, the angle \(111^\circ\) and \(y\) – no, the angle \(111^\circ\) and \(y\) are consecutive? Wait, maybe the angle \(111^\circ\) and \(x\) are consecutive. So \(111 + x = 180\), so \(x = 180 - 111 = 69^\circ\). Then, opposite angles: \(x = z = 69^\circ\), and \(y = 111^\circ\) (since \(y\) is opposite to \(111^\circ\)).

Step3: Calculate \(x\)

Consecutive angles are supplementary. So \(111^\circ + x = 180^\circ\). So \(x = 180 - 111 = 69^\circ\).

Step4: Calculate \(y\)

Opposite angles are equal. So \(y = 111^\circ\) (since \(y\) is opposite to the \(111^\circ\) angle).

Step5: Calculate \(z\)

Opposite angles are equal. So \(z = x = 69^\circ\) (since \(z\) is opposite to \(x\)).

Answer:

\(x = 69\), \(y = 111\), \(z = 69\)