Question

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

(image of a parallelogram with angles labeled: y°, x°, 111°, z°)

answer attempt 1 out of 2

x =

y =

z =

Explanation:

Step1: Recall properties of parallelogram

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 the angle \(z\)? Wait, no, the angle \(111^\circ\) and angle \(y\) are consecutive? Wait, no, let's correct. In a parallelogram, opposite angles are equal. So the angle \(111^\circ\) and angle \(y\) are... Wait, no, the angle \(111^\circ\) and angle \(x\)? Wait, no, let's look at the diagram. The angle \(111^\circ\) and angle \(z\)? Wait, no, in a parallelogram, consecutive angles are supplementary. So the angle \(111^\circ\) and angle \(x\) are consecutive? Wait, no, let's see: in a parallelogram, opposite angles are equal. So the angle \(111^\circ\) and angle \(y\) are... Wait, no, the angle \(111^\circ\) and angle \(x\): wait, no, the angle \(111^\circ\) and angle \(z\) are adjacent? Wait, no, let's recall: in a parallelogram, opposite angles are equal, so \(x = z\) and \(y = 111^\circ\)? Wait, no, consecutive angles are supplementary. So if one angle is \(111^\circ\), the consecutive angle is \(180 - 111 = 69^\circ\). So \(x = 69^\circ\), \(y = 111^\circ\), \(z = 69^\circ\). Wait, let's do step by step.

Step1: Find \(y\)

In a parallelogram, opposite angles are equal. So the angle opposite to \(111^\circ\) is \(y\), so \(y = 111^\circ\).

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

Consecutive angles in a parallelogram are supplementary (sum to \(180^\circ\)). So \(x + 111^\circ = 180^\circ\), so \(x = 180 - 111 = 69^\circ\). Similarly, \(z\) is opposite to \(x\), so \(z = x = 69^\circ\).

Answer:

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