Question

find the measure of the three missing angles in the parallelogram below.
answer attempt 1 out of 3
x =
y =
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 \(x\)

The angle \(69^\circ\) and \(x\) are consecutive? No, wait, in a parallelogram, consecutive angles are supplementary, but also, opposite angles are equal. Wait, the angle given is \(69^\circ\), and \(x\) is adjacent? Wait, no, let's look at the diagram. The angle \(69^\circ\) and \(x\): Wait, in a parallelogram, opposite angles are equal. Wait, maybe the \(69^\circ\) and \(y\) are... Wait, no, let's correct. In a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)), and opposite angles are equal. So if one angle is \(69^\circ\), its consecutive angle is \(180 - 69 = 111^\circ\), and opposite angles are equal.

Wait, let's identify the angles. The angle \(69^\circ\) and \(x\): Wait, maybe \(x\) is opposite to \(z\), and \(69^\circ\) is opposite to \(y\)? No, wait, the diagram: the angle \(69^\circ\) is at the bottom, \(x\) is at the left, \(z\) at the top, \(y\) at the right. So in a parallelogram, opposite angles are equal: so \(x = z\), \(y = 69^\circ\)? Wait, no, that can't be. Wait, no, consecutive angles: \(69^\circ\) and \(x\) are consecutive? Wait, no, let's think again.

Wait, in a parallelogram, adjacent angles (consecutive) are supplementary. So if one angle is \(69^\circ\), the angle adjacent to it (consecutive) is \(180 - 69 = 111^\circ\). And opposite angles are equal. So:

• \(y\) is opposite to \(69^\circ\), so \(y = 69^\circ\) (opposite angles equal).
• \(x\) is opposite to \(z\), and \(x\) is consecutive to \(69^\circ\), so \(x = 180 - 69 = 111^\circ\), so \(z = x = 111^\circ\) (opposite angles equal).

Wait, let's verify:

In a parallelogram, sum of all angles is \(360^\circ\). So \(69 + x + 69 + x = 360\) (since \(y = 69\), \(z = x\)). So \(2*69 + 2x = 360\) → \(138 + 2x = 360\) → \(2x = 360 - 138 = 222\) → \(x = 111\). Then \(z = x = 111\), \(y = 69\).

Wait, let's correct:

Wait, the angle \(69^\circ\) and \(x\) are consecutive? No, maybe \(69^\circ\) and \(x\) are adjacent? Wait, no, let's look at the diagram. The angle \(69^\circ\) is at the bottom, \(x\) at the left, \(z\) at the top, \(y\) at the right. So the sides: bottom and left are adjacent, so the angles at bottom (\(69^\circ\)) and left (\(x\)) are consecutive (adjacent), so they should be supplementary. So \(69 + x = 180\) → \(x = 180 - 69 = 111^\circ\). Then, since opposite angles are equal, \(x = z\) (left and top angles), so \(z = 111^\circ\). And \(y\) is opposite to \(69^\circ\) (bottom and right angles), so \(y = 69^\circ\).

Yes, that makes sense. So:

• \(y = 69^\circ\) (opposite to \(69^\circ\))
• \(x = 180 - 69 = 111^\circ\) (consecutive to \(69^\circ\))
• \(z = x = 111^\circ\) (opposite to \(x\))

Step3: Confirm the values

• \(x = 111^\circ\) (consecutive to \(69^\circ\), supplementary)
• \(y = 69^\circ\) (opposite to \(69^\circ\))
• \(z = 111^\circ\) (opposite to \(x\))

Answer:

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