Question

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

rhombus image with one angle labeled 128°, and other angles labeled x°, y°, z°

answer attempt 1 out of 2
x = box quad y = box quad z = box

Explanation:

Step1: Recall properties of a rhombus

In a rhombus, opposite angles are equal, and adjacent angles are supplementary (their sum is \(180^\circ\)).

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

The given angle is \(128^\circ\). The angle adjacent to \(128^\circ\) (let's say \(x\)) will satisfy \(128^\circ + x^\circ = 180^\circ\). Solving for \(x\):
\(x = 180 - 128 = 52\).
Since opposite angles in a rhombus are equal, \(z = x = 52\)? Wait, no—wait, opposite angles: the angle opposite to \(128^\circ\) is equal to \(128^\circ\), and the angle opposite to \(x\) is equal to \(x\). Wait, let's correct:
Adjacent angles: \(128^\circ\) and \(x\) are adjacent, so \(128 + x = 180\) ⇒ \(x = 52\).
Opposite angles: \(y\) is opposite to \(128^\circ\), so \(y = 128\).
\(z\) is opposite to \(x\), so \(z = x = 52\). Wait, no—wait, the rhombus has angles: let's label the rhombus vertices. Let the angle given be \(128^\circ\), then the angle adjacent (next to it) is \(x\), so \(128 + x = 180\) ⇒ \(x = 52\). Then, the angle opposite to \(128^\circ\) is \(y\), so \(y = 128\). The angle opposite to \(x\) is \(z\), so \(z = x = 52\). Wait, no—wait, in a rhombus, opposite angles are equal. So if one angle is \(128^\circ\), its opposite is \(128^\circ\) (that's \(y\)). The other two angles (adjacent to \(128^\circ\)) are equal to each other, so \(x = z\), and \(128 + x = 180\) ⇒ \(x = 52\), so \(z = 52\), and \(y = 128\).

Step3: Verify

Check: \(128 + 52 = 180\) (supplementary, good). Opposite angles: \(128 = 128\), \(52 = 52\) (good).

Answer:

\(x = \boldsymbol{52}\), \(y = \boldsymbol{128}\), \(z = \boldsymbol{52}\)