Question

find the measure of the three missing angles in the rhombus below.
(an image of a rhombus with one angle labeled 121°, another labeled x°, another y°, another z°)
answer attempt 1 out of 3
x =

y =

z =

Explanation:

Step1: Recall rhombus angle properties

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

Step2: Find \(x\)

The angle adjacent to \(121^\circ\) is \(x\). So \(x + 121^\circ= 180^\circ\), then \(x = 180 - 121 = 59^\circ\)? Wait, no, wait: Wait, in a rhombus, adjacent angles are supplementary. Wait, the given angle is \(121^\circ\), and \(x\) is adjacent? Wait, no, looking at the diagram: the angle labeled \(121^\circ\) and \(x\) – wait, no, in a rhombus, opposite angles are equal, and adjacent angles are supplementary. Wait, the angle \(121^\circ\) and \(x\): wait, maybe I mixed up. Wait, let's re-express: in a rhombus, opposite angles are equal. So the angle opposite to \(121^\circ\) – wait, no, the angle labeled \(121^\circ\) and \(y\) – no, let's look at the diagram. The rhombus has angles: \(z\), \(y\), \(121^\circ\), \(x\). So \(121^\circ\) and \(x\) – wait, no, adjacent angles: \(121^\circ\) and \(x\) are adjacent? Wait, no, in a rhombus, adjacent angles are supplementary. Wait, maybe the angle \(121^\circ\) and \(x\) are adjacent? No, wait, let's correct: in a rhombus, opposite angles are equal, and consecutive (adjacent) angles are supplementary (sum to \(180^\circ\)). So if one angle is \(121^\circ\), its adjacent angle is \(180 - 121 = 59^\circ\)? Wait, no, wait: wait, the angle \(121^\circ\) and \(x\) – maybe \(121^\circ\) and \(x\) are opposite? No, that can't be. Wait, no, let's see: the rhombus has four angles: let's denote them as \(A = z\), \(B = y\), \(C = 121^\circ\), \(D = x\). In a rhombus, \(A = C\)? No, wait, no: opposite angles are equal. So \(A = C\) and \(B = D\)? Wait, no, no: in a parallelogram (rhombus is a parallelogram), opposite angles are equal, and consecutive angles are supplementary. So if angle \(C = 121^\circ\), then angle \(A = z = 121^\circ\)? No, that's not right. Wait, no, consecutive angles: angle \(C\) and angle \(D\) (x) are consecutive, so \(C + D = 180^\circ\), so \(x = 180 - 121 = 59^\circ\). Then angle \(B = y\) is opposite to angle \(C\)? No, angle \(B\) (y) is opposite to angle \(D\) (x)? Wait, no, in a parallelogram, opposite angles are equal. So angle \(A\) (z) is opposite to angle \(C\) (121°), so \(z = 121^\circ\)? No, that can't be, because consecutive angles should be supplementary. Wait, I think I made a mistake. Let's start over.

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

So, let's identify the angles:

- The angle labeled \(121^\circ\) and angle \(x\): are they adjacent? Wait, no, looking at the diagram: the angle \(121^\circ\) is at the bottom-left, \(x\) at the bottom-right, \(y\) at the top-right, \(z\) at the top-left. So in a parallelogram, consecutive angles: bottom-left (\(121^\circ\)) and bottom-right (\(x\)) are consecutive? No, bottom-left and top-left (\(z\)) are consecutive? Wait, no, the sides: the bottom side is between \(121^\circ\) and \(x\), the right side between \(x\) and \(y\), the top side between \(y\) and \(z\), the left side between \(z\) and \(121^\circ\). So consecutive angles: \(121^\circ\) and \(x\) (bottom side), \(x\) and \(y\) (right side), \(y\) and \(z\) (top side), \(z\) and \(121^\circ\) (left side). Wait, no, that's not correct. In a parallelogram, consecutive angles are adjacent (share a side). So \(121^\circ\) and \(x\) share the bottom side? No, \(121^\circ\) is at the bottom-left, \(x\) at the bottom-right: they share the bottom side, so they are consecutive (adjacent) angles. Therefore, they should be supplementary: \(121^\circ + x = 180^\circ\), so \(x = 180 - 121 = 59^\circ\). Then, angle \(y\) is opposite to \(121^\circ\), so \(y = 121^\circ\)? Wait, no, opposite angles: \(121^\circ\) and \(y\) are opposite? Wait, \(121^\circ\) is bottom-left, \(y\) is top-right: yes, they are opposite, so \(y = 121^\circ\). Then angle \(z\) is opposite to \(x\), so \(z = x = 59^\circ\). Wait, that makes sense. Let's verify:

- Consecutive angles: \(121^\circ\) (bottom-left) and \(x\) (bottom-right): \(121 + 59 = 180\), good.
- \(x\) (bottom-right) and \(y\) (top-right): \(59 + 121 = 180\), good.
- \(y\) (top-right) and \(z\) (top-left): \(121 + 59 = 180\), good.
- \(z\) (top-left) and \(121^\circ\) (bottom-left): \(59 + 121 = 180\), good.
- Opposite angles: \(121^\circ\) and \(y\) (121°), equal; \(x\) (59°) and \(z\) (59°), equal. Perfect.

So:

Step1: Find \(x\)

Adjacent to \(121^\circ\), so \(x = 180 - 121 = 59^\circ\).

Step2: Find \(y\)

Opposite to \(121^\circ\), so \(y = 121^\circ\).

Step3: Find \(z\)

Opposite to \(x\), so \(z = x = 59^\circ\).

Answer:

\(x = 59\), \(y = 121\), \(z = 59\)