Question

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

answer

( x = )
( y = )
( z = )

Explanation:

Step1: Recall properties of a rhombus

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

Step2: Find angle \(x\)

The given angle is \(65^\circ\). Angle \(x\) is adjacent to the \(65^\circ\) angle, so \(x + 65^\circ= 180^\circ\). Solving for \(x\), we get \(x = 180 - 65 = 115^\circ\).

Step3: Find angle \(y\)

Angle \(y\) is opposite to the \(x\) angle? No, wait, angle \(y\) is adjacent to the \(65^\circ\) angle? Wait, no, in a rhombus, opposite angles are equal. Wait, the angle given is \(65^\circ\), angle \(z\) is opposite to \(65^\circ\), so \(z = 65^\circ\). Angle \(y\) is opposite to \(x\), so \(y = x = 115^\circ\). Wait, let's re - establish:

1. In a rhombus, opposite angles are equal. So if one angle is \(65^\circ\), its opposite angle (let's say \(z\)) is also \(65^\circ\).
2. Adjacent angles in a rhombus are supplementary (sum to \(180^\circ\)). So the angle adjacent to \(65^\circ\) (let's say \(x\) or \(y\)) will be \(180 - 65=115^\circ\). And since opposite angles are equal, the angle opposite to \(x\) (which is \(y\)) will also be \(115^\circ\), and the angle opposite to \(65^\circ\) (which is \(z\)) will be \(65^\circ\).

So:

• For \(x\): Adjacent to \(65^\circ\), \(x = 180 - 65=115^\circ\)
• For \(y\): Opposite to \(x\), so \(y = x = 115^\circ\)
• For \(z\): Opposite to \(65^\circ\), so \(z = 65^\circ\)

Answer:

\(x = 115\), \(y = 115\), \(z = 65\)