Question

classify quadrilaterals
a rhombus is shown below.
image of a rhombus with angles 127°, 53°, and x°
what is the value of x?
write your answer in the box.
□°

Explanation:

Step1: Recall rhombus angle properties

In a rhombus, opposite angles are equal, and adjacent angles are supplementary (sum to \(180^\circ\)). Also, the sum of all interior angles of a quadrilateral is \(360^\circ\). Alternatively, we can use the property that opposite angles are equal. Here, the angle of \(53^\circ\) and \(x^\circ\) – wait, no, wait: Wait, in a rhombus, opposite angles are equal. Wait, the given angles: one angle is \(127^\circ\), another is \(53^\circ\), and we need to find \(x\). Wait, actually, in a rhombus, opposite angles are equal. So the angle with \(53^\circ\) and \(x^\circ\) – wait, no, let's check the sum. Wait, the sum of interior angles of a quadrilateral is \(360^\circ\). So let's denote the angles: two angles of \(53^\circ\) (if opposite), two angles of \(127^\circ\)? Wait, no, the diagram shows one angle \(127^\circ\), one \(53^\circ\), and \(x\), and the fourth angle. Wait, no, in a rhombus, opposite angles are equal. So the angle labeled \(53^\circ\) and \(x^\circ\) – wait, maybe I made a mistake. Wait, let's use the property that adjacent angles in a rhombus are supplementary. Wait, \(53^\circ\) and \(127^\circ\): \(53 + 127 = 180\), so they are adjacent and supplementary. Then, the angle \(x\) should be equal to \(53^\circ\)? Wait, no, wait the diagram: the rhombus has angles: \(127^\circ\), \(53^\circ\), \(x\), and the fourth angle. Wait, in a rhombus, opposite angles are equal. So the angle with \(53^\circ\) and \(x\) – are they opposite? Wait, looking at the diagram, the angle of \(53^\circ\) and \(x^\circ\) are opposite? Wait, no, maybe the \(53^\circ\) and \(x\) are opposite. Wait, let's calculate the sum. The sum of all angles in a quadrilateral is \(360^\circ\). So \(127 + 53 + x + \text{fourth angle} = 360\). But in a rhombus, opposite angles are equal, so the angle opposite \(53^\circ\) is \(x\), and the angle opposite \(127^\circ\) is the fourth angle. Wait, no, that can't be. Wait, maybe I misread. Wait, the diagram: the rhombus has one angle at the top \(127^\circ\), left angle \(53^\circ\), right angle \(x^\circ\), and bottom angle (unlabeled). Wait, in a rhombus, opposite angles are equal. So left angle (\(53^\circ\)) and right angle (\(x^\circ\)) – are they opposite? Yes, because in a rhombus, opposite angles are equal. Wait, but \(53 + 127 = 180\), which are supplementary (adjacent angles). So the angle opposite \(53^\circ\) should be equal to \(53^\circ\), so \(x = 53\)? Wait, but let's check the sum. \(127 + 127 + 53 + 53 = 127*2 + 53*2 = 254 + 106 = 360\), which matches the sum of interior angles of a quadrilateral. So yes, in a rhombus, opposite angles are equal. So the angle with \(53^\circ\) and \(x^\circ\) are opposite, so \(x = 53\). Wait, but let's confirm with adjacent angles. Adjacent angles in a rhombus are supplementary. So \(53^\circ\) and \(127^\circ\) are adjacent (since \(53 + 127 = 180\)), so they are supplementary. Then the angle opposite \(53^\circ\) is \(x\), so \(x = 53\).

Step2: Confirm with angle sum

Sum of interior angles of quadrilateral: \(360^\circ\). So if two angles are \(127^\circ\) (opposite) and two are \(53^\circ\) (opposite), then \(2*127 + 2*53 = 254 + 106 = 360\), which is correct. Therefore, the angle \(x\) is equal to \(53^\circ\) because it is opposite to the \(53^\circ\) angle.

Answer:

\(53\)