Question

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, alternate interior angles (or opposite angles in terms of congruency) can be considered. Here, we can use the property that opposite angles in a rhombus are equal, or that adjacent angles are supplementary. Let's check the given angles. The angle of \(53^\circ\) and \(x^\circ\) – wait, actually, in a rhombus, opposite angles are equal. Wait, the angle \(53^\circ\) and \(x^\circ\) – wait, no, let's check the other angle. Wait, the angle \(127^\circ\) and the angle opposite to it (the bottom one) should be equal, and the \(53^\circ\) and \(x^\circ\) should be equal? Wait, no, let's calculate using the sum of angles in a quadrilateral. The sum of interior angles in a quadrilateral is \(360^\circ\). So, let's denote the angles: \(127^\circ\), \(127^\circ\) (opposite), \(53^\circ\), and \(x^\circ\) (opposite to \(53^\circ\)). Wait, no, maybe adjacent angles. Wait, in a rhombus, adjacent angles are supplementary. So, if one angle is \(127^\circ\), the adjacent angle should be \(180 - 127 = 53^\circ\)? Wait, no, the given angle is \(53^\circ\), and \(x\) – wait, maybe the \(53^\circ\) and \(x\) are equal? Wait, no, let's do the sum. Let's sum all angles: \(127 + 127 + 53 + x = 360\). Wait, no, that would be \(127 + 127 = 254\), \(53 + x = 106\), so \(x = 53\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, the rhombus has two pairs of equal angles. So, if one angle is \(127^\circ\), the angle opposite to it is also \(127^\circ\), and the other two angles (opposite to each other) are equal. So, sum of angles: \(2 \times 127 + 2 \times x = 360\)? Wait, no, the given angle is \(53^\circ\), so maybe \(53\) is one of the equal angles. Wait, let's recalculate. Sum of interior angles in a quadrilateral: \(360^\circ\). So, \(127 + 127 + 53 + x = 360\). Wait, \(127 + 127 = 254\), \(53 + x = 360 - 254 = 106\), so \(x = 106 - 53 = 53\)? Wait, no, that would mean \(x = 53\). Wait, but let's check adjacent angles. In a rhombus, adjacent angles are supplementary. So, if one angle is \(127^\circ\), the adjacent angle is \(180 - 127 = 53^\circ\). So, the \(53^\circ\) angle is adjacent to \(127^\circ\), and \(x\) is adjacent to \(127^\circ\) as well? Wait, no, the rhombus has two angles of \(127^\circ\) and two angles of \(53^\circ\). So, \(x\) should be \(53^\circ\)? Wait, but the given angle is \(53^\circ\), so \(x = 53\)? Wait, that seems right. Wait, let's check: \(127 + 127 + 53 + 53 = 127 \times 2 + 53 \times 2 = 254 + 106 = 360\), which matches the sum of interior angles in a quadrilateral. So, \(x = 53\).

Step2: Confirm using rhombus properties

In a rhombus, opposite angles are equal. The angle labeled \(53^\circ\) and the angle labeled \(x^\circ\) are opposite angles, so they must be equal. Therefore, \(x = 53\).

Answer:

\(53\)