Question

quadrilateral rstu is a rhombus. what is ( mangle ruv )? ( mangle ruv = square^circ )

Explanation:

Step1: Recall properties of a rhombus

In a rhombus, the diagonals are perpendicular bisectors of each other and they bisect the angles of the rhombus. Also, adjacent angles in a rhombus are supplementary, but here we use the fact that the diagonals form right triangles? Wait, no, first, in triangle \( RVS \) or \( RUV \), we know that in a rhombus, diagonals bisect the angles. Wait, the angle at \( R \) between \( SR \) and \( VR \) is \( 65^\circ \), and since diagonals in a rhombus are perpendicular? Wait, no, actually, in a rhombus, the diagonals are not necessarily perpendicular? Wait, no, wait: In a rhombus, the diagonals are perpendicular bisectors of each other. Wait, no, that's a property: The diagonals of a rhombus are perpendicular. Wait, no, let's correct: In a rhombus, the diagonals bisect the angles and are perpendicular. Wait, no, actually, the diagonals of a rhombus are perpendicular bisectors of each other. So, angle \( \angle RVS \) is \( 90^\circ \)? Wait, no, the diagram shows angle at \( R \) between \( SR \) and \( VR \) is \( 65^\circ \), and we need to find \( \angle RUV \). Wait, let's look at triangle \( RUV \). Since \( RSTU \) is a rhombus, \( RU = RS \)? No, all sides of a rhombus are equal, so \( RU = RS \), but the diagonals bisect the angles. Wait, the diagonal \( SU \) bisects angle \( U \), and diagonal \( RT \) bisects angle \( R \) and angle \( T \). Wait, maybe a better approach: In the rhombus, the diagonals are perpendicular? Wait, no, actually, in a rhombus, the diagonals are perpendicular. Wait, let's check the angles. The angle given is \( 65^\circ \) at \( R \) between \( SR \) and \( VR \). Then, in triangle \( RUV \), we know that one angle is \( 90^\circ \) (since diagonals are perpendicular) and another angle? Wait, no, maybe I made a mistake. Wait, the diagonals of a rhombus bisect the vertex angles. So, if angle \( \angle SRV = 65^\circ \), then since diagonal \( RU \) (wait, no, the diagonals are \( RT \) and \( SU \), intersecting at \( V \). So, \( \angle RVS = 90^\circ \)? Wait, no, the diagonals of a rhombus are perpendicular, so \( \angle RVU = 90^\circ \)? Wait, no, let's start over.

In a rhombus, all sides are equal, and the diagonals bisect the angles. Also, the diagonals are perpendicular to each other. Wait, yes! The diagonals of a rhombus are perpendicular. So, \( \angle RVU = 90^\circ \). Now, in triangle \( RVS \), we have angle at \( R \) is \( 65^\circ \), angle at \( V \) is \( 90^\circ \)? No, that can't be, because the sum of angles in a triangle is \( 180^\circ \). Wait, maybe the angle given is \( \angle SRV = 65^\circ \), and we need to find \( \angle RUV \). Let's consider triangle \( RUV \). Since \( RSTU \) is a rhombus, \( RU = RS \)? No, all sides are equal, so \( RU = RS \), but the diagonals bisect the angles. Wait, the diagonal \( SU \) bisects angle \( S \) and angle \( U \), and diagonal \( RT \) bisects angle \( R \) and angle \( T \). So, angle \( \angle SRV = 65^\circ \), which is half of angle \( \angle SRU \)? Wait, no, angle \( \angle SRU \) is an angle of the rhombus, and the diagonal \( RT \) bisects it, so \( \angle SRV = \angle VRU = 65^\circ \)? No, that would make angle \( \angle SRU = 130^\circ \), and then the adjacent angle would be \( 50^\circ \), but maybe not. Wait, alternatively, in triangle \( RUV \), we know that one angle is \( 90^\circ \) (because diagonals are perpendicular) and another angle is \( 65^\circ \)? No, that would make the third angle \( 25^\circ \). Wait, let's calculate: In triangle \( RUV \), angles sum to \( 180^\circ \). If \( \angle RVU = 90^\circ \) (diagonals perpendicular) and \( \angle VRU = 65^\circ \), then \( \angle RUV = 180^\circ - 90^\circ - 65^\circ = 25^\circ \). Wait, that makes sense. Because the diagonals are perpendicular, so \( \angle RVU = 90^\circ \), and the angle at \( R \) in triangle \( RUV \) is \( 65^\circ \), so the remaining angle \( \angle RUV = 180 - 90 - 65 = 25^\circ \).

Step2: Calculate \( m\angle RUV \)

Using the triangle angle sum property: \( m\angle RUV + m\angle VRU + m\angle RVU = 180^\circ \). We know \( m\angle VRU = 65^\circ \) (given, or from the rhombus diagonal bisecting the angle? Wait, actually, the diagonal \( RT \) bisects angle \( R \), but in the diagram, the angle between \( SR \) and \( VR \) is \( 65^\circ \), so \( \angle VRU = 65^\circ \), and \( \angle RVU = 90^\circ \) (diagonals of rhombus are perpendicular). So:

\( m\angle RUV = 180^\circ - 90^\circ - 65^\circ \)

\( m\angle RUV = 25^\circ \)

Answer:

\( 25 \)