Question

in rhombus vwxy, ( mangle xwz = 62^circ ), ( vy = 17 ) and ( vz = 15 ). find the measures of ( angle vzw ), ( angle xyz ), and ( angle yxz ).
note: the diagram is not drawn to scale.
draw
( mangle vzw = )
( mangle xyz = )
( mangle yxz = )

Explanation:

Step1: Find \( m\angle VZW \)

In a rhombus, the diagonals are perpendicular bisectors of each other, so \( \angle VZW = 90^\circ \) (since diagonals of a rhombus intersect at right angles).

Step2: Find \( m\angle XYZ \)

We know \( m\angle XWZ = 62^\circ \), and in a rhombus, adjacent angles are supplementary? Wait, no. Wait, triangle \( XWZ \) and the properties of the rhombus. Wait, the diagonals bisect the angles. Also, in triangle \( VWZ \), we can check if it's a right triangle. Wait, \( VY = 17 \), so \( VW = 17 \) (sides of rhombus are equal), \( VZ = 15 \). Wait, in triangle \( VZW \), \( VZ = 15 \), \( VW = 17 \), so by Pythagoras, \( ZW = \sqrt{17^2 - 15^2} = \sqrt{289 - 225} = \sqrt{64} = 8 \). But maybe that's not needed. Wait, \( \angle XWZ = 62^\circ \), and \( \angle XYZ \) is adjacent to \( \angle XWZ \)? Wait, no. Wait, in a rhombus, opposite angles are equal, and adjacent angles are supplementary. Wait, \( \angle XWZ \) is part of angle \( \angle XWY \)? Wait, no. Wait, the diagonal \( WY \) and \( VX \) intersect at \( Z \), right angles. So \( \angle VZW = 90^\circ \). Then, \( \angle XWZ = 62^\circ \), so angle \( \angle XWV = 62^\circ \), but the diagonal bisects the angle? Wait, no. Wait, in rhombus, diagonals bisect the angles. So angle \( \angle XWZ = 62^\circ \), and \( \angle XYZ \) is equal to \( 180^\circ - 2 \times 62^\circ \)? Wait, no. Wait, let's think again. The diagonals of a rhombus bisect the angles. So angle \( \angle XWZ = 62^\circ \), which is half of angle \( \angle XWY \)? No, wait, \( Z \) is the intersection of diagonals. So \( \angle XWZ = 62^\circ \), and \( \angle VZW = 90^\circ \), so in triangle \( XWZ \), angle at \( Z \) is \( 90^\circ \), angle at \( W \) is \( 62^\circ \), so angle at \( X \) is \( 28^\circ \)? Wait, no. Wait, \( \angle XWZ = 62^\circ \), \( \angle VZW = 90^\circ \), so \( \angle ZVW = 90^\circ - 62^\circ = 28^\circ \)? Wait, no, triangle \( VZW \): right-angled at \( Z \), so \( \angle VZW = 90^\circ \), \( \angle XWZ = 62^\circ \) (wait, maybe \( \angle XWZ \) is \( \angle W \) in triangle \( XWZ \)). Wait, maybe I messed up. Let's start over.

In rhombus \( VWXY \), diagonals \( VX \) and \( WY \) intersect at \( Z \), so:

1. Diagonals are perpendicular: \( \angle VZW = 90^\circ \).

2. Diagonals bisect the angles: So \( \angle XWZ = 62^\circ \) is half of angle \( \angle XWY \)? No, \( \angle XWZ \) is an angle in triangle \( XWZ \), which is right-angled at \( Z \). Wait, \( \angle XWZ = 62^\circ \), \( \angle XZW = 90^\circ \), so \( \angle WXZ = 180^\circ - 90^\circ - 62^\circ = 28^\circ \). Then, angle \( \angle YXZ \) is equal to \( \angle WXZ = 28^\circ \) (since diagonals bisect the angles, so \( \angle YXZ = \angle WXZ \)). Then, angle \( \angle XYZ \): in the rhombus, adjacent angles are supplementary. Wait, angle \( \angle XWZ \) is \( 62^\circ \), but angle \( \angle XWZ \) is part of angle \( \angle XWY \), which is \( 62^\circ \times 2 = 124^\circ \)? No, wait, no. Wait, \( \angle XWZ = 62^\circ \), and since diagonals are perpendicular, \( \angle VZW = 90^\circ \). Then, angle \( \angle XYZ \): in the rhombus, opposite angles are equal, and adjacent angles are supplementary. Wait, angle \( \angle XWZ \) is \( 62^\circ \), but angle \( \angle XYZ \) is adjacent to angle \( \angle WXY \). Wait, maybe I should use the fact that in triangle \( XWZ \), \( \angle XWZ = 62^\circ \), \( \angle XZW = 90^\circ \), so \( \angle WXZ = 28^\circ \). Then, angle \( \angle YXZ = 28^\circ \) (since diagonal bisects angle \( \angle YXW \)). Then, angle \( \angle XYZ \): since adjacent angles in a rhombus are supplementary, angle \( \angle XYZ = 180^\circ - 2 \times 28^\circ \times 2 \)? No, wait, no. Wait, angle \( \angle YXZ = 28^\circ \), so angle \( \angle YXW = 2 \times 28^\circ = 56^\circ \), then angle \( \angle XYZ = 180^\circ - 56^\circ = 124^\circ \). Wait, that makes sense. Because adjacent angles in a rhombus are supplementary. So \( \angle YXW = 56^\circ \), so \( \angle XYZ = 180^\circ - 56^\circ = 124^\circ \). Then, \( \angle YXZ = 28^\circ \), \( \angle VZW = 90^\circ \).

Let's verify:

- \( \angle VZW \): diagonals of rhombus are perpendicular, so \( 90^\circ \).

- \( \angle XYZ \): adjacent to \( \angle YXW \), which is \( 2 \times 28^\circ = 56^\circ \), so \( 180 - 56 = 124^\circ \).

- \( \angle YXZ \): in triangle \( XYZ \), which is right-angled? Wait, no, diagonals are perpendicular, so \( \angle XZY = 90^\circ \). Wait, \( \angle YXZ = 28^\circ \), as calculated from triangle \( XWZ \) (since \( \angle XWZ = 62^\circ \), \( \angle XZW = 90^\circ \), so \( \angle WXZ = 28^\circ \), and \( \angle YXZ = \angle WXZ \) because diagonal bisects the angle).

So:

- \( m\angle VZW = 90^\circ \)

- \( m\angle XYZ = 124^\circ \) (since adjacent angles are supplementary: \( 180 - 2 \times 62 = 180 - 124 = 56 \)? Wait, no, wait, I think I messed up. Wait, \( \angle XWZ = 62^\circ \), which is angle at \( W \) in triangle \( XWZ \). Then, angle at \( X \) in triangle \( XWZ \) is \( 90 - 62 = 28^\circ \). Then, angle \( \angle YXW \) is \( 2 \times 28 = 56^\circ \) (since diagonal bisects the angle). Then, angle \( \angle XYZ \) is supplementary to \( \angle YXW \), so \( 180 - 56 = 124^\circ \). That's correct.

- \( m\angle YXZ = 28^\circ \)

Step3: Verify each angle

- \( \angle VZW \): diagonals of rhombus intersect at right angles, so \( 90^\circ \).

- \( \angle XYZ \): adjacent to \( \angle YXW \) (which is \( 56^\circ \)), so \( 180 - 56 = 124^\circ \).

- \( \angle YXZ \): half of \( \angle YXW \), which is \( 28^\circ \).

Answer:

\( m\angle VZW = \boxed{90^\circ} \)

\( m\angle XYZ = \boxed{124^\circ} \)

\( m\angle YXZ = \boxed{28^\circ} \)