Question

find the value of ( x ), ( y ), and ( z ) in the rhombus below.

answer attempt 1 out of 3
( x = square ) ( y = square ) ( z = square )

Explanation:

Step1: Recall properties of a rhombus

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

Step2: Solve for \(x\)

The angle \(71^\circ\) and \((-x - 8)^\circ\) are opposite? Wait, no, wait. Wait, in a rhombus, opposite angles are equal. Wait, the angle \(71^\circ\) and \((-6y + 1)^\circ\)? Wait, no, let's look at the angles. Wait, the given angles: one is \(71^\circ\), another is \((-6y + 1)^\circ\), another is \((-x - 8)^\circ\), another is \((-4z + 5)^\circ\). Wait, actually, in a rhombus, opposite angles are equal, and adjacent angles are supplementary. So first, let's identify opposite angles. Let's see: the angle \(71^\circ\) and \((-x - 8)^\circ\)? Wait, no, maybe \(71^\circ\) and \((-6y + 1)^\circ\) are not. Wait, wait, adjacent angles: \(71^\circ\) and \((-6y + 1)^\circ\) are adjacent? No, wait, the rhombus has four angles: let's label them as \(A = 71^\circ\), \(B = (-6y + 1)^\circ\), \(C = (-x - 8)^\circ\), \(D = (-4z + 5)^\circ\). In a rhombus, \(A = C\) and \(B = D\), and \(A + B = 180^\circ\) (adjacent angles supplementary).

Wait, first, let's check \(A\) and \(C\): \(71^\circ = (-x - 8)^\circ\)? Wait, no, that would give \(71 = -x - 8\), so \(x = -79\), but that seems odd. Wait, maybe \(A\) and \(B\) are adjacent? Wait, no, maybe I made a mistake. Wait, in a rhombus, opposite angles are equal, so \(A = C\) and \(B = D\), and adjacent angles are supplementary (\(A + B = 180^\circ\)).

Wait, let's re-express:

Angle \(A = 71^\circ\), angle \(B = (-6y + 1)^\circ\), angle \(C = (-x - 8)^\circ\), angle \(D = (-4z + 5)^\circ\).

Since opposite angles are equal: \(A = C\) and \(B = D\), and \(A + B = 180^\circ\) (adjacent angles).

Wait, so \(A + B = 180^\circ\): \(71 + (-6y + 1) = 180\)? No, that would be \(72 - 6y = 180\), so \(-6y = 108\), \(y = -18\), which is negative, that can't be. Wait, maybe \(A\) and \(D\) are adjacent? Wait, maybe I mixed up the angles. Wait, let's look at the diagram again. The rhombus: one angle is \(71^\circ\), then the next is \((-6y + 1)^\circ\), then \((-x - 8)^\circ\), then \((-4z + 5)^\circ\). So the order is \(71^\circ\), \((-6y + 1)^\circ\), \((-x - 8)^\circ\), \((-4z + 5)^\circ\) going around the rhombus. So adjacent angles: \(71^\circ\) and \((-6y + 1)^\circ\) are adjacent, so they should be supplementary: \(71 + (-6y + 1) = 180\)? Wait, \(72 - 6y = 180\) → \(-6y = 108\) → \(y = -18\), which is negative. That can't be. So maybe \(71^\circ\) and \((-x - 8)^\circ\) are opposite? Wait, no, opposite angles are equal. Wait, maybe \(71^\circ\) and \((-4z + 5)^\circ\) are adjacent? Wait, no, let's think again.

Wait, maybe the angle \(71^\circ\) and \((-x - 8)^\circ\) are adjacent? Wait, no, let's use the property that in a rhombus, opposite angles are equal. So:

1. \(71^\circ = (-x - 8)^\circ\) → \(71 = -x - 8\) → \(x = -79\). But that seems odd. Wait, maybe I got the opposite angles wrong. Wait, maybe \(71^\circ\) and \((-6y + 1)^\circ\) are opposite? Then \(71 = -6y + 1\) → \(-6y = 70\) → \(y = -\frac{35}{3}\), which is also odd. Wait, maybe the angle \(71^\circ\) and \((-4z + 5)^\circ\) are adjacent? Wait, no, let's check the other pair. Wait, maybe the angle \(71^\circ\) and \((-x - 8)^\circ\) are adjacent, so they should be supplementary. So \(71 + (-x - 8) = 180\) → \(63 - x = 180\) → \(-x = 117\) → \(x = -117\). No, that's not right. Wait, maybe I made a mistake in the diagram. Wait, the diagram shows:

Top left: \(71^\circ\)

Top right: \((-6y + 1)^\circ\)

Bottom right: \((-x - 8)^\circ\)

Bottom left: \((-4z + 5)^\circ\)

So the angles are in order: top left (71), top right (-6y+1), bottom right (-x-8), bottom left (-4z+5), then back to top left. So in a rhombus, consecutive angles are adjacent, so top left (71) and top right (-6y+1) are adjacent, so they should be supplementary: \(71 + (-6y + 1) = 180\) → \(72 - 6y = 180\) → \(-6y = 108\) → \(y = -18\). Then top right (-6y+1) and bottom right (-x-8) are opposite? Wait, no, top right and bottom left are opposite? Wait, no, in a rhombus, opposite angles are: top left and bottom right, top right and bottom left. So:

Top left (71) and bottom right (-x - 8) are opposite: so \(71 = -x - 8\) → \(x = -79\).

Top right (-6y + 1) and bottom left (-4z + 5) are opposite: so \(-6y + 1 = -4z + 5\).

But we also know that adjacent angles are supplementary, so top left (71) and top right (-6y + 1) are adjacent: \(71 + (-6y + 1) = 180\) → \(72 - 6y = 180\) → \(y = -18\), as before. Then top right angle: \(-6(-18) + 1 = 108 + 1 = 109^\circ\). Then bottom left angle should be equal to top right: \(-4z + 5 = 109\) → \(-4z = 104\) → \(z = -26\). Then bottom right angle: \(-x - 8\) should be equal to top left: \(71 = -x - 8\) → \(x = -79\). Wait, but let's check the sum of all angles: 71 + 109 + 71 + 109 = 360, which is correct (sum of angles in a quadrilateral is 360). So that works.

Wait, but let's verify again:

1. Solve for \(y\):

Adjacent angles (71° and (-6y + 1)°) are supplementary:

\(71 + (-6y + 1) = 180\)

\(72 - 6y = 180\)

\(-6y = 180 - 72 = 108\)

\(y = 108 / (-6) = -18\)

2. Solve for \(x\):

Opposite angles (71° and (-x - 8)°) are equal:

\(71 = -x - 8\)

\(-x = 71 + 8 = 79\)

\(x = -79\)

3. Solve for \(z\):

Opposite angles (top right (-6y + 1)° and bottom left (-4z + 5)°) are equal:

\(-6y + 1 = -4z + 5\)

We know \(y = -18\), so substitute:

\(-6(-18) + 1 = -4z + 5\)

\(108 + 1 = -4z + 5\)

\(109 = -4z + 5\)

\(-4z = 109 - 5 = 104\)

\(z = 104 / (-4) = -26\)

Let's check all angles:

- Top left: 71°

- Top right: \(-6(-18) + 1 = 108 + 1 = 109°\)

- Bottom right: \(-(-79) - 8 = 79 - 8 = 71°\) (correct, opposite to top left)

- Bottom left: \(-4(-26) + 5 = 104 + 5 = 109°\) (correct, opposite to top right)

Adjacent angles: 71 + 109 = 180 (supplementary), 109 + 71 = 180, etc. So that works.

Step1: Solve for \(x\)

Opposite angles in a rhombus are equal. The angle \(71^\circ\) (top left) and \((-x - 8)^\circ\) (bottom right) are opposite. So:

\(71 = -x - 8\)

Add 8 to both sides: \(71 + 8 = -x\) → \(79 = -x\) → \(x = -79\)

Step2: Solve for \(y\)

Adjacent angles in a rhombus are supplementary. The angle \(71^\circ\) (top left) and \((-6y + 1)^\circ\) (top right) are adjacent. So:

\(71 + (-6y + 1) = 180\)

Simplify: \(72 - 6y = 180\)

Subtract 72 from both sides: \(-6y = 180 - 72 = 108\)

Divide by -6: \(y = \frac{108}{-6} = -18\)

Step3: Solve for \(z\)

Opposite angles in a rhombus are equal. The angle \((-6y + 1)^\circ\) (top right) and \((-4z + 5)^\circ\) (bottom left) are opposite. We already found \(y = -18\), so substitute \(y\) into \(-6y + 1\):

\(-6(-18) + 1 = 108 + 1 = 109\)

So set \(-4z + 5 = 109\):

Subtract 5 from both sides: \(-4z = 109 - 5 = 104\)

Divide by -4: \(z = \frac{104}{-4} = -26\)

Answer:

\(x = -79\), \(y = -18\), \(z = -26\)