Question

find the value of x, y, and z in the rhombus below.
(2z+9)°
(x+8)°
107°
(-y+10)°
answer attempt 2 out of 2
x =
y =
z =

Explanation:

Step1: Recall rhombus angle properties

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

Step2: Solve for \(x\)

The angle \((x + 8)^\circ\) is opposite to \(107^\circ\)? No, wait, adjacent? Wait, no: in a rhombus, opposite angles are equal. Wait, the angle \(107^\circ\) and \((2z + 9)^\circ\) – wait, no, let's check the angles. Wait, the angle \((x + 8)^\circ\) and \((-y + 10)^\circ\) – no, wait, adjacent angles: in a rhombus, adjacent angles are supplementary. Wait, the angle \(107^\circ\) and \((x + 8)^\circ\) – are they adjacent? Wait, no, let's look at the rhombus. The angles: \((x + 8)^\circ\), \((2z + 9)^\circ\), \(107^\circ\), \((-y + 10)^\circ\). In a rhombus, opposite angles are equal. So, \((x + 8)^\circ\) should be equal to \(107^\circ\)? Wait, no, wait: wait, maybe \((x + 8)^\circ\) and \(107^\circ\) are adjacent? Wait, no, let's think again. Wait, in a rhombus, opposite angles are equal, and adjacent angles are supplementary (sum to \(180^\circ\)). Wait, the angle \(107^\circ\) and \((2z + 9)^\circ\) – are they opposite? Wait, no, the angle \((x + 8)^\circ\) and \(107^\circ\) – maybe \((x + 8)^\circ\) is equal to \(107^\circ\)? Wait, no, wait: let's check the other angle. Wait, the angle \((-y + 10)^\circ\) and \((2z + 9)^\circ\) – maybe \((-y + 10)^\circ\) is equal to \(107^\circ\)? No, that can't be. Wait, maybe I made a mistake. Wait, in a rhombus, opposite angles are equal. So, the angle \((x + 8)^\circ\) and \(107^\circ\) – are they opposite? Wait, no, the angle \((2z + 9)^\circ\) and \((-y + 10)^\circ\) – maybe. Wait, no, let's look at the given angles. Wait, the angle \(107^\circ\) is one angle, \((x + 8)^\circ\) is another, \((2z + 9)^\circ\) is another, \((-y + 10)^\circ\) is the fourth. So, opposite angles: \((x + 8)^\circ\) and \(107^\circ\) – if they are opposite, then \(x + 8 = 107\), so \(x = 107 - 8 = 99\)? Wait, no, that can't be, because adjacent angles would be supplementary. Wait, maybe \((x + 8)^\circ\) and \(107^\circ\) are adjacent, so they sum to \(180^\circ\). So, \(x + 8 + 107 = 180\). Then \(x + 115 = 180\), so \(x = 180 - 115 = 65\)? Wait, that makes more sense. Wait, I confused opposite and adjacent. Let's correct: in a rhombus, opposite angles are equal, adjacent angles are supplementary (sum to \(180^\circ\)). So, the angle \((x + 8)^\circ\) and \(107^\circ\) are adjacent, so they sum to \(180^\circ\). So:

\(x + 8 + 107 = 180\)

\(x + 115 = 180\)

\(x = 180 - 115 = 65\)

Then, the angle \((-y + 10)^\circ\) and \(107^\circ\) – are they opposite? Wait, no, the angle \((-y + 10)^\circ\) and \((2z + 9)^\circ\) – maybe \((-y + 10)^\circ\) is equal to \(107^\circ\)? No, that can't be. Wait, no, the angle \((-y + 10)^\circ\) and \((x + 8)^\circ\) – are they opposite? Wait, no, let's see: the angle \((x + 8)^\circ\) and \((-y + 10)^\circ\) – are they adjacent? Wait, no, in a rhombus, opposite angles are equal. So, the angle \((x + 8)^\circ\) should be equal to \((2z + 9)^\circ\)? No, wait, no. Wait, the angle \(107^\circ\) and \((2z + 9)^\circ\) – are they opposite? So, \(2z + 9 = 107\)? Then \(2z = 107 - 9 = 98\), so \(z = 49\). Then, the angle \((-y + 10)^\circ\) and \((x + 8)^\circ\) – are they opposite? Wait, no, \((x + 8)^\circ = 65 + 8 = 73^\circ\), so \((-y + 10)^\circ\) should be equal to \(73^\circ\)? Wait, no, that can't be. Wait, I think I messed up. Wait, let's start over.

In a rhombus, opposite angles are equal, and adjacent angles are supplementary (sum to \(180^\circ\)). So, let's identify the opposite angles. The angle \((x + 8)^\circ\) and \(107^\circ\) – are they adjacent? Wait, the rhombus has four angles: let's label them as A, B, C, D in order. So, angle A: \((x + 8)^\circ\), angle B: \((2z + 9)^\circ\), angle C: \(107^\circ\), angle D: \((-y + 10)^\circ\). Then, angle A and angle C are opposite, so they should be equal. Angle B and angle D are opposite, so they should be equal. Also, angle A and angle B are adjacent, so they sum to \(180^\circ\).

So, angle A = angle C: \((x + 8)^\circ = 107^\circ\)? Wait, no, that would make \(x + 8 = 107\), so \(x = 99\), but then angle A is \(107^\circ\), angle B would be \(180 - 107 = 73^\circ\), so angle B = \(73^\circ\), so angle D (which is opposite angle B) would be \(73^\circ\), so \((-y + 10)^\circ = 73^\circ\), so \(-y + 10 = 73\), so \(-y = 63\), so \(y = -63\), which doesn't make sense. So, that's wrong.

Wait, maybe angle A and angle C are adjacent. No, in a rhombus, opposite angles are equal. So, angle A (x+8) and angle C (107) are opposite, so x+8 = 107? Then x = 99. Then angle B (2z+9) and angle D (-y+10) are opposite, so 2z+9 = -y+10. Also, angle A and angle B are adjacent, so (x+8) + (2z+9) = 180. So, (99 + 8) + (2z + 9) = 180 → 107 + 2z + 9 = 180 → 2z + 116 = 180 → 2z = 64 → z = 32. Then, angle D (-y + 10) = angle B (2*32 + 9 = 73) → -y + 10 = 73 → -y = 63 → y = -63. That's negative, which is impossible. So, my initial assumption is wrong.

Wait, maybe angle A (x+8) and angle D (-y+10) are opposite, and angle B (2z+9) and angle C (107) are opposite. So, angle B = angle C: 2z + 9 = 107 → 2z = 98 → z = 49. Then, angle A and angle D are opposite: x + 8 = -y + 10. Also, angle A and angle B are adjacent: (x + 8) + (2z + 9) = 180. We know z = 49, so 2z + 9 = 107, so (x + 8) + 107 = 180 → x + 115 = 180 → x = 65. Then, angle A = 65 + 8 = 73°, so angle D (opposite angle A) should be 73°, so -y + 10 = 73 → -y = 63 → y = -63. Wait, still negative. That can't be.

Wait, maybe the angle (-y + 10)° is adjacent to 107°, so they sum to 180°. So, (-y + 10) + 107 = 180 → -y + 117 = 180 → -y = 63 → y = -63. Still negative. That can't be. So, maybe the angle (-y + 10)° is equal to (x + 8)°, and 107° is equal to (2z + 9)°. Let's try that.

So, 107° = (2z + 9)° → 2z + 9 = 107 → 2z = 98 → z = 49.

Then, (x + 8)° = (-y + 10)° → x + 8 = -y + 10 → x + y = 2.

Also, adjacent angles sum to 180°, so (x + 8) + 107 = 180 → x + 115 = 180 → x = 65.

Then, from x + y = 2, 65 + y = 2 → y = 2 - 65 = -63. Still negative. Hmm.

Wait, maybe the angle (-y + 10)° is 107°, so -y + 10 = 107 → -y = 97 → y = -97. No, that's worse.

Wait, maybe I made a mistake in the properties. Wait, in a rhombus, opposite angles are equal, and adjacent angles are supplementary. So, if one angle is 107°, then the opposite angle is also 107°, and the adjacent angles are 180 - 107 = 73°. So, the angles are 107°, 73°, 107°, 73°.

So, let's match the given angles:

- (x + 8)° should be 73° → x + 8 = 73 → x = 65.

- (-y + 10)° should be 73° → -y + 10 = 73 → -y = 63 → y = -63. Wait, but y is negative? That seems odd, but maybe it's correct.

- (2z + 9)° should be 107° → 2z + 9 = 107 → 2z = 98 → z = 49.

So, x = 65, y = -63, z = 49.

Let's check:

- x + 8 = 65 + 8 = 73°, which is adjacent to 107°, so 73 + 107 = 180°, correct.

- -y + 10 = -(-63) + 10 = 63 + 10 = 73°, which is adjacent to 107°, so 73 + 107 = 180°, correct.

- 2z + 9 = 2*49 + 9 = 98 + 9 = 107°, which is opposite to 107°, correct.

So, even though y is negative, it's mathematically correct.

Step1: Solve for \(x\)

Adjacent angles in a rhombus sum to \(180^\circ\). So, \((x + 8)^\circ + 107^\circ = 180^\circ\).

\(x + 8 + 107 = 180\)

\(x + 115 = 180\)

\(x = 180 - 115 = 65\)

Step2: Solve for \(z\)

Opposite angles in a rhombus are equal. So, \((2z + 9)^\circ = 107^\circ\).

\(2z + 9 = 107\)

\(2z = 107 - 9 = 98\)

\(z = \frac{98}{2} = 49\)

Step3: Solve for \(y\)

Adjacent angles in a rhombus sum to \(180^\circ\). So, \((-y + 10)^\circ + 107^\circ = 180^\circ\).

\(-y + 10 + 107 = 180\)

\(-y + 117 = 180\)

\(-y = 180 - 117 = 63\)

\(y = -63\)

Answer:

\(x = 65\), \(y = -63\), \(z = 49\)