Question

find the value of x, y, and z in the parallelogram below.
(10x+6)°
74°
(-y-6)°
(-z+8)°
answer attempt 1 out of 2
x =
y =
z =

Explanation:

Step1: 利用平行四边形对角相等

在平行四边形中，对角相等。所以\(10x + 6 = 74\)，解方程：
\(10x = 74 - 6\)
\(10x = 68\)
\(x = \frac{68}{10} = 6.8\)？不对，重新算：\(74 - 6 = 68\)？不，\(74 - 6 = 68\)？不对，\(10x + 6 = 74\)，应该是\(10x = 74 - 6 = 68\)？不，平行四边形邻角互补？哦，我错了，平行四边形的邻角互补，对角相等。所以\(10x + 6\)和\(74^{\circ}\)是邻角吗？不，看图形，\((10x + 6)^{\circ}\)和\(74^{\circ}\)的位置，应该是对角？不，重新看，平行四边形的对角相等，所以\((-y - 6)^{\circ}\)和\(74^{\circ}\)是对角，所以\(-y - 6 = 74\)？不对，\((10x + 6)^{\circ}\)和\((-z + 8)^{\circ}\)是对角？不，正确的平行四边形性质：对角相等，邻角互补（和为\(180^{\circ}\)）。所以\(10x + 6 + 74 = 180\)？因为邻角互补。对，平行四边形邻角互补，所以\(10x + 6\)和\(74^{\circ}\)是邻角，和为\(180^{\circ}\)。所以：

Step1: 求x（邻角互补）

\(10x + 6 + 74 = 180\)
\(10x + 80 = 180\)
\(10x = 180 - 80 = 100\)
\(x = 10\)

Step2: 求y（对角相等）

对角相等，所以\(-y - 6 = 74\)？不，\((-y - 6)^{\circ}\)和\(74^{\circ}\)是对角，所以\(-y - 6 = 74\)？解方程：
\(-y = 74 + 6 = 80\)
\(y = -80\)？不对，应该是\((-y - 6)^{\circ}\)和\((10x + 6)^{\circ}\)的对角？不，重新看，\((-y - 6)^{\circ}\)和\(74^{\circ}\)是对角，所以\(-y - 6 = 74\)，解得\(y = -80\)？或者\((-y - 6)^{\circ}\)和\((-z + 8)^{\circ}\)的邻角？不，正确的，平行四边形对角相等，所以\((-y - 6)^{\circ} = 74^{\circ}\)，所以：
\(-y - 6 = 74\)
\(-y = 80\)
\(y = -80\)

Step3: 求z（对角相等）

\((-z + 8)^{\circ}\)和\((10x + 6)^{\circ}\)是对角，因为\(x = 10\)，所以\(10x + 6 = 106\)，所以\(-z + 8 = 106\)？不对，邻角互补？不，\((-z + 8)^{\circ}\)和\(74^{\circ}\)的邻角？哦，我之前错了，重新整理：

平行四边形的性质：
1. 对角相等：\(\angle A = \angle C\)，\(\angle B = \angle D\)
2. 邻角互补：\(\angle A + \angle B = 180^{\circ}\)

从图中，设四个角为：\(\angle 1 = (10x + 6)^{\circ}\)，\(\angle 2 = 74^{\circ}\)，\(\angle 3 = (-y - 6)^{\circ}\)，\(\angle 4 = (-z + 8)^{\circ}\)

所以\(\angle 1\)和\(\angle 3\)是对角？不，\(\angle 1\)和\(\angle 4\)是邻角，\(\angle 2\)和\(\angle 3\)是邻角，\(\angle 1\)和\(\angle 2\)是邻角，\(\angle 3\)和\(\angle 4\)是邻角？

哦，正确的，平行四边形中，\(\angle 1\)和\(\angle 3\)是对角，\(\angle 2\)和\(\angle 4\)是对角；\(\angle 1\)和\(\angle 2\)是邻角，\(\angle 2\)和\(\angle 3\)是邻角，等等。

所以\(\angle 1 = \angle 3\)，\(\angle 2 = \angle 4\)；\(\angle 1 + \angle 2 = 180^{\circ}\)，\(\angle 2 + \angle 3 = 180^{\circ}\)等。

现在，\(\angle 2 = 74^{\circ}\)，\(\angle 4 = (-z + 8)^{\circ}\)，所以\(\angle 2 = \angle 4\)（对角相等），所以\(-z + 8 = 74\)？不对，\(74^{\circ}\)和\((-z + 8)^{\circ}\)是对角？不，\(\angle 1 = (10x + 6)^{\circ}\)和\(\angle 3 = (-y - 6)^{\circ}\)是对角，\(\angle 2 = 74^{\circ}\)和\(\angle 4 = (-z + 8)^{\circ}\)是对角。同时，\(\angle 1\)和\(\angle 2\)是邻角，所以\(\angle 1 + \angle 2 = 180^{\circ}\)，即\(10x + 6 + 74 = 180\)，解得：

\(10x + 80 = 180\)
\(10x = 100\)
\(x = 10\)，正确。

然后，\(\angle 3 = (-y - 6)^{\circ}\)和\(\angle 2 = 74^{\circ}\)是邻角，所以\(\angle 3 + \angle 2 = 180^{\circ}\)？不，\(\angle 3\)和\(\angle 1\)是对角，所以\(\angle 3 = \angle 1 = 10x + 6 = 106^{\circ}\)，所以\(-y - 6 = 106\)，解得：

\(-y = 106 + 6 = 112\)
\(y = -112\)？不对，之前错了，\(\angle 3\)和\(\angle 1\)是对角，所以\(\angle 3 = \angle 1 = 106^{\circ}\)，所以\(-y - 6 = 106\)，\(-y = 112\)，\(y = -112\)

然后，\(\angle 4 = (-z + 8)^{\circ}\)和\(\angle 2 = 74^{\circ}\)是邻角，所以\(\angle 4 + \angle 2 = 180^{\circ}\)？不，\(\angle 4\)和\(\angle 3\)是邻角，所以\(\angle 4 + \angle 3 = 180^{\circ}\)，因为\(\angle 3 = 106^{\circ}\)，所以\(\angle 4 = 180 - 106 = 74^{\circ}\)？不对，\(\angle 4\)和\(\angle 2\)是对角，所以\(\angle 4 = \angle 2 = 74^{\circ}\)，所以\(-z + 8 = 74\)？不，\(\angle 4 = (-z + 8)^{\circ}\)，如果\(\angle 4 = 74^{\circ}\)，则\(-z + 8 = 74\)，\(-z = 66\)，\(z = -66\)？这显然有问题，说明我对角度的位置判断错误。

重新看图形：四个角的位置，\((10x + 6)^{\circ}\)和\((-z + 8)^{\circ}\)是对角，\(74^{\circ}\)和\((-y - 6)^{\circ}\)是对角。邻角互补，所以\((10x + 6) + 74 = 180\)（因为是邻角），所以\(10x + 6 = 106\)，\(x = 10\)，正确。

然后，\(74^{\circ}\)和\((-y - 6)^{\circ}\)是对角，所以\(-y - 6 = 74\)，\(-y = 80\)，\(y = -80\)

然后，\((10x + 6)^{\circ} = 106^{\circ}\)和\((-z + 8)^{\circ}\)是对角，所以\(-z + 8 = 106\)，\(-z = 98\)，\(z = -98\)？这显然不对，说明我邻角和对角的判断错误。

哦，天啊，我搞反了，平行四边形的对角相等，邻角互补。所以正确的：

如果\(\angle A = (10x + 6)^{\circ}\)，\(\angle B = 74^{\circ}\)，\(\angle C = (-y - 6)^{\circ}\)，\(\angle D = (-z + 8)^{\circ}\)

那么\(\angle A = \angle C\)，\(\angle B = \angle D\)

\(\angle A + \angle B = 180^{\circ}\)（邻角互补）

所以：

1. \(\angle A + \angle B = 180\)：\(10x + 6 + 74 = 180\) → \(10x = 100\) → \(x = 10\)

2. \(\angle B = \angle D\)：\(74 = -z + 8\) → \(-z = 66\) → \(z = -66\)？不对，\(-z + 8 = 74\) → \(-z = 66\) → \(z = -66\)

3. \(\angle A = \angle C\)：\(10x + 6 = -y - 6\) → 代入\(x = 10\)，\(106 = -y - 6\) → \(-y = 112\) → \(y = -112\)

现在检查：\(\angle A = 106^{\circ}\)，\(\angle B = 74^{\circ}\)，\(\angle C = 106^{\circ}\)，\(\angle D = 74^{\circ}\)，和为\(106 + 74 + 106 + 74 = 360^{\circ}\)，正确。所以计算是对的。

Answer:

\(x = 10\)，\(y = -112\)，\(z = -66\)