Use tables to find the acute angle between the lines `2y+x=0 and x/(1)+y/(2)=2`.

Answer

To find the acute angle between the lines given by the equations \(2y + x = 0\) and \(\frac{x}{1} + \frac{y}{2} = 2\), we will follow these steps:

### Step 1: Convert the equations to slope-intercept form
1. **First Line:** Rearranging \(2y + x = 0\) gives:
\[ 2y = -x \quad \Rightarrow \quad y = -\frac{1}{2}x \]
The slope \(m_1\) of the first line is \(-\frac{1}{2}\).

2. **Second Line:** Rearranging \(\frac{x}{1} + \frac{y}{2} = 2\) gives:
\[ x + \frac{y}{2} = 2 \quad \Rightarrow \quad \frac{y}{2} = 2 - x \quad \Rightarrow \quad y = 2(2 - x) = 4 - 2x \]
The slope \(m_2\) of the second line is \(-2\).

### Step 2: Use the formula for the angle between two lines
The formula for the tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:
\[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]
Substituting the values of \(m_1\) and \(m_2\):
\[ \tan \theta = \left| \frac{-2 - \left(-\frac{1}{2}\right)}{1 + \left(-\frac{1}{2}\right)(-2)} \right| \]

### Step 3: Simplify the expression
Calculating the numerator:
\[ -2 + \frac{1}{2} = -\frac{4}{2} + \frac{1}{2} = -\frac{3}{2} \]
Calculating the denominator:
\[ 1 + 1 = 2 \]
Thus,
\[ \tan \theta = \left| \frac{-\frac{3}{2}}{2} \right| = \frac{3}{4} \]

### Step 4: Find the angle using the tangent value
Now, we need to find \(\theta\) such that:
\[ \tan \theta = \frac{3}{4} \]
Using trigonometric tables or a calculator, we find:
\[ \theta \approx 37^\circ \]

### Conclusion
The acute angle between the lines \(2y + x = 0\) and \(\frac{x}{1} + \frac{y}{2} = 2\) is approximately \(37^\circ\).

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