Find the acute angles between the st. Lines :
`2x-y+3=0` and `x+y-2=0`

Answer

To find the acute angle between the two straight lines given by the equations \(2x - y + 3 = 0\) and \(x + y - 2 = 0\), we will follow these steps:

### Step 1: Convert the equations to slope-intercept form
We need to find the slopes of both lines by rewriting them in the form \(y = mx + b\), where \(m\) is the slope.

**For the first line \(2x - y + 3 = 0\):**
\[ -y = -2x - 3 \implies y = 2x + 3 \]
Thus, the slope \(m_1 = 2\).

**For the second line \(x + y - 2 = 0\):**
\[ y = -x + 2 \]
Thus, the slope \(m_2 = -1\).

### 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:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

### Step 3: Substitute the slopes into the formula
Substituting \(m_1 = 2\) and \(m_2 = -1\):
\[ \tan \theta = \left| \frac{2 - (-1)}{1 + 2 \cdot (-1)} \right| = \left| \frac{2 + 1}{1 - 2} \right| = \left| \frac{3}{-1} \right| = 3 \]

### Step 4: Find the angle \(\theta\)
To find \(\theta\), we take the arctangent:
\[ \theta = \tan^{-1}(3) \]

### Step 5: Determine if the angle is acute
Since \(\tan \theta\) is positive, \(\theta\) is acute. Thus, the acute angle between the two lines is:
\[ \theta = \tan^{-1}(3) \]

### Final Answer
The acute angle between the lines \(2x - y + 3 = 0\) and \(x + y - 2 = 0\) is \(\tan^{-1}(3)\).

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