     Find the size of the acute angle formed by the intersecting lines 3x + 2y = 12 and x -

2y = -2, to the nearest tenth of a degree. Do you need to find the intersection point?

To find the acute angle between two intersecting lines, we don't need to find the intersection point itself. Instead, we can use the slopes of the lines to calculate the angle between them.

### Step-by-step approach:

1. **Find the slopes of the lines**:  
   The general equation of a line is \( y = mx + b \), where \( m \) is the slope.

   - For the first line, \( 3x + 2y = 12 \):
     \[
     2y = -3x + 12 \quad \Rightarrow \quad y = -\frac{3}{2}x + 6
     \]
     So, the slope \( m_1 = -\frac{3}{2} \).

   - For the second line, \( x - 2y = -2 \):
     \[
     2y = x + 2 \quad \Rightarrow \quad y = \frac{1}{2}x + 1
     \]
     So, the slope \( m_2 = \frac{1}{2} \).

2. **Use the formula for the angle between two lines**:
   The formula to calculate the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is:
   \[
   \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|
   \]
   Substituting the values \( m_1 = -\frac{3}{2} \) and \( m_2 = \frac{1}{2} \):
   \[
   \tan \theta = \left| \frac{-\frac{3}{2} - \frac{1}{2}}{1 + \left(-\frac{3}{2}\right) \left(\frac{1}{2}\right)} \right|
   = \left| \frac{-2}{1 - \frac{3}{4}} \right|
   = \left| \frac{-2}{\frac{1}{4}} \right|
   = \left| -8 \right| = 8
   \]

3. **Find the acute angle**:
   To find \( \theta \), we take the arctangent (inverse tangent) of 8:
   \[
   \theta = \tan^{-1}(8) \approx 82.9^\circ
   \]

   Since we need the acute angle, we subtract from \( 90^\circ \):
   \[
   90^\circ - 82.9^\circ = 7.1^\circ
   \]

### Answer:
The acute angle formed by the intersecting lines is approximately \( \boxed{7.1^\circ} \).

Would you like any further details?

Here are 5 related questions:
1. How would the angle change if the slopes were steeper?
2. How do parallel lines affect the angle between them?
3. Can two lines with negative slopes form an acute angle?
4. How do you determine if the angle formed is obtuse instead of acute?
5. How do you solve for the angle if one of the lines is vertical?

**Tip**: When using slopes to find angles, the tangent function is key, and it's always wise to consider whether the angle formed is acute or obtuse.

Find the size of the acute angle formed by the intersecting lines 3x + 2y = 12 and x - 2y = -2, to the nearest tenth of a degree. Do you need to find the intersection point?

Learn how to calculate the acute angle formed by the intersecting lines 3x + 2y = 12 and x - 2y = -2 using slopes and trigonometry. This problem covers key algebraic and trigonometric concepts, including calculating slopes and using the tangent function. Suitable for high school students in Grades 10-12.

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