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To find the smallest angle between the two given lines, we need to calculate the angle between them using the slopes derived from the equations. Let's go through the steps.

### Step 1: Identify the line equations and convert them to slope-intercept form
The equations given are:
1. \( 3x + 2y - 5 = 0 \)
2. \( 6x + 4y + 19 = 0 \)

Rewrite each equation to find the slope (\( m \)).

#### For the first equation:
\[ 3x + 2y - 5 = 0 \Rightarrow 2y = -3x + 5 \Rightarrow y = -\frac{3}{2}x + \frac{5}{2} \]
So, the slope of the first line (\( m_1 \)) is \( -\frac{3}{2} \).

#### For the second equation:
\[ 6x + 4y + 19 = 0 \Rightarrow 4y = -6x - 19 \Rightarrow y = -\frac{3}{2}x - \frac{19}{4} \]
The slope of the second line (\( m_2 \)) is also \( -\frac{3}{2} \).

### Step 2: Determine the angle between the lines
The angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) can be found using:
\[
\tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 \cdot m_2} \right|
\]

Since \( m_1 = m_2 = -\frac{3}{2} \), we have:
\[
\tan(\theta) = \left| \frac{-\frac{3}{2} + \frac{3}{2}}{1 + \left(-\frac{3}{2}\right)^2} \right| = \left| \frac{0}{1 + \frac{9}{4}} \right| = 0
\]

Thus, \( \theta = 0^\circ \), which indicates that the lines are parallel.

### Conclusion:
Since the lines are parallel, the smallest angle between them is \( 0^\circ \). However, if the question requires selecting from the provided options (45°, 60°, 90°, or 30°), and assuming the parallel case was overlooked, none of these options would technically be correct. For parallel lines, we should select \( 0^\circ \), which is not listed here.

If you have any questions or need further clarification, feel free to ask!

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Here are some related questions you might find interesting:
1. What is the method to find angles between non-parallel lines?
2. How do we interpret cases where \( m_1 \cdot m_2 = -1 \) for perpendicular lines?
3. Why is the angle between parallel lines considered to be \( 0^\circ \)?
4. Can this method be used for three-dimensional planes, and if so, how?
5. How would we solve if the lines had distinct slopes?

**Tip:** Always check if lines are parallel by comparing slopes before calculating the angle.

The smallest angle between the lines given by 3x + 2y - 5 = 0 and 6x + 4y + 19 = 0 in degrees is:

Discover how to find the smallest angle between two given lines, 3x + 2y - 5 = 0 and 6x + 4y + 19 = 0. This problem involves key concepts in linear algebra, including slope calculation and angle determination between lines. Ideal for undergraduate students studying linear algebra.

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