Solution: The slope of the line from $(2, 3)$ to $(5, 11)$ is $\frac{11 - 3}{5 - 2} = \frac{8}{3}$. Since $\tan \theta$ equals the slope, $\tan \theta = \frac{8}{3}$. \boxed{\dfrac{8}{3}}

["Understanding Slope and Angle of Intersection: A Clear Solution", "In coordinate geometry, one of the fundamental concepts is the slope of a line, which describes its steepness and direction. This measure not only helps define how a line rises or falls but also plays a key role in understanding angles formed by intersecting lines. Let’s explore how to determine the slope between two points and connect it to the trigonometric tangent of the angle it makes with the x-axis.", "### Calculating the Slope Between Two Points\nConsider two points on a plane, ( A(2, 3) ) and ( B(5, 11) ). To find the slope of the line passing through these points, use the standard slope formula:", "[\n\ ext{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 3}{5 - 2} = \frac{8}{3}\n]", "This value—( \frac{8}{3} )—represents how much the y-coordinate increases per unit increase in the x-coordinate along this line.", "### Relating Slope to Angle Using Tangent\nA crucial insight in geometry is that the tangent of the angle ( \ heta ) that the line makes with the positive x-axis equals the slope of the line. Therefore:", "[\n\ an \ heta = \ ext{slope} = \frac{8}{3}\n]", "This direct relationship simplifies many problems in trigonometry and coordinate analysis, linking algebraic computation with geometric interpretation.", "### Why This Matters\nUnderstanding the slope-to-tangent connection is essential when solving problems related to angles, design, engineering, or navigation. For example, architects and engineers use this concept to calculate inclines and ensure structural integrity based on precise angular measurements.", "### Final Takeaway\nThe slope from point ((2, 3)) to ((5, 11)) is ( \frac{8}{3} ), confirming that ( \ an \ heta = \frac{8}{3} ). Mastering this relationship bridges algebra and geometry, empowering clearer problem-solving across STEM disciplines.", "\boxed{\dfrac{8}{3}}"]