Calculating Line Slope Angle

To find the angle of the slope, we first need to determine the slope ($m$) of the line passing through the points $(3, 4)$ and $(4, 5)$.

Finding the Slope

The formula for the slope ($m$) given two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$ m = \frac{y_2 - y_1}{x_2 - x_1} $

Substituting the given points:

$ m = \frac{5 - 4}{4 - 3} $

$ m = \frac{1}{1} $

$ m = 1 $

Determining the Angle of Slope

The slope ($m$) of a line is also equal to the tangent of the angle of inclination ($\theta$) it makes with the positive x-axis.

$ m = \tan(\theta) $

We found the slope $m = 1$. Therefore:

$ \tan(\theta) = 1 $

To find the angle $\theta$, we take the arctangent (inverse tangent) of 1:

$ \theta = \arctan(1) $

The angle whose tangent is 1 is $45^\circ$.

$ \theta = 45^\circ $