Which of these can be used to show that the slope of a non-vertical line is the same anywhere along the line? \( \begin{array}{ll}\text { A) acute triangles } & \text { (C) right triangles } \\ \text { (B) obtuse triangles } & \text { (D) any triangles }\end{array} \)

Consider any nonvertical line on the coordinate plane. The slope of the line is given by \[ \text{slope} = \frac{\Delta y}{\Delta x}. \] If we take any two distinct pairs of points on the line, we can form two right triangles by drawing horizontal and vertical segments between the points. Each right triangle has a horizontal leg of length \(\Delta x\) and a vertical leg of length \(\Delta y\). Since the line is straight, the ratio \[ \frac{\Delta y}{\Delta x} \] remains the same regardless of which pair of points is chosen. Furthermore, observe that the construction of these triangles always results in one right angle (where the horizontal and vertical segments meet) and another angle that is the same as the angle the line makes with the horizontal axis. This common acute angle implies that all such right triangles are similar by the AA (Angle-Angle) similarity criterion. The similarity of these right triangles ensures that the ratio of their corresponding legs (i.e., the slope) is constant along the entire line. Therefore, the property of right triangles showing the constancy of the ratio \(\frac{\Delta y}{\Delta x}\) demonstrates that the slope of a nonvertical line is the same anywhere along the line. \[ \boxed{\text{(C) right triangles}} \]