Final answer:
To prove that the slopes of two perpendicular lines are opposite reciprocals, we can use a pair of right triangles formed by the lines. These triangles have proportional sides as per the Angle-Angle similarity postulate. By finding the proportions of corresponding sides, we demonstrate the relationship between the slopes.
Step-by-step explanation:
To show that the slopes of two perpendicular lines are opposite reciprocals using similar triangles, we look for a pair of triangles that share one angle (making them similar by Angle-Angle similarity) and whose sides are along the lines in question. These triangles would typically be right triangles formed by the lines intersecting at a right angle. We use the properties of these triangles in conjunction with the definitions of sine, cosine, and tangent in terms of side lengths, and the Pythagorean Theorem, to establish the relationship between the side lengths and therefore the slopes of the lines.
Let's say we have two perpendicular lines, forming a right angle at their intersection. The lines create two right triangles with one common angle, besides the right angle. The sides of the first triangle can be denoted as 'a' and 'b', with hypotenuse 'c'. For the other triangle, let the sides be 'd' and 'e', with 'f' as the hypotenuse. The side 'a' of the first triangle corresponds to the rise (or fall) and the side 'b' corresponds to the run for the first line, so the slope (m1) is a/b. Similarly, the side 'd' of the second triangle corresponds to the rise (or fall), and 'e' to the run for the second line, so the slope (m2) is d/e. Because the lines are perpendicular, the triangles are similar by the Angle-Angle similarity postulate, and the following proportions can be written: a/c = e/f and b/c = d/f. Simplifying these proportions, we find that m1 = a/b = -e/d = -1/m2, showing that the slopes are indeed opposite reciprocals.