To prove that two lines perpendicular to the same line are parallel to each other, we need to understand the properties of perpendicular lines and parallel lines.



Two lines are said to be perpendicular if they intersect at a right angle (90 degrees). In other words, the slopes of the lines are negative reciprocals of each other.



Let's consider a line AB and two lines CD and EF, both of which are perpendicular to line AB.



We can prove that CD and AB are perpendicular by showing that their slopes are negative reciprocals of each other.

Assume the slope of AB is m. Since CD is perpendicular to AB, its slope will be the negative reciprocal of m, which we can denote as -1/m.

Therefore, the slope of CD is -1/m.



Similarly, assume the slope of AB is m. Since EF is also perpendicular to AB, its slope will be the negative reciprocal of m, denoted as -1/m.

Therefore, the slope of EF is -1/m.



Since both CD and EF have slopes of -1/m, we can conclude that their slopes are equal.



By the definition of parallel lines, two lines are parallel if and only if their slopes are equal.

Since the slopes of CD and EF are equal, we can conclude that CD and EF are parallel to each other.



Therefore, we have proven that two lines perpendicular to the same line are parallel to each other. This is because the slopes of the perpendicular lines are negative reciprocals of the same slope, leading to equal slopes and parallel lines.