In Relationship To Parallel Lines
If two lines (a and b) are both perpendicular to a third line (c), all of the angles formed along the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.
In the figure at the right, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines a and b are parallel, any of the following conclusions leads to all of the others:
- One of the angles in the diagram is a right angle.
- One of the orange-shaded angles is congruent to one of the green-shaded angles.
- Line 'c' is perpendicular to line 'a'.
- Line 'c' is perpendicular to line 'b'.
Read more about this topic: Perpendicular
Famous quotes containing the words relationship, parallel and/or lines:
“Sometimes in our relationship to another human being the proper balance of friendship is restored when we put a few grains of impropriety onto our own side of the scale.”
—Friedrich Nietzsche (1844–1900)
“The universe expects every man to do his duty in his parallel of latitude.”
—Henry David Thoreau (1817–1862)
“Indeed, I believe that in the future, when we shall have seized again, as we will seize if we are true to ourselves, our own fair part of commerce upon the sea, and when we shall have again our appropriate share of South American trade, that these railroads from St. Louis, touching deep harbors on the gulf, and communicating there with lines of steamships, shall touch the ports of South America and bring their tribute to you.”
—Benjamin Harrison (1833–1901)