Parallel and Perpendicular Lines in Polygons
Fine, polygons are everywhere. They're unavoidable. But what do they have to do with parallel and perpendicular lines?
Well, let's have a look-see. Squares are made up of two sets of parallel line segments, and their four 90° angles mean that those segments also happen to be perpendicular to one another. Did we blow your mind?
Many polygons have parallel and perpendicular sides. Rectangles, right trapezoids, and loads of other polygons have perpendicular line segments (including right triangles, which are special enough to have an entire chapter named after them). Parallel lines are equally popular, since every regular polygon with an even number of sides is made up of sets of parallel line segments.
Do non-regular polygons have parallel or perpendicular sides?
Maybe. Maybe not. Lots of polygons will have no parallel or perpendicular sides, but some will have some.
As we mentioned before, right triangles have perpendicular sides, rectangles have both perpendicular and parallel sides, but other quadrilaterals might not. A regular pentagon has no parallel or perpendicular sides, but a non-regular pentagon might have parallel and perpendicular sides. It all depends on the polygon.
How many sets of parallel and perpendicular lines are there in a regular octagon?
A regular octagon is made up of eight sides of the same length, and eight congruent angles (all of which measure 135°). If we extend the sides out, we can see clearly how the segments are related to each other.
We can see that lines a and d are perpendicular to both e and h. Just the same, lines c and f are perpendicular to b and g. So perpendicular lines managed to sneak their way into shapes that don't even have 90° angles. Those crafty little weasels.
If two lines are perpendicular to the same line, we know that they're parallel. If we take another look at the perpendicular lines, we'll see that we have four sets of parallel lines here as well: a || d, b || g, c || f, and e || h.
Seeing these relationships among segments and angles makes it possible to find angle measures and side lengths in polygons.
What is the total measure of all interior angles of this regular hexagon?
Since it's a regular hexagon (six-sided polygon), we know it's made up of sets of parallel lines. Even if we don't know much about hexagons, we sure know about parallel lines and transversals, so let's use what we know. First, we can extend these side lengths to better see the parallel lines at play here.
We know that lines l and m are parallel and crossed by transversal n, so alternate interior angles are congruent. In other words, ∠1 has a measure of 60° also. The interior angle of the hexagon is supplementary to ∠1 because they form a linear pair, so the measure of one interior angle of the hexagon is 180 – m∠1, or 120°.
Almost done! Since we know that all angles in a regular polygon are congruent and there are 6 angles in a hexagon (count 'em if you don't believe us), we know that the sum of all interior angles in the hexagon is 6(120°) = 720°.
By the way, that's true for any hexagon, not just the regular ones. We can double-check that because a polygon with n sides has a total interior angle sum of 180(n – 2). Substituting 6 for n would give us 180(6 – 2) = 180(4) = 720 too.
Don't forget these important properties of parallel lines because we'll use them again when we talk about different polygons. In fact, there's a quadrilateral whose name reeks of love for all things parallel. (If you haven't guessed it, it's "parallelogram.")