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Shapes have most likely caused you quite a bit of trauma in the past. Seriously, fitting a square peg into a circular hole is no easy task, especially when your whole kindergarten class is watching. We don't blame you for crumbling under the pressure. We'd have been bawling too if we were in your shoes.
But you know what? That was years ago. All that therapy couldn't have been for nothing (and if it was, you owe your parents a big apology and an even bigger check). You're emotionally stable enough now to confront those shapes again. Take a deep breath, count to three, and face your fears.
Really, though, there's no reason to be afraid of shapes—especially polygons. They're completely and totally safe, assuming they don't shoot off the page and whirl around like two-dimensional ninja stars or something. But really, what are the chances of that happening?
A polygon is a closed two-dimensional shape that's made up of only straight line segments. We'll write that again in list form. In order to be a polygon, a shape must be:
Triangles, squares, rectangles, pentagons, and other more complicated shapes like the ones below are all examples of polygons.
These shapes are not polygons. Can you figure out why?
Even splitting up shapes into categories like "polygon" and "non-polygon" leaves a lot of room for uncertainty. There are two things mathematicians simply can't stand: uncertainty and waiting in line at the DMV. Of course, that last one isn't specific to mathematicians.
If polygons have sides that are all equal in length, angles that are all equal in measure, and daily trips to the loo, we call them regular. Regular polygons include shapes like equilateral triangles and squares. For any shape that has more than 4 sides, just put "regular" in front of the name (regular pentagon, regular hexagon, etc.) to indicate that it has routine bowel movements.
We've seen regular polygons all our lives, from that triangle in music class, to the friendly red octagon around the corner. Don't forget to stop by and say hello when you pass it. Seriously, it's illegal not to, and traffic school ain't all it's cracked up to be.
If we need to get more specific with describing polygons, we usually do so by the number of sides they have. Triangles have 3 sides, quadrilaterals have 4, pentagons have 5, hexagons have 6, and so on. You already know this stuff, so we won't bore you with it.
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.")