Do Pentagons Have Parallel Sides

Fine, polygons are everywhere. They're unavoidable. But what do they have to do with parallel and perpendicular lines?

Well, let'due south accept a look-meet. Squares are made up of 2 sets of parallel line segments, and their four 90° angles hateful that those segments also happen to be perpendicular to i some other. 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 take an entire chapter named later on them). Parallel lines are equally popular, since every regular polygon with an even number of sides is fabricated up of sets of parallel line segments.

Sample Problem

Practice non-regular polygons have parallel or perpendicular sides?

Maybe. Perhaps not. Lots of polygons volition have no parallel or perpendicular sides, merely some volition accept some.

As we mentioned before, right triangles take 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.

Sample Trouble

How many sets of parallel and perpendicular lines are there in a regular octagon?

A regular octagon is made upwards of viii sides of the same length, and viii congruent angles (all of which mensurate 135°). If we extend the sides out, we can encounter clearly how the segments are related to each other.

We can come across 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 style into shapes that don't even have 90° angles. Those crafty trivial weasels.

If two lines are perpendicular to the same line, nosotros know that they're parallel. If we have some other look at the perpendicular lines, we'll see that we have iv sets of parallel lines here as well: a || d, b || g, c || f, and e || h.

Seeing these relationships amid segments and angles makes information technology possible to find angle measures and side lengths in polygons.

Sample Trouble

What is the total mensurate of all interior angles of this regular hexagon?

Since it's a regular hexagon (six-sided polygon), we know it's made upwards of sets of parallel lines. Even if nosotros don't know much most hexagons, nosotros sure know about parallel lines and transversals, so let's use what nosotros know. First, we tin extend these side lengths to amend see the parallel lines at play here.

We know that lines l and m are parallel and crossed past transversal n, so alternate interior angles are congruent. In other words, ∠1 has a measure of 60° too. The interior bending of the hexagon is supplementary to ∠1 considering they grade a linear pair, so the mensurate of one interior angle of the hexagon is 180 – m∠one, or 120°.

Virtually washed! Since nosotros know that all angles in a regular polygon are congruent and there are six angles in a hexagon (count 'em if yous don't believe us), we know that the sum of all interior angles in the hexagon is 6(120°) = 720°.

Past the way, that's true for any hexagon, not merely the regular ones. We can double-bank 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(six – 2) = 180(4) = 720 too.

Don't forget these important backdrop of parallel lines because we'll apply them over again when we talk about different polygons. In fact, there'due south a quadrilateral whose name reeks of love for all things parallel. (If y'all haven't guessed it, information technology'southward "parallelogram.")