Fine, polygons space everywhere. They"re unavoidable. But what execute they need to do v parallel and also perpendicular lines?

Well, let"s have a look-see. Squares are comprised of 2 sets of parallel heat segments, and also their 4 90° angles typical that those segments additionally happen to it is in perpendicular to one another. Did us blow her mind?

Many polygons have actually parallel and perpendicular sides. Rectangles, best trapezoids, and loads of various other polygons have actually perpendicular line segments (including best triangles, which room special enough to have an entire chapter named after them). Parallel lines are equally popular, due to the fact that every continual polygon with an even variety of sides is made up of set of parallel heat segments.

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Sample Problem

Do non-regular polygons have actually parallel or perpendicular sides?

Maybe. Possibly not. Many polygons will have actually no parallel or perpendicular sides, yet some will have some.

As we discussed before, right triangles have actually perpendicular sides, rectangles have actually both perpendicular and also parallel sides, but other quadrilaterals could not. A continuous pentagon has no parallel or perpendicular sides, however a non-regular pentagon could have parallel and also perpendicular sides. The all depends on the polygon.

Sample Problem

How plenty of sets that parallel and also perpendicular lines are there in a consistent octagon?


A continuous octagon is made up of eight political parties of the same length, and eight congruent angle (all the which measure up 135°). If we prolong the political parties out, we have the right to see plainly how the segments are pertained to each other.


We can see the lines a and d space perpendicular come both e and h. Simply the same, present c and f are perpendicular come b and g. For this reason perpendicular lines regulated to sneak their method into shapes that don"t also have 90° angles. Those crafty tiny weasels.

If 2 lines are perpendicular come the very same line, we recognize that they"re parallel. If us take another look in ~ the perpendicular lines, we"ll view that we have four sets of parallel lines below as well: a || d, b || g, c || f, and also e || h.

Seeing these relationships amongst segments and also angles renders it possible to uncover angle measures and side lengths in polygons.

Sample Problem

What is the complete measure the all internal angles that this consistent hexagon?


Since it"s a constant hexagon (six-sided polygon), we know it"s comprised of sets of parallel lines. Even if we don"t know much about hexagons, we certain know around parallel lines and transversals, for this reason let"s usage what we know. First, us can prolong these side lengths to much better see the parallel lines in ~ play here.


We recognize that present l and also m room parallel and also crossed through transversal n, so alternative interior angles space congruent. In other words, ∠1 has actually a measure up of 60° also. The inner angle of the hexagon is supplementary come ∠1 due to the fact that they kind a linear pair, so the measure of one interior angle the the hexagon is 180 – m∠1, or 120°.

Almost done! due to the fact that we understand that all angles in a continual polygon space congruent and also there space 6 angle in a hexagon (count "em if friend don"t believe us), we know that the amount of all inner angles in the hexagon is 6(120°) = 720°.

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By the way, that"s true because that any hexagon, not simply the consistent ones. We deserve to double-check that due to the fact that a polygon through n sides has actually a total interior angle amount of 180(n – 2). Substituting 6 for n would provide us 180(6 – 2) = 180(4) = 720 too.

Don"t forget these essential properties that parallel lines because we"ll use them again as soon as we talk about different polygons. In fact, there"s a quadrilateral whose name reeks of love because that all things parallel. (If girlfriend haven"t guessed it, it"s "parallelogram.")