Fine, polygons space everywhere. They"re unavoidable. But what carry out they need to do through parallel and perpendicular lines?
Well, let"s have actually a look-see. Squares are made up of 2 sets that parallel line segments, and their 4 90° angles average that those segments additionally happen to be perpendicular come one another. Did us blow your mind?
Many polygons have actually parallel and also perpendicular sides. Rectangles, right trapezoids, and also loads of various other polygons have perpendicular line segments (including ideal triangles, which are special enough to have whole chapter named after them). Parallel lines room equally popular, since every consistent polygon v an even number of sides is consisted of of set of parallel heat segments.
You are watching: A hexagon with two sides perpendicular
Do non-regular polygons have parallel or perpendicular sides?
Maybe. Possibly not. Numerous polygons will have no parallel or perpendicular sides, but some will have some.
As we stated before, best triangles have actually perpendicular sides, rectangles have actually both perpendicular and also parallel sides, but other quadrilaterals could not. A constant pentagon has actually no parallel or perpendicular sides, yet a non-regular pentagon could have parallel and perpendicular sides. It all depends on the polygon.
How numerous sets of parallel and perpendicular lines room there in a continual octagon?
A constant octagon is made up of eight sides of the exact same length, and eight congruent angles (all that which measure 135°). If we prolong the sides out, we can see plainly how the segments are related to each other.
We deserve to see that lines a and d are perpendicular come both e and h. Just the same, present c and f room perpendicular to b and also g. For this reason perpendicular lines regulated to sneak their method into shapes that don"t even have 90° angles. Those crafty tiny weasels.
If two lines space perpendicular to the same line, we recognize that they"re parallel. If us take another look at the perpendicular lines, we"ll view 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 uncover angle measures and also side lengths in polygons.
What is the complete measure of all interior angles that this regular hexagon?
Since it"s a regular hexagon (six-sided polygon), we understand it"s consisted of of sets of parallel lines. Even if we don"t understand much around hexagons, we certain know about parallel lines and transversals, so let"s usage what us know. First, us can expand these side lengths to far better see the parallel lines in ~ play here.
We understand that present l and m room parallel and also crossed by transversal n, so alternate interior angles are congruent. In various other words, ∠1 has actually a measure of 60° also. The internal angle that the hexagon is supplementary come ∠1 due to the fact that they type a direct pair, therefore the measure of one internal angle that the hexagon is 180 – m∠1, or 120°.
Almost done! due to the fact that we understand that all angle in a continuous polygon are congruent and there room 6 angle in a hexagon (count "em if you don"t think us), we know that the sum of all interior 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 have the right to double-check that because a polygon with n sides has actually a complete interior angle sum of 180(n – 2). Substituting 6 for n would offer us 180(6 – 2) = 180(4) = 720 too.
Don"t forget these essential properties of parallel lines due to the fact that we"ll usage them again as soon as we talk around different polygons. In fact, there"s a square whose surname reeks of love for all things parallel. (If you haven"t guessed it, it"s "parallelogram.")