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Constructions in Geometry, a la Shmoop Some people are happy being famous, even with
all the bad press they may get.
Some people like helping others and making a difference in the world, regardless of how
much money they earn.
And some people find true inner peace...
...just by sitting on top of a construction scaffolding with hard hats and day-old turkey
sandwiches. Of course, construction isn't all about hammering
up drywall and playing with power tools.
It's also about being creative. In geometry, a construction is a drawing that
you make using only a straightedge, a compass, and a pencil.
A straightedge is anything you can use to make a straight line...
...like a ruler or the back of your geometry textbook or a 2 by 4.
A compass might point north if you're venturing through the Amazon...
...but in geometry we're talking about this thing, which preserves distance and draws
a perfect circle.
Make sure to pack the right compass next time you're going on safari.
We use a pencil because mistakes happen...
...and whiteout is expensive. Using only these three tools, we can construct
practically anything!
A perpendicular bisector, a congruent angle, an equilateral triangle, or parallel lines.
But what good is all this talk of construction if we can't do anything with it?
Let's try a relatively simple construction. How do we construct an angle congruent to
this given angle? We'll start by drawing a ray, which will be
one of the sides of our new angle. Now, we can use our compass to mark an arc
length within the given angle...
... with the center at the endpoint of the angle.
Without changing the measurement of the compass, let's draw an identical arc on our ray.
We adjust our compass to be the distance between the intersections...
...between the arc and the sides of the given angle.
If we take this distance and apply it to the intersection between the arc and the ray...
...we can draw a small arc that intersects with the first one.
Two points are enough to draw another ray from the endpoint of the angle through the
point where the two arcs intersect. And there you have it... a congruent angle,,
constructed using only a pencil, a straightedge, and a compass.
It's no Eiffel tower, but
it'll do.