Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
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Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
An angle is formed by two rays with a common endpoint. The angle bisector is a ray or line segment that bisects the angle, creating two congruent angles. To construct an angle bisector you need a compass and straightedge. Bisectors are very important in identifying corresponding parts of similar triangles and in solving proofs.
The key points to an angle bisector
is that it does -- almost lost it
-- it does a couple things.
The first thing is it bisects the
angle, creating two congruent angles.
So if I have an angle that's in blue here
and the red ray is my angle bisector,
then it has created two
congruent angles.
So notice that this red is a ray.
That's another key thing.
Now, it also could be a line segment, if
you're talking about something in an
isosceles triangle perhaps, we could
say line segment, and every point
along this bisector is the same distance
from the two rays that make up
the sides.
But how do we measure distance?
Well, the shortest distance from a point
on this ray to a ray that forms the
angle is along a perpendicular.
So if you're to construct the perpendicular
from the angle bisector to a side,
and if you did that down here, then you
would say that these two segments
are congruent.
So that's the key parts
to an angle bisector.
Is that it bisects the angle creating two
congruent angles, it's a line-- a ray
or line segment, and that every point
on this ray is the same distance
measured along the perpendicular from
the rays that make up your angle.
But how do we actually construct that?
To do that let's grab our compass and our
straight edge and head over to this
angle right here.
So we know that we're going to create a
ray that creates two congruent angles.
So the first thing you're going to do is
you're going to swing an arc just like
if you were duplicating an angle.
So from the vertex I'm going to swing
an arc so that I create two points of
intersection.
Now I want to create a point out here that
is the same distance from these two
points of intersection.
So if you want to, you can change your compass,
but you don't have to, for the
sake of argument I will.
And you're going to swing an arc
from each of these end points.
So there's one arc from this intersection.
Here's another point of intersection
that I'm going to swing an arc from.
Now, this point right here is the same distance
from both of these end points.
So I'm going to connect this point
of intersection with my vertex.
Thereby creating my angle bisector.
So I'm going to draw this, connect my vertex,
and that point of intersection,
and what we've created are two congruent
angles that when they sum you
get the angle that you started with.
Unit
Constructions