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Dividing Angles

Angles can be divided just like ordinary numbers. An angle can only be
divided by a ray on the interior of the
angle, though. Such a ray that divides an angle into two equal angles is called
an angle bisector. Likewise, two rays that divide an angle into three
congruent angles are called angle trisectors.

Figure %: An angle bisected and trisected

On the left, angle ABC is bisected by the ray BD. To know this, we must know
that angle ABD and angle CBD are congruent. On the right, angle ABC is
trisected by ray BE and ray BF. In this case, the three angles ABE, EBF, and
FBC are congruent.

With angle bisectors and trisectors, it also holds true that any of the new
angles created by the bisector or trisector is equal to exactly one-half or one-third
of the original angle, depending on whether the angle has been bisected or
trisected.

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Dividing Segments

A segment is divided into two equal segments
only when a line or segment passes through
the midpoint of the original segment. The midpoint of a segment is the
point lying in the segment that is exactly
halfway from each endpoint of the segment.

Figure %: The midpoint of a segment

In the above figure, the segment AB is divided into two segments, AM and MB.
Point M is the midpoint of segment AB, thus AM and MB are of the same length:
one-half the length of AB.

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Bisectors

When a line or segment passes through the midpoint of another segment, that line
or segment is a bisector of the other segment. There are an infinite number
of bisectors for every segment, depending on the angle at which the incoming
segment or line bisects the other segment.

Figure %: A segment being bisected by many different lines and segments

The segment AB, with midpoint M, is bisected by segment CD, line EF, and segment
GH.

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Perpendicular Bisectors

If a bisector is perpendicular to the segment it bisects, it is called the
perpendicular bisector of that segment. Because there exists only one line
perpendicular to a line at a given point, a segment has only one perpendicular
bisector: the perpendicular line that passes through the midpoint of the
segment.

Figure %: A perpendicular bisector

The line CD contains the midpoint of segment AB, and forms a right angle
with the segment. Therefore, it is the perpendicular bisector of segment AB.

Just as there are bisectors for segments, there are trisectors, too.
*Segment
trisectors* divide a segment into three equal segments.