SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Payment Details

Payment Summary

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thanks for creating a SparkNotes account! Continue to start your free trial.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

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.

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.

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.

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.

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.

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

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.

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.