Division of Segments and Angles.
Level 1.
In the following series of videos we will learn how to identify midpoints, bisectors
of segments, trisection points, trisectors of segments, adjacent angles, angle bisectors
and angle trisectors.
We will start this video series with Midpoints and Bisectors of Segments.
In geometry we often work with segments that are divided in half.
A point (or segment, ray, line or plane) that divides a segment into two congruent segments
bisects the segment.
“Bi” is a prefix that means “two” and “sect” means “to cut”.
In this sense the bisector is the “thing” doing the cutting.
The bisection point is called the midpoint of the segment.
The midpoint represents the middle of the line segment.
This point is exactly half way between both endpoints.
It is important to note that only segments can have midpoints.
It does not make sense to say that a ray or a line has a midpoint.
Remember that segments are measurable and have a definite length.
Rays and lines have an indefinite length.
Recall that rays extend infinitely in one direction and lines extend infinitely in two
directions so they have no midpoint.
Midpoints are defined for segments only.
For example the midpoint of segment PQ is represented by point M. Point M is said to
bisect segment PQ this means that point M divides segment PQ into two congruent segments.
We can conclude that the length of segment PM is equal to the length of segment MQ in
other words; segment PM is congruent to segment MQ.
A point is not the only figure that can bisect a segment in fact another segment, ray, line
or plane can also bisect a segment.
For example in the following figure segment XY bisects segment AC at point B.
We can conclude that point B is the midpoint of segment AC and that segment AB is congruent
to segment BC.
In this figure, point D is the midpoint of segment FE.
We can conclude that segment FD is congruent to segment DE because point D bisects segment
FE.
We can also conclude that ray DG bisects segment FE.
In the next figure segment OK is congruent to segment KP we can conclude that point K
is the midpoint of segment OP, we can also conclude that line JM is a bisector of segment
OP since it is collinear with the midpoint K.
As a final example in the following diagram plane X bisects segment AB at its midpoint
P. You can think of this plane as slicing segment AB at point P creating two congruent
segments in this case segment AP and segment PB.
So a point, segment, ray, line or plane can bisect a line segment creating a midpoint
which divides the segment in half creating two congruent segments.
Next let’s talk about trisection points and trisectors of segments.
A segment divided into three congruent parts is said to be trisected.
“Tri” is a prefix that means “three”.
Two points (or segments, rays, lines or planes) that divide a segment into three congruent
segments trisect the segment.
The two points where the segment is divided are called the trisection points of the segment.
Recall that midpoints are associated with segments, in the same manner trisection points
is also associated with segments; this means that rays and lines do not have trisection
points.
In the following figure segment AR is congruent to segment RS which is congruent to segment
SC.
From this information we can conclude that point R and point S are trisection points
of segment AC this means that segment AC is trisected by points R and S.
In the next figure if point E and point F are trisection points of segment DG then we
can conclude that segment DE is congruent to segment EF which is congruent to segment
FG we can also conclude that segment HE and HF are trisectors of segment DG since these
figures cut segment DG into 3 congruent segments.
Alright before we talk about angle bisectors, let’s quickly talk about adjacent angles.
Adjacent angles are two angles in a plane that have a common vertex and a common side
but no common interior points.
For example in the diagram shown angle 1 and angle 2 are adjacent angles.
The next diagrams illustrate examples of non adjacent angles notice that angle 3 and angle
4 are not adjacent angles.
When we describe adjacent angles we are picturing diagrams like the ones on the left.
Alright now let’s talk about angle bisectors.
An angle, like a segment, can be bisected.
A ray that divides an angle into two congruent adjacent angles bisects the angle.
The dividing ray is called the bisector of the angle.
In the following figure the measurement of angle ABD is equal to the measurement of angle
DBC in other words, angle ABD is congruent to angle DBC from this information we can
conclude that ray BD is the bisector of angle ABC.
As an additional example if angle NOP is congruent to angle POR and ray OQ bisects angle POR,
then we can conclude that ray OP is the bisector of angle NOR.
We can also conclude that angle 1 is congruent to angle 2.
Alright let’s end the video by going over angle trisectors.
Two rays can divide an angle into three equal parts.
Two rays that divide an angle into three congruent adjacent angles trisect the angle.
The two dividing rays are called trisectors of the angle.
In the following figure angle ABC is congruent to angle CBD which is congruent to angle DBE
from this information we can conclude that ray BC and ray BD trisect angle ABE.
As a final example, if ray SV and ray SX are trisectors of angle TSY then we can conclude
that angle TSV is congruent to angle VSX which is also congruent to angle XSY.
Alright so to recap, segments and angles can be divided.
The special ways to divide segments include segment bisectors and segment trisectors.
Segment bisectors intersect a segment at a point called the midpoint dividing the segment
into two congruent segments.
Segment trisectors intersect a segment creating two points called trisection points which
divide the segment into 3 congruent segments.
The special ways to divide angles include angle bisectors and angle trisectors.
Angle bisectors divide an angle into two congruent adjacent angles.
Angle trisectors divide an angle into three congruent adjacent angles.
Alright in our next video we will start applying these new concepts and go over some basic
examples.