Subsets of a line & Different Kinds of Angles

SUBSETS OF A LINE
and
ANGLES
Mr. Jhon Paul A. Lagumbay
Math Teacher
St. Agnes’ Academy

GOALS:
a. Illustrates subsets of a line
b. Use some postulates and theorems that relate
points, lines, and planes
c. Distinguish between segments, rays and lines
d. Find the distance between two points on a
number line
e. Find the coordinate of the midpoint of a
segment
f. Identify opposite rays and angles
g. Measure, classify and identify types of angles.

A B
Segment AB, denoted by 𝐴𝐵 or 𝐵𝐴 is the
union of points A, B and all the points between
them. A and B are called the endpoints of the
segment.
Definition of a Segment
A segment is a subset of a line.
The length of the segment is the distance between its
endpoints.

A BP
Point P is said to be between A and B if
and only if A, P, and B are distinct points of
the same line and 𝐴𝑃 + 𝑃𝐵 = 𝐴𝐵.
Definition of Between

A BP
Ray AP, denoted by 𝐴𝑃 is the union of
(a) 𝐴𝑃 and (b) all points B such that P is
between A and B .
Definition of a Ray
A ray is another subset of a line.
A ray starts at one point of a line and goes on indefinitely
in one direction.

Ray AP, denoted by 𝐴𝑃 is the union of
(a) 𝐴𝑃 and (b) all points B such that P is
between A and B .
Definition of a Ray
𝑃𝐵 and 𝑃𝐴 are described as opposite rays if and
only if they are subsets of the same line and have a
common endpoint.
A BP

Relationships Among
Points, Lines and Planes

A postulate is a statement which is accepted as true
without proof.
A statement that needs to be proven is called a
theorem.
A corollary is a theorem whose justification follows
from another theorem.

A line contains at least two distinct points. A plane
contains at least three noncollinear points. Space
contains at least four noncoplanar points.
Postulate 1

If two distinct points are given, then a unique line
contains them.
Postulate 2 – Line Postulate
A B
l
The points A and B determine exactly one line l.
This means that there is one and only one line l that
contains points A and B.

If two distinct lines intersect, then they intersect at
exactly one point.
Theorem 1
Lines l and m
intersect at K.
m
l
K

Three collinear points are contained in at least
one plane and three noncollinear points are contained in
exactly one plane.
Postulate 3
The noncollinear
points A, B, and C are
contained in exactly one
plane P whereas the
collinear points D, E, and F
in at least one plane.

If two distinct planes intersect, then their
intersection is a line.
Postulate 4
S
T
l

If two points are in a plane, then the line that
contains those points lies entirely in the plane.
Postulate 5
A B
l
E
A line that lies in a plane divides the plane into two
subsets, each of which is called a half-plane. The dividing line
is called the edge.

If a line not contained in a plane intersects the
plane, then the intersection contains only one point.
Theorem 2
If line l and plane E
intersect two points A and B,
then line l lies in plane E by
Postulate 5. But this could
not be since line l is not
contained in plane E.
A
B
E
l

If two distinct lines intersect, then they lie in
exactly one plane.
Theorem 3
m
l
K

If there is a line and a point not in the line, then
there is exactly one plane that contains them.
Theorem 4
A B
l
ER

Given any two points there is a unique distance
between them.
Postulate 6
A
0-4 5
B

There is one-to-one correspondence between the
points of a line and the set of real numbers such that the
distance between two distinct points of the line is the
absolute value of the difference of their coordinates.
Postulate 7 – The Ruler Postulate
A
0 3 10
B

S T
Distance ST
The length or measure , ST,
of a segment, 𝑆𝑇 , is the
distance between S and T.
0 5
A B
3
C D
8
Two segments are congruent if and only if they have equal
measures. 𝐴𝐵 ≅ 𝐶𝐷 if and only if 𝐴𝐵 = 𝐶𝐷.

A point of a segment is its midpoint if and only if divides the
segment into two congruent segments. M is the midpoint of 𝑆𝑇 if
and only if 𝑆𝑀 ≅ 𝑀𝑇 .
S TM
On a ray there is exactly one point that is at a
given distance from the endpoint of the ray.
Theorem 5

Each segment has exactly one midpoint.
Corollary 1
X YM
R
T
k
Z
Any line, segment, ray,
or plane that intersects
a segment at its
midpoint is called a
bisector of the
segment.
If M is the midpoint of
𝑋𝑌, then the line 𝑘,
plane 𝑍, 𝑀𝑅 and 𝑀𝑇 all
bisect 𝑋𝑌.

If M is the midpoint of a segment AB, denoted as
𝐴𝐵, then
2𝐴𝑀 = 𝐴𝐵 and 2𝑀𝐵 = 𝐴𝐵
𝐴𝑀 =
1
2
𝐴𝐵 𝑀𝐵 =
1
2
𝐴𝐵
Theorem 6 – Midpoint Theorem
A BM

EXAMPLE:
I
0-3 3
J
6
H
A
F
D
E
G
C
B
GIVEN: 𝐷𝐵 ≅ 𝐵𝐸, 𝐴𝐵 ≅
𝐵𝐶, 𝐹𝐵 ≅ 𝐵𝐺, 𝐴𝐵 = 3, 𝐹𝐵 =
2, and 𝐷𝐵 = 1
a. What is the
midpoint of 𝐹𝐺 ?
b. Name four
bisectors of 𝐹𝐺.
c. What is the
midpoint of 𝐼𝐵 ?
d. What segment is
congruent to 𝐻𝐽 ?
e. 𝐼𝐵 + 𝐵𝐷 = _____
Is B between I
and D?

A figure is an angle if and only if it is the union of
two noncollinear rays with a common endpoint.
Definition
X
Z
Y
noncollinear rays - SIDES
common endpoint - VERTEX
SIDES: 𝑌𝑋 , 𝑌𝑍
VERTEX: 𝑌
ANGLE: ∠𝑋𝑌𝑍 or ∠𝑍𝑌𝑋

B
Q
A
R
C
P
Z
1
interior
exterior
An angle in a plane
separates it into three sets
of points:
a. the points in the interior
of the angle;
b. the points in the exterior
of the angle; and
c. the points on the angle
itself.
Thus, R is an interior point, P is an exterior point and Q is
a point on the angle.
An angle can also be named by a number or by its
vertex.

Two coplanar angles are adjacent if and only if
they satisfy three conditions: (1) they have a common
vertex , (2) they have a common side , and (3) they have
no common interior points.
Definition

EXAMPLE:
Use the figure to name the following:
a. An angle named by one
letter.
b. The sides of ∠3
c. ∠1 and ∠2 with letters
d. An angle adjacent to ∠1
YZ
X
C
B A
12
3

Classifying Angles According to Measures
Angle
Name of the
Angle
Measure of
the Angles
Classification
∠1 Less than 90° Acute Angle
∠2 Equal to 90° Right Angle
∠3
Greater than
90° but less
than 180°
Obtuse Angle
2
1
3

∠𝐴 is an acute angle if and only if the measure
of 𝑚∠𝐴 is greater than 0 but less than 90. In symbol,
𝟎 < 𝒎∠𝑨 < 𝟗𝟎
∠𝐴 is a right angle if and only if the measure of
𝑚∠𝐴 is 90. In symbol, 𝒎∠𝑨 = 𝟗𝟎
∠𝐴 is an obtuse angle if and only if the measure
of 𝑚∠𝐴 is greater than 90 but less than 180. In
symbol, 𝟗𝟎 < 𝒎∠𝑨 < 𝟏𝟖𝟎
Definition

Given any angle, there is a unique real number
between 0 and 180 known as its degree measure.
Postulate 8 – Angle Measurement Postulate
In a half-plane with edge 𝐴𝐵 any point S between
A and B, there exists a one-to-one correspondence
between the rays that originate at S in that half-plane
and the real numbers between 0 and 180
To measure an angle formed by two of these rays,
find the absolute value of the difference of the
corresponding real numbers.
Postulate 9 – The Protractor Postulate

In a half-plane, through the endpoint of ray lying
in the edge of the half-plane, there is exactly one other
ray such that the angle formed by two rays has a given
measure between 0 and 180.
Postulate 10 – The Angle Construction Postulate

Two angles are congruent if and only if they have
equal measures. In symbols, ∠𝑋 ≅ ∠𝑌 if and only if
𝑚∠𝑋 ≅ 𝑚∠𝑌
Definition
All right angles are congruent.
Theorem 7

GIVEN: three coplanar rays
𝑂𝐴, 𝑂𝑇, and 𝑂𝐵, 𝑂𝑇 is between
𝑂𝐴 and 𝑂𝐵 if and only if
𝑚∠𝐴𝑂𝑇 + 𝑚∠𝑇𝑂𝐵 = 𝑚∠𝐴𝑂𝐵
A
O
B
T
A ray is a bisector of an angle if and only if it
divides the angle into two congruent angles, thus angles
of equal measure.
Definition

If 𝑂𝑋 is a bisector of ∠𝐴𝑂𝐵, then
2𝑚∠𝐴𝑂𝑋 = 𝑚∠𝐴𝑂𝐵
𝑚∠𝐴𝑂𝑋 =
1
2
𝑚∠𝐴𝑂𝐵
and
2𝑚∠𝑋𝑂𝐵 = 𝑚∠𝐴𝑂𝐵
𝑚∠𝑋𝑂𝐵 =
1
2
𝑚∠𝐴𝑂𝐵
Theorem 8 – Angle Bisector Theorem

If T is in the interior of ∠𝐴𝑂𝐵, then
𝑚∠𝐴𝑂𝐵 = 𝑚∠𝐴𝑂𝑇 + 𝑚∠𝐵𝑂𝑇
Postulate 11 – Angle Addition Postulate
A
O
B
T

ILLUSTRATE & ANSWER THE
FOLLOWING:
1. If 𝑚∠𝐴𝑂𝐵 = 25 and ∠𝐴𝑂𝐵 ≅ ∠𝐶𝑂𝐷,
what can you conclude about ∠𝐶𝑂𝐷?
2. If 𝑚∠𝐴𝑂𝐶 = 90 and 𝑚∠𝐴𝑂𝐵 = 20,
what can you conclude about ∠𝐵𝑂𝐶?

That in all things,
God may be
Glorified!!!

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