SUBSETS OF A LINEand

ANGLESMr. 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.

SUBSETS OF A LINE

A B

Segment AB, denoted by or 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 .

Definition of Between

A BP

Ray AP, denoted by is the union of (a) 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 is the union of (a) and (b) all points B such that P is between A and B .

Definition of a Ray

and 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 Bl

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 Bl

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.

AB

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 Bl

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, , 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. if and only if .

A point of a segment is its midpoint if and only if divides the

segment into two congruent segments. M is the midpoint of if

and only if .

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

kZ

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 , then the line , plane , and all bisect .

If M is the midpoint of a segment AB, denoted as , then

and

Theorem 6 – Midpoint Theorem

A BM

EXAMPLE:

I

0-3 3

J

6

H

A

F

D

E

GC

B

GIVEN: , , , , , and

a. What is the midpoint of ?

b. Name four bisectors of .

c. What is the midpoint of ?

d. What segment is congruent to ?

e. Is B between I

and D?

Angles

A figure is an angle if and only if it is the union of two noncollinear rays with a common endpoint.

Definition

X

ZY

noncollinear rays - SIDES

common endpoint - VERTEX

SIDES: ,

VERTEX:ANGLE: or

B

Q

A

R

C

PZ

1interior

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; andc. 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

c. and with lettersd. An angle adjacent to

YZ

X

C

B A

12

3

Classifying Angles According to Measures

AngleName of the Angle

Measure of the Angles

Classification

Less than Acute Angle

Equal to Right Angle

Greater than but less than

Obtuse Angle

2

1

3

is an acute angle if and only if the measure of is greater than 0 but less than 90. In symbol,

is a right angle if and only if the measure of is 90. In symbol,

is an obtuse angle if and only if the measure of is greater than 90 but less than 180. In symbol,

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 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, if and only if

Definition

All right angles are congruent.

Theorem 7

GIVEN: three coplanar rays, , and , is between and if and only if

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 is a bisector of , then

and

Theorem 8 – Angle Bisector Theorem

If T is in the interior of , then

Postulate 11 – Angle Addition Postulate

A

O B

T

ILLUSTRATE & ANSWER THE FOLLOWING:

1. If and , what can you conclude about ?

2. If and , what can you conclude about ?

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