Points, Lines, Planes and Angles

Points, Lines and Planes These basic concepts of geometry are theoretical and cannot be precisely defined. They do not exist in the physical world.

Concept Dimension RepresentationA point is a location in space.

It has no dimensions, no length, no width, and no thickness.

Points A, B, and M

A line is a set of connected points. It is assumed to be straight and is determined by two specific points.

It has one dimension, an infinite length, but no width or thickness.

Line a and line PQ

A plane is an infinite large flat surface.

It has two dimensions, an infinite length and infinite width, but no thickness.

Plane

B

M

A

QP

a

A line segment is a portion of a line between two points called endpoints. A line segment is named using its endpoints.

Line segment AB

A ray is half a line and its endpoint. It is a part of a line that has only one endpoint and goes on forever in one direction. A ray is named by using the endpoint and some other point on the ray. The endpoint is always mentioned first.

Ray PQ

A B

QP

Some Basic Concepts of Angles

P is said to be the interior of the AOB whereas Q is exterior to the angle.

An angle is the union of two rays with a common endpoint. The rays are called sides and the common point is called the vertex. An angle can be named in different ways:

O

B

A

O

B

A

O

B

Q

P

Oor AOBor BOA

Forming an angle by rotation:

OB is called the initial side and OA is called the terminal side.

Given OA, OB on the same line and extended in the same direction.If OA is rotated counterclockwise about O, we say that BOA is an angle of rotation generated by OA.

in itia l

A

term inal

BO

When the terminal side rotates until it falls on the initial side,We say that it has made one complete revolution.

Equal angles are angles formed by the same amount of rotation, and they are also called congruent angles.

Angles formed by a counterclockwise rotation are called positive and those formed by a clockwise rotation are called negative angles.

B

A

positive

B

A

positive

O B

A

negative

Units of measurement:- degree – 1/360 of a complete revolution- minute – 1/60 of a degree- second – 1/60 of a minute

The notation for the measure of angle m OO is

Classification of Angles

Acute Anglehas a measure between 0 and 90.

Right Anglehas a measure of 90 . A right angle is angle formed by a quarter of a revolution.

Obtuse Anglehas a measure between 90 and 180.

Straight Anglehas a measure of 180. A straight angle is an angle formed by one half of a revolution.The sides of a straight angle extended in opposite direction, forming a straight line.

Reflex Anglehas a measure between 180 and 360.

Adjacent Anglestwo angles with same vertex and a common side between them.

2

1

Complementary Anglestwo angles whose some is a right angle. 2

1

Supplementary Anglestwo angles whose some is a straight angle. 2 1

Vertical Anglestwo nonadjacent angles, each less than a straight angle, formed by the same measure.Vertical angles have the same measure.Angles a,b and c,d are vertical angles.

a

d

b

c

Names of Angles Pairs Formed by a Transversal Intersecting Parallel Lines

Parallel lines never meet.The distance between them is always the same. p

l

pl

Intersecting lines cross at exactly one point. aP

Perpendicular lines are intersecting lines that form right angle. a

b

ba A line that intersects two other lines at different points is called transversal.

Names of Angles Pairs Formed By Transversal Intersecting Parallel Lines

Name Description Sketch Angle Pair Described

Property

Alternate interior angles

Interior angles that do not have a common vertex on alternate sides of transversal

Alternate interior angles have the same measure.

Alternate exterior angles

Exterior angles that do not have a common vertex on alternate sides of transversal

Alternate exterior angles have the same measure.

Corresponding

angles

One interior and one exterior angle on the same side of the transversal

Corresponding angles have the same measure.

l2

4 3

8 7

1

5 6 p

t

l2

4 3

8 7

1

5 6 p

t

l2

4 3

8 7

1

5 6 p

t

4 6

3 5

and

and

1 7

2 8

and

and

1 5

2 6

3 7

4 8

and

and

and

and

m m

m m

4 6

3 5

m m

m m

1 7

2 8

m m

m m

m m

m m

1 5

2 6

3 7

4 8

• Identify each pair of complementary angles.

Exercises

75

D

E

B

C

25

65

15

AO

• Identify each pair of supplementary angles.

D

A

Q

C

B14040

40

140

AOBand BOC

CODand DOE

AQBand BQC

CQDand DQE

• Identify the angles that are congruent and their measure .

Exercises

• Find the measure of all angles in each figure.

B

F

4242C D

A

E

63

?? 13 72

3

2

11

2 3

6 7

11 2

4 5

2 7 1 6 7

ABCand BCD

m ABC m BCD

ABFand ECD

m ABF m ECD

4 2

1 3 8

m m

m

1 3 1 0 8

2 7 2m m m m

m m m

1 2 5 6 6 8

3 4 7 11 2

Exercises

• Given m A=(3x+15) and m B=(2x-5) find x if • a) the two angles are complementary

3x+15+2x-5=90• b) the two angles supplementary

3x+15+2x-5=180

x=18

x=34

Self Check

Identify each pair of complementary angles.

76

D

E

B

C

20

70

14

AO

Identify each pair of supplementary angles.

A

Q

C

B150

30

30

150

Given m A=100 and m B=(3x-7).

Find x if A and B are supplementary angles.

Find the measure of 1 if line l is parallel to line p.

40110

1

l

p