9.1 – Points, Line, Planes and Angles Definitions: A

point

has no magnitude and no size.

A

line

has no thickness and no width and it extends indefinitely in two directions.

A

plane

is a flat surface that extends infinitely.

A D

m

E



9.1 – Points, Line, Planes and Angles Definitions: A point divides a line into two

half-lines

, one on each side of the point.

A

ray

is a half-line including an initial point.

A

line segment

includes two endpoints.

N



F D E G

9.1 – Points, Line, Planes and Angles Summary:

Name

Line

AB

or

BA

Half-line

AB

Half-line

BA

Ray

AB

Ray

BA

Segment

AB

or Segment

BA A A A

Figure

A B A A B B B B B

Symbol

AB BA AB AB BA AB BA BA

9.1 – Points, Line, Planes and Angles Definitions:

Parallel lines

lie in the same plane and never meet.

Two distinct

intersecting lines

meet at a point.

Skew lines

do not lie in the same plane and do not meet.

Parallel Intersecting Skew

9.1 – Points, Line, Planes and Angles Definitions:

Parallel planes

never meet.

Two distinct

intersecting planes

meet and form a straight line.

Parallel Intersecting

9.1 – Points, Line, Planes and Angles Definitions: An

angle

is the union of two rays that have a common endpoint.

A

Vertex

B 1 C

An angle can be named using the following methods: – with the letter marking its vertex,  B – with the number identifying the angle,  1 – with three letters,  ABC. 1) the first letter names a point one side; 2) the second names the vertex; 3) the third names a point on the other side.

9.1 – Points, Line, Planes and Angles

Angles

are measured by the amount of rotation in degrees. Classification of an angle is based on the degree measure.

Measure

Between 0° and 90° 90° Greater than 90° but less than 180° 180°

Name

Acute Angle Right Angle Obtuse Angle Straight Angle

9.1 – Points, Line, Planes and Angles When two lines intersect to form right angles they are called

perpendicular

.

Vertical angles

are formed when two lines intersect.

D A B E C

 ABC and  DBE are one pair of vertical angles.

 DBA and  EBC are the other pair of vertical angles.

Vertical angles have equal measures.

9.1 – Points, Line, Planes and Angles Complementary Angles and Supplementary Angles If the sum of the measures of two acute angles is 90°, the angles are said to be complementary. Each is called the

complement

of the other. Example: 50° and 40° are complementary angles.

If the sum of the measures of two angles is 180°, the angles are said to be

supplementary

. Each is called the

supplement

of the other. Example: 50° and 130° are supplementary angles

9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below.

(3

x

+ 10)° (5

x

– 10)° Vertical angels are equal.

3

x

+ 10 = 5

x

– 10 2

x

= 20

x

= 10 Each angle is 3(10) + 10 = 40°.

9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below.

(2

x

+ 45)° (

x

– 15)° Supplementary angles.

2

x

+ 45 +

x

– 15 = 180 3

x

+ 30 = 180 3

x

= 150

x

= 50 2(50) + 45 = 145  50 – 15 = 35  35° + 145° = 180 

9.1 – Points, Line, Planes and Angles Parallel Lines cut by a Transversal line create 8 angles 5 6 7 8 1 2 3 4 5 4 (also 3 and 6) 1 Alternate interior angles Angle measures are equal.

8 (also 2 and 7) Alternate exterior angles Angle measures are equal.

9.1 – Points, Line, Planes and Angles 1 2 3 4 5 6 7 8 4 6 (also 3 and 5) Same Side Interior angles Angle measures add to 180°.

2 6 Corresponding angles Angle measures are equal.

(also 1 and 5, 3 and 7, 4 and 8)

9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below.

(

x

+ 70)° (3

x

– 80)° Alternate interior angles.

x

+ 70 = 3

x

– 80 2

x

= 150

x

= 75 x + 70 = 75 + 70 = 145°

9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below.

(4

x

– 45)° (2

x

– 21)° Same Side Interior angles.

4x – 45 + 2x – 21 = 180 6x – 66 = 180 6x = 246 x = 41 4( 41 ) – 45 164 – 45 119° 2( 180 – 119 = 61° 41 ) – 21 82 – 21 61°