9.1 – Points, Line, Planes and AnglesDefinitions:
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
DE
m
9.1 – Points, Line, Planes and AnglesDefinitions: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.
D
E
N
F
G
Name Figure Symbol
9.1 – Points, Line, Planes and Angles
Summary:
A B AB BA
AB
BA
AB
BA
A B
A B
Line AB or BA
Half-line AB
Half-line BA
Ray AB
Ray BA
Segment AB or Segment BA
A B
A B
A B AB BA
9.1 – Points, Line, Planes and AnglesDefinitions:
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 AnglesDefinitions:Parallel planes never meet.
Parallel Intersecting
Two distinct intersecting planes meet and form a straight line.
9.1 – Points, Line, Planes and AnglesDefinitions:An angle is the union of two rays that have a common endpoint.
Vertex BSide
Side
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.
C
A
1
9.1 – Points, Line, Planes and AnglesAngles are measured by the amount of rotation in degrees.
Classification of an angle is based on the degree measure.
Measure NameBetween 0° and 90° Acute Angle
90° Right Angle
Greater than 90° but less than 180°
Obtuse Angle
180° Straight Angle
9.1 – Points, Line, Planes and AnglesWhen two lines intersect to form right angles they are called perpendicular.
Vertical angles are formed when two lines intersect.A
CB
D
E
Vertical angles have equal measures.
ABC and DBE are one pair of vertical angles.
DBA and EBC are the other pair of vertical angles.
9.1 – Points, Line, Planes and AnglesComplementary 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 AnglesFind the measure of each marked angle below.
(3x + 10)° (5x – 10)°
3x + 10 = 5x – 10
Each angle is 3(10) + 10 = 40°.
Vertical angels are equal.
2x = 20
x = 10
9.1 – Points, Line, Planes and AnglesFind the measure of each marked angle below.
(2x + 45)° (x – 15)°
2x + 45 + x – 15 = 180
35° + 145° = 180
Supplementary angles.
3x + 30 = 180
3x = 150
x = 50
2(50) + 45 = 14550 – 15 = 35
9.1 – Points, Line, Planes and Angles
1 23 4
5 67 8
Alternate interior angles
Alternate exterior angles
Angle measures are equal.
Angle measures are equal.
1
5 4
8
(also 3 and 6)
(also 2 and 7)
Parallel Lines cut by a Transversal line create 8 angles
9.1 – Points, Line, Planes and Angles
1 23 4
5 67 8
Same Side Interior angles
Corresponding angles
Angle measures are equal.
Angle measures add to 180°.4
6
2
6
(also 3 and 5)
(also 1 and 5, 3 and 7, 4 and 8)
9.1 – Points, Line, Planes and AnglesFind the measure of each marked angle below.
(x + 70)°(3x – 80)°
Alternate interior angles.
x + 70 = 3x – 80
2x = 150
x = 75 145°
x + 70 =
75 + 70 =
9.1 – Points, Line, Planes and AnglesFind the measure of each marked angle below.
(2x – 21)°
(4x – 45)°
Same Side Interior angles.
4x – 45 + 2x – 21 = 180
6x – 66 = 180
6x = 246 119°
4(41) – 45
164 – 45
x = 41
180 – 119 = 61°
61°
2(41) – 21
82 – 21
