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SECTION 9-1
• Points, Lines, Planes, and Angles
Slide 9-1-1
POINTS, LINES, PLANES, AND ANGLES
• The Geometry of Euclid
• Points, Lines, and Planes
• Angles
Slide 9-1-2
THE GEOMETRY OF EUCLID
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.
Slide 9-1-3
POINTS, LINES, AND PLANES
A capital letter usually represents a point. A line
may named by two capital letters representing
points that lie on the line or by a single letter such
as l. A plane may be named by three capital letters
representing points that lie in the plane or by a
letter of the Greek alphabet such as  ,  , or  .

l
A
D
E
Slide 9-1-4
HALF-LINE, RAY, AND LINE SEGMENT
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.
Slide 9-1-5
HALF-LINE, RAY, AND LINE SEGMENT
Name
Line AB or BA
Half-line AB
A
AB or BA
B
A
AB
B
A
Ray BA
Segment AB or
segment BA
Symbol
A
Half-line BA
Ray AB
Figure
B
BA
AB
B
A
B
BA
AB or BA
A
B
Slide 9-1-6
PARALLEL AND INTERSECTING LINES
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
Slide 9-1-7
PARALLEL AND INTERSECTING PLANES
Parallel planes never meet.
Two distinct intersecting planes meet and form a
straight line.
Parallel
Intersecting
Slide 9-1-8
ANGLES
An angle is the union of two rays that have a
common endpoint. An angle can be named with
the letter marking its vertex, B , and also with
three letters: ABC- the first letter names a point
on the side; the second names the vertex; the third
names a point on the other side.
A
Vertex
B
C
Slide 9-1-9
ANGLES
Angles are measured by the amount of rotation.
360° is the amount of rotation of a ray back onto
itself.
45°
90°
10°
150°
360°
Slide 9-1-10
ANGLES
Angles are classified and named with reference to
their degree measure.
Measure
Name
Between 0° and 90°
Acute Angle
90°
Right Angle
Greater than 90° but
less than 180°
180°
Obtuse Angle
Straight Angle
Slide 9-1-11
PROTRACTOR
A tool called a protractor can be used to measure
angles.
Slide 9-1-12
INTERSECTING LINES
When two lines intersect to form right angles
they are called perpendicular.
Slide 9-1-13
VERTICAL ANGLES
In the figure below the pair ABC and DBE
are called vertical angles. DBA and EBC
are also vertical angles.
A
D
B
E
C
Vertical angles have equal measures.
Slide 9-1-14
EXAMPLE: FINDING ANGLE MEASURE
Find the measure of each marked angle below.
(3x + 10)°
(5x – 10)°
Solution
3x + 10 = 5x – 10 Vertical angles are equal.
2x = 20
x = 10
So each angle is 3(10) + 10 = 40°.
Slide 9-1-15
COMPLEMENTARY AND
SUPPLEMENTARY ANGLES
If the sum of the measures of two acute angles is 90°,
the angles are said to be complementary, and each is
called the complement of the other. For 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, and each is
called the supplement of the other. For example, 50°
and 130° are supplementary angles
Slide 9-1-16
EXAMPLE: FINDING ANGLE MEASURE
Find the measure of each marked angle below.
(2x + 45)°
(x – 15)°
Solution
2x + 45 + x – 15 = 180
Supplementary angles.
3x + 30 = 180
3x = 150
x = 50
Evaluating each expression we find that the angles
are 35° and 145°.
Slide 9-1-17
ANGLES FORMED WHEN PARALLEL
LINES ARE CROSSED BY A
TRANSVERSAL
The 8 angles formed will be discussed on the
next few slides.
1
3
5
7
2
>
4
6
>
8
Slide 9-1-18
ANGLES FORMED WHEN PARALLEL
LINES ARE CROSSED BY A
TRANSVERSAL
Name
Alternate
interior
angles
5
4
Angle
measures
are equal.
(also 3 and 6)
1
Alternate
exterior
angles
Angle
measures
are equal.
8
(also 2 and 7)
Slide 9-1-19
ANGLES FORMED WHEN PARALLEL
LINES ARE CROSSED BY A
TRANSVERSAL
Name
Interior
angles on
same side of
transversal
Corresponding
angles
6
4
(also 3 and 5)
2
6
(also 1 and 5, 3
and 7, 4 and 8)
Angle
measures
add to 180°.
Angle
measures are
equal.
Slide 9-1-20
EXAMPLE: FINDING ANGLE MEASURE
Find the measure of each marked angle below.
(x + 70)°
(3x – 80)°
Solution
x + 70 = 3x – 80 Alternating interior angles.
2x = 150
x = 75
Evaluating we find that the angles are 145°.
Slide 9-1-21