Download Points, Lines, Planes, and Angles

Chapter 10
Geometry
1. Points, Lines, Planes, and Angles
Definitions

Point
• has no length, width, or thickness
• has position

Line
• determined by 2 points
• has no thickness
• infinite length

Plane
• 2 dimensional
• has no thickness, no boundaries
Definitions (cont.)

Line – named by any two of its points
A

B
Ray—a half-line with its endpoint
included
B
A

Line segment—portion of line joining
2 points
A
B
Lines

In the diagrams below, a closed circle
indicates that point is included.
An open circle indicates that the point
is not included.
Angles

Angle
• formed by the union of 2 rays
• Measure amount of rotation from one ray
to the other
• vertex—endpoint of rays
Measuring Angles

360° in full rotation (circle)

1° is 1/360 of a complete rotation

E.g.: When the hour hand of a clock
moves from 12 to 2 o’clock, how many
degrees does it move?
• Solution:
2/12 or 1/6 of complete revolution.
1
6
x 360 = 60°
Classifying Angles
Special Pairs of Angles

Complementary Angles—2 angles whose
sum is 90°
The complement of 70° is
90° - 70° = 20°

Supplementary angles—2 angles whose sum
is 180°
The supplement of 110° is
180° - 110° = 70°
Example

Find m∠DBC

Solution:
m∠DBC = 90° - 62 ° = 28°
Example
mABD is 66° greater than mDBC and they are
supplementary angles.
Find the measure of each angle.
Solution:
mDBC + mABD = 180°
Let x = mDBC
x + (x + 66°) = 180°
2x + 66° = 180°
2x =114°
x = 57°
m  DBC = 57°
m  ABD =57° + 66° = 123°
Vertical Angles
 When
two lines intersect, the opposite
angles formed are called Vertical Angles.
Vertical Angles are equal.
 The angle on the left measures 68°.
Find the other angles.
 Solution:
1 = 68°
1 + 2 = 180°
2 = 180° − 68°
= 112°
3 = 2 = 112°
Special Line Relationships

Parallel Lines– Lines that
line in the same plane and
have no points in common

Intersecting lines—Two
lines that are not parallel
and have single point in
common

Transversal—A line that
intersects two parallel
lines
Angle Pairs Formed by a
Transversal Intersecting 2 || lines
Name
Description
Sketch
Angle Pairs
Described
Property
Alternate
interior angles
Interior angles that do
not have a common
vertex, and are on
alternate sides of the
transversal
3 and 6
4 and 5
Alternate interior angles
have the same measure.
3 = 6
4 = 5
Alternate
exterior angles
Exterior angles that do
not have a common
vertex, and are on
alternate sides of the
transversal
1 and 8
2 and 7
Alternate exterior angles
have the same measure.
1 = 8
2 = 7
Corresponding
angles
One interior and one
exterior angle on the
same side of the
transversal
1 and 5
2 and 6
3 and 7
4 and 8
Corresponding angles have
the same measure.
1 = 5
2 = 6
3 = 7
4 = 8
Parallel Lines and Angle Pairs

If 2 parallel lines are intersected by a
transversal:
• Alternate interior angles are equal.
• Alternate exterior angles are equal.
• Corresponding angles are equal.

Conversely, if 2 lines are intersected by a
transversal:
• If alternate interior angles are equal, the lines are
parallel.
• If alternate exterior angles are equal, the lines are
parallel.
• If corresponding angles are equal, the lines are
parallel.
Example
Given m8 = 35°,
find the measure
of all angles.
m1
m6
m7
m2
m3
m5
m4
=
=
=
=
=
=
=
35°
180° − 35° = 145°
145
35°
145°
35°
180° − 35° = 145°