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Transcript
1.3 Segments, Rays and Angles
Copyright © 2014 Pearson Education, Inc.
Slide 1-1
Undefined Terms
Description
A line segment or simply segment is part of a line.
It consists of two endpoints and all the points
between them.
How to Name It
Name a segment by its end points: AB (segment AB)
or BA (segment BA)
Example
AB BA
Copyright © 2014 Pearson Education, Inc.
Slide 1-2
Postulate 1.13 Segment Addition Postulate
If point B is between points A and C, then
AB + BC = AC. Also, if AB + BC = AC, then point B
is between points A and C.
Copyright © 2014 Pearson Education, Inc.
Slide 1-3
Using the Segment Addition
Example
Postulate
If MP = 47 units, find MN and NP.
Solution
Use the value of x to find
MN and NP.
MN = 5x + 1 = 5(3) + 1 = 16
NP = 11x – 2 = 11(3) – 2 = 31
Check: See that MP = 47.
16 + 31 = 47
Copyright © 2014 Pearson Education, Inc.
Slide 1-4
Using the Segment Addition
Example
Postulate
If MP = 47 units, find MN and NP.
Solution
MN +
NP = MP
(5x + 1) + (11x – 22) = 47
16x – 1 = 47
16x = 48
16 x 48

16 16
x3
Copyright © 2014 Pearson Education, Inc.
Slide 1-5
Terms
Description
A ray is part of a line. It consists of an endpoint and
all points of a line on one side of the endpoint.
How to Name It
Name a ray by its endpoint and any other point on
the ray. Here, the order of points is important—list
the endpoint first. AB (ray AB)
Example
BA
AB
Copyright © 2014 Pearson Education, Inc.
Slide 1-6
Example
Naming Segments and Rays
Use the given figure to find the following.
a. Name the segments in the figure.
The three segments are
DE or ED EF or FE DF or FD
b. Name the rays in the figure.
ED, EF
DE or DF
FD or FE
c. Which of the rays in part b are opposite rays?
ED, EF
Copyright © 2014 Pearson Education, Inc.
Slide 1-7
Example
Drawing Segments, and Rays
a. Draw
NL
b. Draw
LM
Copyright © 2014 Pearson Education, Inc.
Slide 1-8
Defined Terms
Definition
An angle consists of two different
rays with a common endpoint.
The rays are the sides of the angle.
The common endpoint is the vertex of the angle.
Point A is the vertex.
The sides are rays AC and AB.
Ways to name the angle:
A, 1, CAB, BAC
Copyright © 2014 Pearson Education, Inc.
Slide 1-9
Example
Identifying and Naming Angles
a. How many different angles are in the diagram?
b. Write two other ways to name ∠1.
Solution
a. There are three different angles in the diagram:
∠1, ∠2, and ∠MPQ.
b. ∠1 can also be named as ∠MPN or ∠NPM.
Copyright © 2014 Pearson Education, Inc.
Slide 1-10
Protractor
The instrument shown is called a protractor. It can
be used to measure angles in units called degrees
(°). For example, the measure of ∠A (also denoted
by m∠A) is 30°.
Copyright © 2014 Pearson Education, Inc.
Slide 1-11
Example
Measuring and Classifying Angles
Find m∠RQM, m∠RQS, and m∠RQN. Then classify
each angle as acute, right, obtuse, or straight.
Solution
mRQM | 45  0 | 45
acute angle
mRQS | 90  0 | 90
right angle
mRQN |165  0 | 165
obtuse angle
Copyright © 2014 Pearson Education, Inc.
Slide 1-12
Classifying Angles
Acute Angle
0° < m∠A < 90°
Right Angle
m∠B = 90°
Copyright © 2014 Pearson Education, Inc.
Slide 1-13
Classifying Angles
Obtuse Angle
90° < m∠C < 180°
Straight Angle
m∠D = 180°
Copyright © 2014 Pearson Education, Inc.
Slide 1-14
Special Angle Pairs
Definition
Two angles are complementary if their measures
have a sum of 90°. Each angle is called the
complement of the other.
∠1 and ∠2 are
complementary angles.
∠A and ∠B are
complementary angles.
Copyright © 2014 Pearson Education, Inc.
Slide 1-15
Special Angle Pairs
Definition
Two angles are supplementary if their measures
have a sum of 180°. Each angle is called the
supplement of the other.
∠3 and ∠4 are
supplementary angles.
∠C and ∠D are
supplementary angles.
Copyright © 2014 Pearson Education, Inc.
Slide 1-16
Defined Terms
The interior of an angle contains all points between
the two sides of the angle.
The exterior of an angle contains all points that are
not in the interior of the angle and are not on the
angle.
Copyright © 2014 Pearson Education, Inc.
Slide 1-17
Special Angle Pairs
Definition
Two angles are called adjacent angles if they share
a common side and a common vertex, but have no
interior points in common.
∠1 and ∠2 are
adjacent angles.
∠3 and ∠4 are
adjacent angles.
Copyright © 2014 Pearson Education, Inc.
Slide 1-18
Postulate 1.14 Angle Addition Postulate
If P is in the interior of ∠ABC, then
m∠ABP + m∠PBC = m∠ABC.
Copyright © 2014 Pearson Education, Inc.
Slide 1-19