1.4 Angles and Their Measures
GOAL 1 Use Angle Postulates
GOAL 2 Classify angles as acute, right, obtuse, or straight.
What you should learn
To solve real-life problems about angles, such as the field of vision of a horse wearing blinkers.
Why you should learn it
GOAL 1 USING ANGLE POSTULATES
1.4 Angles and Their Measures
An _____ consists of two different rays that have the same initial point.
angle
The rays are the _____ of the angle and the initial point is the ______.
sidesvertex
A
C
B
The three names for this angle are , , and .CAB BAC A
The sides are and .AB AC����������������������������
The vertex is point A.
EXAMPLE 1 Did you study Example 1???
Extra Example 1
Name the angles in the figure.
M N O
P
Click for the answers.
or MNP PNM or PNO ONP
1.
2.
3. or ONM MNO
should not be used to name any angle in the figure. Why not?
N
All three angles have N as the vertex, so could mean any of the angles.
N
Measuring AnglesThe expression is read as “the ________ of angle A.”m A measure
IMPORTANT!!! Note the difference in notation between an angle and its measure. Always use the correct notation!!!
The tool used to measure angles is called a _________.protractor
The units used to measure angles are called _______, and the symbol for them is a _.
degrees
Measuring Angles
Q
R
S
Let’s measure some angles.
____m QSR 70
____m MST M
T
45
____m QSY Y
45
Since we say that the angles are _________.
,m MST m QSY congruent
Remember: Angles are congruent, measures are equal.
m MST m QSY MST QSY andCorrect:
Incorrect: and MST QSY m MST m QSY
PROTRACTOR POSTULATE
For any point A on one side of , can be matched one to one with the real numbers from 0 to ___.
OB�������������� �
OA��������������
180
and OA OB����������������������������
.AOBThe absolute value of the difference between the real numbers for is the ________ of measure
B
O
A
Check your understanding of the Protractor Postulate by finding Use either scale on the protractor to find it, but use the same one for both rays.
.m AOB
60 140 80m AOB
120 40 80m AOB or
Click to see two solutions.
To understand the next postulate, you must understand some vocabulary:
A point that is between points that lie on each side of an angle is in the _______ of the angle.
A point that is not on an angle or in its interior is in the ________ of the angle.
interior
exterior
exteriorBinterior
A
Z
In the above diagram, A is in the interior of the angle and B is in the exterior of the angle.
ANGLE ADDITION POSTULATE
If P is in the interior of then
,RST
m RSP m PST m RST
R
T
PS
Does it make sense? Study the figure to be sure you understand before going on!
EXAMPLE 2
Extra Example 2
The backyard of a house is illuminated by a light fixture that has two bulbs. Each bulb illuminates an angle of 120°. If the angle illuminated only by the right bulb is 35°, what is the angle illuminated by both bulbs?
Click for a picture.
35°
120°
120°
What is the measure of the angle that is shaded green?
Click for the solution.
120° - 35° = 85°
Do you see the application of the Angle Addition Postulate?
85°
Checkpoint1. Name the angles in the figure.
Click for the solution.
CF
D
E
or CDE EDC or EDF FDE or CDF FDC
2. In the figure above, andFind the measure of Click for the solution.
62m CDE 18 .m EDF .CDF
m CDF m CDE m EDF 62 18 80m CDF
GOAL 2 CLASSIFYING ANGLES
1.4 Angles and Their Measures
We can classify all angles by their measure as follows:
Angles with measures between 0 and 90 degrees: _____
90 degree angles: _____
Angles with measures between 90 and 180 degrees: ______
180 degree angles: _______
acute
right
obtuse
straight
Important!!! Anytime this box appears at the vertex of an angle, it means that the angle is a right angle. You MUST use it
when you want to show a right angle.
EXAMPLE 3
Can you match each angle with its description?Click to check your answer.
A
A
A
A
0 90m A acute:
90m A right:
90 180m A obtuse:
180m A straight:
Extra Example 3
Plot the points A(-3, -1), B(-1, 1), C(2, 4), D(2, 1), and E(2, -2).
Then measure and classify the following angles as acute, right, obtuse, or straight. Click for each answer.
a.
b.
c.
d.
DBE
EBC
ABC
ABD
45°, acute
90°, right
180°, straight
135°, obtuse
Reminder: Be sure you understand the material before going on to the next slide. If you need to
review, do so NOW!
EXAMPLE 4
Two angles that share a common vertex and side, but have no common interior points, are called ________ angles.adjacent
AB
CD Name the two adjacent angles in the diagram. Click to check.
( ) and ( )ADB BDA CDB BDC
The common vertex is __ and the common side is ___. D DB��������������
Extra Example 4Use a protractor to draw two adjacent angles and so that is acute and is straight.Click to see a sample answer.
LMNLMN
NMO LMO
L
N
M O
Classify as acute, right, obtuse, or straight: NMO obtuse
Checkpoint
Draw 5 points A, B, C, D, and E so that all four statements are true:
are adjacent. is obtuse.
D is in the exterior of is a right angle.
Click to see a sample answer.
and AEC BEC AEB
.AEBDEC
Does your solution meet the
requirements above?
B
C
E
A
D
QUESTIONS?
