Transcript Angle - frauvan.org

Copyright © 2005 Pearson Education, Inc.
Introduction to
Trigonometry
Angle Relationships and
Similar Triangles
Copyright © 2005 Pearson Education, Inc.
Basic Terms continued

Angle-formed by rotating
a ray around its endpoint.

The ray in its initial
position is called the
initial side of the angle.

The ray in its location
after the rotation is the
terminal side of the
angle.
Copyright © 2005 Pearson Education, Inc.
Slide 1-3
Basic Terms continued

Positive angle: The
rotation of the terminal
side of an angle
counterclockwise.
Copyright © 2005 Pearson Education, Inc.

Negative angle: The
rotation of the terminal
side is clockwise.
Slide 1-4
Standard Position

An angle is in standard position if its vertex is
at the origin and its initial side is along the
positive x-axis.

Angles in standard position having their terminal
sides along the x-axis or y-axis, such as angles
with measures 90, 180, 270, and so on, are
called quadrantal angles.
Copyright © 2005 Pearson Education, Inc.
Slide 1-5
Coterminal Angles

A complete rotation of a ray results in an angle
measuring 360. By continuing the rotation,
angles of measure larger than 360 can be
produced. Such angles are called coterminal
angles.
Copyright © 2005 Pearson Education, Inc.
Slide 1-6
Angles and Relationships
q
m
n
Name
Angles
Rule
Alternate interior angles
4 and 5
3 and 6
Angles measures are equal.
Alternate exterior angles
1 and 8
2 and 7
Angle measures are equal.
Interior angles on the same
side of the transversal
4 and 6
3 and 5
Angle measures add to 180.
Corresponding angles
2 & 6, 1 & 5,
3 & 7, 4 & 8
Angle measures are equal.
Copyright © 2005 Pearson Education, Inc.
Slide 1-7
Conditions for Similar Triangles

Corresponding angles must have the same
measure.

Corresponding sides must be proportional.
(That is, their ratios must be equal.)
Copyright © 2005 Pearson Education, Inc.
Slide 1-8
Example: Finding Angle Measures

Triangles ABC and DEF
are similar. Find the
measures of angles D
and E.


D
Since the triangles are
similar, corresponding
angles have the same
measure.
Angle D corresponds to
angle A which = 35
A
112
35
F
C
112
33
Copyright © 2005 Pearson Education, Inc.
E

Angle E corresponds to
angle B which = 33
B
Slide 1-9
Example: Finding Side Lengths

Triangles ABC and DEF
are similar. Find the
lengths of the unknown
sides in triangle DEF.

32 64

16 x
32 x  1024
x  32
D
A

16
112
35
64
F
32
C
112
33
48
Copyright © 2005 Pearson Education, Inc.
To find side DE.
B
E
To find side FE.
32 48

16 x
32 x  768
x  24
Slide 1-10
Example: Complementary Angles


Find the measure of each angle.
Since the two angles form a right
angle, they are complementary
angles. Thus,
k  20  k  16  90
k +20
k  16
2k  4  90
2 k  86
k  43
Copyright © 2005 Pearson Education, Inc.
The two angles have measures of
43 + 20 = 63 and 43  16 = 27
Slide 1-11
Example: Coterminal Angles




Find the angles of smallest possible positive
measure coterminal with each angle.
a) 1115
b) 187
Add or subtract 360 as may times as needed to
obtain an angle with measure greater than 0 but
less than 360.
o
o
o
a) 1115  3(360 )  35
b) 187 + 360 = 173
Copyright © 2005 Pearson Education, Inc.
Slide 1-12
Example: Finding Angle Measures

Find the measure of each
marked angle, given that
lines m and n are parallel.
(6x + 4)
(10x  80)
m


n


The marked angles are
alternate exterior angles,
which are equal.
Copyright © 2005 Pearson Education, Inc.
6 x  4  10 x  80
84  4 x
21  x
One angle has measure
6x + 4 = 6(21) + 4 = 130
and the other has measure
10x  80 = 10(21)  80 =
130
Slide 1-13