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Geometry
Geometry:Part III
By
Dick Gill, Julia Arnold and
Marcia Tharp
for
Elementary Algebra Math 03 online
Table of Contents
Converting Degrees to Degrees and Minutes
Converting Degrees and Minutes to decimal degrees
Vertical Angles
Straight Angles
Parallel Lines
Problems involving the above and similar triangles.
Fractions of Angles
One degree is an awfully small angle,
but when we need to talk about a
fraction of a degree, we can do so with
decimals or with minutes and seconds.
One degree can be divided into 60
minutes and one minute can be divided
into 60 seconds. The notation for
minutes and seconds will look like this:
Which brings up the question, how do
you convert from decimal fractions to
fractions with minutes and seconds?
1o = 60’
1’ = 60”
25.5o = 25o 30’
Converting from Decimals to Minutes
To convert 57.4o degrees to degrees and minutes we can write the
degrees as 570 + .4o . Next we find 0.4 of 60’minutes by
multiplying 0.4(60’) = 24’minutes . Therefore, 57.4o = 57o 24’.
Try a few of these on your own.
Convert each of the following to degrees and minutes:
123.8o
= 123o + 0.8(60’) = 123o + 48’ = 123o 48’
12.3o
= 12o + 0.3(60’) = 12o + 18’ = 12o 18’
82.33o
= 82o + 0.33(60’) = 82o + 19.8’ which rounds to
82o 20’ Though we won’t focus on it in this
course we could convert 19.8’ to 19’ 48”.
Converting from Degrees and Minutes to
Decimal Degrees
To convert the minutes to a decimal it is helpful to remember that
minutes represent a fraction of a degree. We need to get that
fraction into decimal form. For example in 120o 45’ the 45’ are
the fraction of a degree: (45/60)o = .75o so 120o 45’ = 120.75o.
See what you can do with the following conversions. Round to
the nearest one hundredth of a degree.
12o 52’
= 12o + (52/60)o = 12o + .866o rounding to 12.87o
135o 45’
= 135o + (45/60)o = 135o + .75o = 135.75o
60o 25’
= 60o + (25/60)o = 60o + .416o rounding to 60.42o
Practice Problems
1. Convert 1680 15’ to degrees.
2. Convert 145.880 to degrees and minutes.
Work out the answers then click to the next slide to check.
Answers
1. Convert 1680 15’ to degrees.
1680 +15/60=168 + .25 = 168.250
2. Convert 145.880 to degrees and minutes.
1450 + .88*60= 1450 52.8’
Table of Contents
Vertical Angles and Parallel Lines
Vertical Angles
When two lines intersect they form vertical angles.
Angles 3 and 4
form a pair of
vertical angles
Angles 1 and 2
form a pair of
vertical angles.
3
2
1
4
Straight Angles
A straight angle is an angle formed by a straight
line and has measure 1800.
1800
Let’s explore
Suppose we have
two intersecting
lines with angle
measures as shown.
120
60
What do these two
angles add up to?
Do you see the straight
angle?
Let’s explore
What should the
measure of
the ? angle be?
120
60
? 120
What measure is left
for the remaining
angle?
Let’s explore
Can you come up with
a property about
vertical angles after
looking at this
120
example?
60
60
120
The property is
Vertical angles are
always equal to each
other.
Look at the following picture and give
the measure of all unknown angles.
100
80
80
100
Table of Contents
Parallel lines
Two lines are parallel if they do not intersect.
A line which crosses the parallel
lines is called a transversal.
Parallel lines
Look at all the angles formed.
1
4
5
8
6
7
2
3
Let’s explore
If I give the measure of angle 1 as 1200, how
many other angles can I find?
1 20 2
4
5
8
3
6
7
See how many angles you can
find then click to the next slide.
Names of equal angles in picture
120
60
60
120
60
120
60 120
The red angles are called
alternate interior angles.
What other pair of angles are also alternate interior angles?
Names of equal angles in picture
The red 600 angles are also alternate interior
angles.
120
60
60
120
60 120
60
120
Names of equal angles in picture
The red angles pictured are called
Alternate exterior angles.
120
60
60
120
60
120
60 120
Can you find another pair of alternate exterior
angles?
Names of equal angles in picture
The red angles pictured are called
Alternate exterior angles.
120
60
60
120
60
120
60 120
Do you see that the interior angles are between the parallel lines and
the exterior angles are on the outside of the parallel lines?
Names of equal angles in picture
The red angles pictured are called
Corresponding angles.
120
60
60
120
60
120
60 120
There are 3 more pairs of corresponding angles. Can you find them?
Names of equal angles in picture
The corresponding angles are pictured in
matching colors.
120
60
60
120
60 120
60
120
Properties
If two lines are parallel and cut by a
transversal then the alternate interior angles
are equal, the alternate exterior angles are
equal, and the corresponding angles are equal.
Note: Corresponding angles are always on the
same side of the transversal.
Can you fill in the missing
angles in the following picture
and state the reason why?
2
1
4
3
100o
5
7
6
1
Table of Contents
Ang 7= 100 because of vertical angles.
Ang 5 = 80 because 7 & 5 form a straight
angle.
Ang 6 is 80 because 5 and 6 are vertical
and therefore equal.
Ang 4 = 100 because alternate interior
angles are equal
Ang 2 = 100 because
of corresponding
2
angles or because of
vertical angles.
3
4
Angle 3 = 80 because
of straight angles or
because 3 & 5 are alt.
Int. angles
o
100 Ang. 1 = 80 because 3
5
& 1 are vertical or 1&5
7
6 are corresponding or 1
and 6 are alt ext
angles.
Problems involving similar triangles,
parallel lines, vertical angles, and
equal angles.
C
In this triangle
DB || EA
This makes CA and
CE transversals.
D
B
What angles
are equal?
CBD =
CDB =
E
CAE
CEA
A
These are two pairs of corresponding angles.
Note: The similar tick marks indicate equal angles.
C
What triangles
are similar?
BCD
D
E
B
___
ACE
Which is correct?
A
ABC
ECA
EBA
AEC
ACE
C
Answer the following:
AE = 10
DB = 6
1. If DC=7 find CE
D
E
B
A
When you’ve worked it out click here to check.
C
Answer the following:
AE = 10
DB = 6
2. If CA=27 find CB
D
E
B
A
When you’ve worked it out click here to check
C
Answer the following:
AE = 10
DB = 6
3. If CB=10 find BA
D
E
B
A
When you’ve worked it out click here to check
C
Answer the following:
AE = 10
DB = 6
4. If CE=24 find DE
D
E
B
A
When you’ve worked it out click here to check
Name the equal angles in this figure.
L
K=
N (Both are right angles)
LMK =
L=
OMN (Vertical angles)
O
M
N
K
How do you write the
similarity of the triangles?
NOM
KLM
Think before you click.
O
Click once for the
problem.
What are the corresponding
sides?
KM
MN
L
LM
5. If KM = 6
MN = 9
MO = 12
Find LM
MO
x
M
K
9
6
N
12
When you’ve worked it out click here to check
O
A
6.
A = DBC
AC = 12
BC = 8
BD = 5
FIND AB
12
D
5
B
8
C
When you’ve worked it out click here to check
B
D
7. BD = 5
BE = 8
BA = 10
FIND BC
E
A
When you’ve worked it out click here to check
C
C
Answer the following:
AE = 10
DB = 6
1. If DC=7 find CE
D
E
B
AE CE

DB DC
10 CE

6
7
A 6CE=70
CE = 35/3
Back to problem2
C
Answer the following:
AE = 10
DB = 6
2. If CA=27 find CB
D
B
AE CA

DB CB
10 27

6 CB
A 10CB=162
E
Back to problem3
CB=162/10
CB = 81/5
C
Answer the following:
AE = 10
DB = 6
3. If CB=10 find BA
D
Back to problem4
E
B
AE CA

DB CB
10 CA

6 10
We could not
A use BA
because BA is
6CA=100
not a side.
CA = 100/6=50/3
Now CA - CB = BA or 50/3-10 = (50-30)/3=20/3
BA = 20/3
C
Back to problem5
D
E
Answer the following:
AE = 10
DB = 6
4. If CE=24 find DE
B
AE CE

DB DC
10 24

6 DC
A
10DC=144
DC = 144/10=72/5
Now CE - DC = DE or 24 -72/5= (120-72)/5=48/5
DE = 48/5
5. If KM = 6
MN = 9
MO = 12
Find LM
L
x
M
K
6 x

9 12
9 x  72
x 8
9
6
N
12
Back to problem6
O
A
6.
A = DBC
AC = 12
BC = 8
BD = 5
FIND AB
12
First we must
find the similar
triangles.
D
What angle is
in two triangles?
5
B
8
C
A = DBC
AC = 12
BC = 8
BD = 5
FIND AB
A
12
D
5
B
First we must
find the similar
triangles.
What angle is
in two triangles?
c
8
C
Imagine separating
the two triangles.
ABC is flipped up,
while BDC is rotated
so DC is horizontal.
B
x
12
12
A
D
B
D
5
A
8
C
8
B
C
8
ABC
C
BDC
x 5

12 8
ABC
A
12
x
D
5
B
BDC
x 5

12 8
8 x  60
60 15
x

8
2
Back to problem7
8
C
B
7. BD = 5
BE = 8
BA = 10
FIND BC
DBE
D
A
First, what
two triangles
are similar?
ABC
E What angle is
in both triangles?
C
B
5
10
D
7. BD = 5
BE = 8
BA = 10
8 FIND BC
E
x
DBE
5 10

8 x
5 x  80
x  16
A
Table of Contents
Return to Problems
ABC
End show
C
You are now ready for the last
geometry topic: Area and Volume
Go to Geometry: Part IV