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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

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geometry part iii by dick gill julia arnold and marcia tharp for elementary algebra math 03 online
Geometry:Part IIIByDick Gill, Julia Arnold and Marcia Tharpfor Elementary Algebra Math 03 online
table of contents
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
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:

1o = 60’

1’ = 60”

25.5o = 25o 30’

Which brings up the question, how do you convert from decimal fractions to fractions with minutes and seconds?

converting from decimals to minutes
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’

= 12o + 0.3(60’) = 12o + 18’ = 12o 18’

12.3o

= 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”.

82.33o

converting from degrees and minutes to decimal degrees
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 in120o 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 + (25/60)o = 60o + .416o rounding to 60.42o

60o 25’

slide7

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.

slide8

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

slide10

Vertical Angles

When two lines intersect they form vertical angles.

Angles 3 and 4

form a pair of

vertical angles

3

Angles 1 and 2

form a pair of

vertical angles.

2

1

4

slide11

Straight Angles

A straight angle is an angle formed by a straight line and has measure 1800.

1800

slide12

Let’s explore

Suppose we have

two intersecting

lines with angle

measures as shown.

60

120

What do these two

angles add up to?

Do you see the straight

angle?

slide13

Let’s explore

What should the measure of

the ? angle be?

60

120

?

120

What measure is left

for the remaining

angle?

slide14

Let’s explore

Can you come up with

a property about

vertical angles after

looking at this

example?

60

120

120

60

The property is

Vertical angles are

always equal to each

other.

slide15

Look at the following picture and give

the measure of all unknown angles.

100

80

80

100

Table of Contents

slide16

Parallel lines

Two lines are parallel if they do not intersect.

A line which crosses the parallel

lines is called a transversal.

slide17

Parallel lines

Look at all the angles formed.

1

2

4

3

5

6

8

7

slide18

Let’s explore

If I give the measure of angle 1 as 1200, how

many other angles can I find?

20

1

2

4

3

5

6

8

7

See how many angles you can

find then click to the next slide.

slide19

Names of equal angles in picture

120

60

120

60

60

120

60

120

The red angles are called

alternate interior angles.

What other pair of angles are also alternate interior angles?

slide20

Names of equal angles in picture

The red 600 angles are also alternate interior angles.

120

60

120

60

60

120

60

120

slide21

Names of equal angles in picture

The red angles pictured are called

Alternate exterior angles.

120

60

120

60

60

120

60

120

Can you find another pair of alternate exterior

angles?

slide22

Names of equal angles in picture

The red angles pictured are called

Alternate exterior angles.

120

60

120

60

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?

slide23

Names of equal angles in picture

The red angles pictured are called

Corresponding angles.

120

60

120

60

60

120

60

120

There are 3 more pairs of corresponding angles. Can you find them?

slide24

Names of equal angles in picture

The corresponding angles are pictured in matching colors.

120

60

60

120

60

120

60

120

slide25

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.

slide26

Can you fill in the missing

angles in the following picture and state the reason why?

2

1

3

4

100o

5

7

6

slide27

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 angles or because of vertical angles.

Angle 3 = 80 because of straight angles or because 3 & 5 are alt.

Int. angles

Ang. 1 = 80 because 3 & 1 are vertical or 1&5 are corresponding or 1 and 6 are alt ext angles.

2

1

3

4

100o

5

7

6

Table of Contents

slide28

Problems involving similar triangles,

parallel lines, vertical angles, and

equal angles.

slide29

C

In this triangle

DB || EA

This makes CA and CE transversals.

What angles

are equal?

B

D

CBD = CAE

CDB = CEA

A

E

These are two pairs of corresponding angles.

Note: The similar tick marks indicate equal angles.

slide30

C

What triangles

are similar?

BCD

___

ACE

Which is correct?

B

D

ABC

ECA

EBA

AEC

ACE

A

E

slide31

C

Answer the following:

AE = 10

DB = 6

1. If DC=7 find CE

B

D

A

E

When you’ve worked it out click here to check.

slide32

C

Answer the following:

AE = 10

DB = 6

2. If CA=27 find CB

B

D

A

E

When you’ve worked it out click here to check

slide33

C

Answer the following:

AE = 10

DB = 6

3. If CB=10 find BA

B

D

A

E

When you’ve worked it out click here to check

slide34

C

Answer the following:

AE = 10

DB = 6

4. If CE=24 find DE

B

D

A

E

When you’ve worked it out click here to check

slide35

Name the equal angles in this figure.

L

K = N (Both are right angles)

LMK = OMN (Vertical angles)

L = O

N

M

K

How do you write the

similarity of the triangles?

NOM

KLM

Think before you click.

O

slide36

Click once for the problem.

What are the corresponding

sides?

KM

MN

5. If KM = 6

MN = 9

MO = 12

Find LM

LM

MO

L

x

M

9

N

6

K

12

When you’ve worked it out click here to check

O

slide37

6.

A = DBC

AC = 12

BC = 8

BD = 5

FIND AB

A

12

D

5

8

C

B

When you’ve worked it out click here to check

slide38

B

BD = 5

BE = 8

BA = 10

FIND BC

7.

E

D

A

C

When you’ve worked it out click here to check

slide39

C

Answer the following:

AE = 10

DB = 6

1. If DC=7 find CE

B

D

6CE=70

CE = 35/3

A

E

Back to problem2

slide40

C

Answer the following:

AE = 10

DB = 6

2. If CA=27 find CB

B

D

10CB=162

CB=162/10

CB = 81/5

A

E

Back to problem3

slide41

C

Answer the following:

AE = 10

DB = 6

3. If CB=10 find BA

B

D

Back to problem4

We could not use BA

because BA is not a side.

A

E

6CA=100

CA = 100/6=50/3

Now CA - CB = BA or 50/3-10 = (50-30)/3=20/3

BA = 20/3

slide42

C

Answer the following:

AE = 10

DB = 6

4. If CE=24 find DE

Back to problem5

B

D

A

E

10DC=144

DC = 144/10=72/5

Now CE - DC = DE or 24 -72/5= (120-72)/5=48/5

DE = 48/5

slide43

5. If KM = 6

MN = 9

MO = 12

Find LM

L

x

N

M

9

6

K

12

Back to problem6

O

slide44

6.

A = DBC

AC = 12

BC = 8

BD = 5

FIND AB

A

First we must

find the similar

triangles.

12

What angle is

in two triangles?

D

5

8

C

B

slide45

A = DBC

AC = 12

BC = 8

BD = 5

FIND AB

A

First we must

find the similar

triangles.

12

What angle is

in two triangles?

D

5

c

8

C

B

slide46

B

Imagine separating

the two triangles.

ABC is flipped up,

while BDC is rotated

so DC is horizontal.

C

8

x

12

12

D

C

A

5

5

8

A

B

D

ABC BDC

8

C

B

slide47

ABC BDC

A

12

x

D

5

Back to problem7

8

C

B

slide48

First, what

two triangles

are similar?

B

BD = 5

BE = 8

BA = 10

FIND BC

7.

DBE

ABC

What angle is

in both triangles?

E

D

A

C

slide49

DBE

ABC

B

BD = 5

BE = 8

BA = 10

FIND BC

7.

8

5

E

x

10

D

A

C

Table of Contents

Return to Problems

End show

slide50

You are now ready for the last

geometry topic: Area and Volume

Go to Geometry: Part IV