Geometry G Name______________________________Notes: Angle Notation.

Definition: An angle is formed by two rays that share a common endpoint.

1. The point that the two rays intersect is called the ________________________.

2. The two rays are called the ______________ of the angle.

3. When naming angles, it is typical to use one or three letters. Sometimes one cannot use one letter. When using three letters, the _________________ must be the letter in the middle. Other times one uses numbers to name the angles as below.

4. Name an angle using one letter. _________

5. Name three different angles. _________, __________, _________

6. IRC can also be named in what two other ways? ,

An angle breaks up a plane into three regions: the exterior of the angle the interior of the angle points on

the angle.

7. Name the points on the interior of FAB , , ,

8. Name the points on FAB. , , ,

1

M

A

X

R

IC

K

M

A

X1 2

S

M

A R T

F

B Y

N

2

3

Name______________________________________________________________________ Date__________________ Hour_____________

3.2 Notes – Angle MeasureGeometry G

In geometry, angles are measured in units called ____________________. The symbol is _______.

There are 4 different types of angles:Name of

angleDefinition Example

ACUTE

OBTUSE

RIGHT

STRAIGHT

Example 1:Use a protractor to measure . What kind of angle is it?

4P R

O

Example 2:Find the measure of each angle and classify it.

a) b) c) d)

I

V

L

E D S

Example 3:Use a protractor to draw an angle having each measurement. Then classify each angle.

a) b)

5

Example 4:The measure of B is 138. Solve for x.

(5x – 7)

B

Geometry G Name: ____________________________Protractor Practice Worksheet 1 Date: ____________

Goal: To measure angles using a protractor.

When using a protractor, you must use it correctly. What are the two things you need to do when using a protractor?

6

1. _______________________________________________________________________________

2. _______________________________________________________________________________

7

Geometry G Name: ____________________________Protractor Practice Worksheet 2 Date: ____________

Measure each angle to the nearest whole degree. You may have to extend the sides of your angle to do the measurement.

1. 2.

3. 4.

5. 6.

8

Draw each angle using a Protractor.

7. 8.

9. 10.

11. 12.

9

Review 3.1 – 2

1] Name the angle in 4 ways:

2] What points are in the interior, exterior or on the angle?

Interior:

Exterior:

On:

3] Use a protractor to measure each angle.

ABG = EBC =

ABF = FBC =

ABD = FBD =

4] Us a protractor to draw an angle having each of the following measurements:

50 125

90 158

10

B

AT

2

B

RI

CK

A B C

DEF

G

Name_____________________________________________________ Date________________ Hour_______

3.3 Notes – The Angle Addition PostulateGeometry G

Suppose mKNL = 110 and mLNM = 25. What would you do to find the mKNM?

Suppose mMNK = 155 and mLNM is 30. What would you do to find the mLNK?

Angle Addition PostulateFor any ABC, if D is in the interior of ABC, then mABD + mDBC = mABC.

Draw a diagram below to show this.

VocabularyA _______ that divides an angle into ____________ angles of equal ________________

is called the ___________________________________.

11

K N

LM

K N

LM

12

13

Geometry G Name________________________Worksheet 3.3

SHOW ALL YOUR WORK!!

1. Find m1 if mCUB = 78. 2. Find m2 if mWHI = 160.

3. mSOX = 160 m1 = x + 14 m2 = 3x – 10 Find m2

4. mBEA = 71. Find mREA.

5. mWOV = 12x. Find mLOV.

14

S

O

W

1 2 X

B U

SC

148

W

T

H

E

I

104242

A

B

E R(5x + 8)

2x

W

O

L

V76(5x + 1)

6. mFIE = 3x, mRIE = 42, mFIR = 5x Find mFIR.

7. mHAK = 4x – 2, mKAW = 2x – 5, and mHAW = 77. Find mHAK and mKAW.

8. US bisects BUL, mBUS = 2x + 10, and mSUL = 3x – 18. Find mBUL.

9. mTRI = 3x – 5, mIRB = x + 27, and mTRB = 86. Does RI bisect TRB?

10. Find the measure of each angle.

a. mNEO = _______ b. mDES = _______

c. mDEO = _______ d. mSEO = _______

15

H

A

K

W

B

U

S

L

T

R

B

I

S E C

DN

O27

18

FE

IR

Section 3.4 – Adjacent Angles and Linear Pairs

In the diagrams below, 1 and 2 are …

Not Adjacent Angles Adjacent Angles Not Adjacent Angles Adjacent Angles

What can you conclude about Adjacent Angles?

Adjacent Angles are angles that have a shared ____________ and the same ________________, but no interior points in common.

Try these…

Determine whether 1 and 2 are adjacent angles.

In the diagrams below, 1 and 2 are …

a linear pair a linear pair not a linear pair

What can you conclude about a Linear Pair?

Linear Pair consists of 2 angles that are ________________ and their noncommon sides are _________________ ________________.

16

1 2

E

H1 2 2

1

O

L

T

1 2 1 2 2

1

1 21

2

21

1 2

K

17

3-5 – Complementary and Supplementary Angles

Two angles whose measures add up to ______ are called __________________ __________.

They can also be called a _____________ __________ if together they form a straight angle.

In the picture above, _______ and _______ are ______________________ ___________.

Two angles whose measures add up to ______ are _____________________ ___________.

In the diagram above, ______ and ______ are _____________________ ___________.

Use the figure on the right to name each of the following.

1. Name a pair of complementary angles.

2. Name a pair of supplementary angles.

3. Name a different pair of supplementary angles.

4. Name a linear pair.

Find the measure of each angle

5. 6. 7.

18

C

B

A D

50°40°

RT

L

Q

P O

N

M

56° 56°

L

E

F

A

Find the measure of each angle in the diagram.

DAB is a right angleADE is a right angle1 = 53

m 1 = m 123 = 555 = 88

m 4 = m 9ABE = 100DEB = 80

19

A

E

D

C

B

12

65

43

87

9

1211

10

Geometry G Name_________________________Worksheet 3.5

Supplementary and Complementary AnglesFind the measures of angles 1 through 22. Mark them in your diagram.

23) Find mDBC.

24) Find mDBC.20

571

71 2

756287

6

91181920

25

B CAx

D

8x

(4x – 20)

xA B

CD

1213

122

34 5

104

17161514 73

4211

109

70

21

22

25) 1 and 2 are complementary. m1 = 2x + 7 and m2 = 4x – 19. Find the measure of each angle.

26) 3 and 4 are supplementary. m3 = 5x + 22 and m4 = 7x + 2. Find the measure of each angle.

27) Use the diagram on the right to name:

a) two complementary angles

b) a linear pair

c) two adjacent angles

21

A

D

CB

E

F

G

Name_________________________________________ Date________ Hour____

3.6 – Vertical AnglesGeometry G

Vertical Angles:

THEOREM:

Examples:

1) Find x, y, and z x 51 Y z

2) Given: m4 = (2x + 5) m5 = (x +30)

Find: m6

3) Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or linear pair.

a) 1 and 2 b) 3 and 4

c) 5 and 4

d) 3 and 5

22

12

34

4 56

1

2

3 45

4) Find x and y if CBD is congruent to FDG.

5) Find each of the following:

a) x

b) mLAT

c) mTAO

d) mPAO

23

Vocabulary Words:

Complementary Angles Supplementary Angles Right AnglesAngle Bisector Linear Pair Vertical

AnglesAdjacent Angles.

ST bisects

1. In the pictures above, FOH and GOH are called _____________________________.

2. FOH and GOH are also called ____________________________________________.

3. Further, FOH and GOH are _____________________________________________.

4. In the pictures above, ACB and DCE are called ______________________________.

5. In the pictures above, JPK and KPL are called ______________________________.

6. JPK and KPL are also called ___________________________________.

7. Name the vertical angle ACD to ___________________________________.

8. What do you know about RST and TSW? ___________________________________

9. What do you call LPM? ___________________________________

24

In the figure, GA and GD, and GB and GE are opposite rays.

10] Which angle forms a linear pair with ? ________

11] Do and form a linear pair? ________

12] Name two angles that are adjacent to . ________ ________

13] Name two angles that form a linear pair with . ________ ________

14] Name three angles adjacent to . ________ ________ ________

15] Do and form a linear pair? ________

16] Name the vertical angle to . _______________

17] Name another pair of vertical angles. ____________ and _______________

25

Name:

Name:

1] a linear pair

2] a pair of supplementary angles

3] a pair of complementary angles

4] a pair of adjacent angles

5] a pair of vertical angles

6] two right angles

Write each pair of angles that you named above into the proper column of the table below.

Angle Relationships

Equals Equals 180 Equals 90

26

A

B

C

E

FG

D

27

Determine the relationship in the diagram.

Are the angles complementary or is it a right angle? The angles add to 90.

Are the angles supplementary or are they a linear pair? The angles add to 180.

Do you have an angle bisector? The two angles are congruent.

Do you have vertical angles? The two angles are congruent.

Write the equation and then solve the equation.

1. 2.

Equation: _______________________ Equation: _______________________

x = ______ x = ______

3. 4.

Equation: _______________________ Equation: _______________________

x = ______ x = ______28

5. 6.

Equation: _______________________ Equation: _______________________

x = ______ x = ______

7. 8.

Equation: _______________________ Equation: _______________________

x = ______ x = ______

3.7 – Perpendicularity Name____________________________29

Geometry G Date___________________ Hour_____NOTES

Perpendicularity, _____________________, and __________ measurements go together.

Definition: If lines, rays or segments form right angles, then they are perpendicular( ).

What would be the converse of the definition?

Examples:

a b

What conclusions would I be able to make if given the following:

1)

2)

30

D

E

F

a

b

A

BC

Example 1: True or False?

1. PRN is acute.

2. 4 8

3. m5 + m6 = 90

4.

5. 7 is obtuse

Example 2:

Find x.

Example 3:

Find mDBC.

31

Geometry G Name____________________________Section 2.5. Worksheet 3

Warm – Up:

1. 2.

ST bisects ,

3. 4.

& BD bisects

5.AB CDCE bisects

6.

BC bisects

32

7.

8. BD bisects and

9.

10. AB CDHE bisects

33

Given l p

Determine the relationship in the diagram. Are the angles complementary or is it a right angle? The angles add to

90.Are the angles supplementary or are they a linear pair? The angles add

to 180.Do you have a angle bisector? The two angles are

congruent.Do you have vertical angles? The two angles are

congruent.

Write the equation and then solve the equation.

1. 2.

Equation: _______________________ Equation: _______________________

x = ______ x = ______

3. 4.

Equation: _______________________ Equation: _______________________

x = ______ x = ______

34

Geometry G Name____________________________Section 2.5. Worksheet 6

5. 6.

Equation: _______________________ Equation: _______________________

x = ______ x = ______

Note: Picture is not drawn to scale.

7. BD bisects 8.

Equation: _______________________ Equation: _______________________

x = ______ x = ______

35

9.Equation: _______________________

x = ________

10. 11.

Equation:_________________________ Equation:_________________________

x = ________ x = ________

36

Geometry G Name____________________________Section 2.5. Worksheet 7

Determine the relationship in the diagram. Are the angles complementary or is it a right angle? The angles add to

90.Are the angles supplementary or are they a linear pair? The angles add

to 180.Do you have an angle bisector? The two angles are

congruent.Do you have vertical angles? The two angles are

congruent.

Write the equation and then solve the equation.

1. 2.

Equation: _______________________ Equation: _______________________

x = ______ x = ______

3. 4.

Equation: _______________________ Equation: _______________________

x = ______ x = ______

37

8x (7x + 10)

(6x + 12)

(16x + 4) (18x +

4)

18x(16x + 4)

(7x - 12)

(5x + 18)

38