Download Chapter 3.angles

Geometry G
Notes: Angle Notation.
Name______________________________
Definition: An angle is formed by two rays
that share a common endpoint.
M
X
A
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.
M

C

I
X

K
1 2
A
R
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.
M
7. Name the points on the interior of  FAB
8. Name the points on  FAB.
,
,
,
,
,
F
,
N
A
T
B
S
R
Y
1
2
Name______________________________________________________________________ Date__________________ Hour_____________
3.2 Notes – Angle Measure
Geometry G
In geometry, angles are measured in units called ____________________.
The symbol is _______.
There are 4 different types of angles:
Name of angle
Definition
Example
ACUTE
OBTUSE
RIGHT
STRAIGHT
Example 1:
Use a protractor to measure PRO . What kind of angle is it?
O
P
R
3
Example 2:
Find the measure of each angle and classify it.
a) VDS
b) SDL
c) IDS
d) SDE
I
V
L
E
D
S
Example 3:
Use a protractor to draw an angle having each measurement. Then classify each angle.
a) 115
b) 25
Example 4:
The measure of  B is 138. Solve for x.
(5x – 7)
B
4
Geometry G
Protractor Practice Worksheet 1
Name: ____________________________
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?
1. _______________________________________________________________________________
2. _______________________________________________________________________________
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Geometry G
Protractor Practice Worksheet 2
Name: ____________________________
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.
6
Draw each angle using a Protractor.
7.
A  35
8.
B  100
9.
C  150
10.
D  50
11.
F  90
12.
G  180
7
Review 3.1 – 2
1]
B
Name the angle in 4 ways:
2
A
2]
B
Interior:
K
Exterior:
R
4]
Use a protractor to measure each angle.
ABG =
EBC =
ABF =
FBC =
ABD =
FBD =
C
I
On:
3]
T
What points are in the interior, exterior or on the angle?
F
E
D
G
A
B
Us a protractor to draw an angle having each of the following measurements:
50
125
90
158
C
8
Name_____________________________________________________ Date________________ Hour_______
3.3 Notes – The Angle Addition Postulate
Geometry G
Suppose mKNL = 110 and mLNM = 25. What would you do to find the mKNM?
L
M
K
N
Suppose mMNK = 155 and mLNM is 30. What would you do to find the mLNK?
L
M
K
N
Angle Addition Postulate
For any ABC, if D is in the interior of ABC, then mABD + mDBC = mABC.
Draw a diagram below to show this.
Vocabulary
A _______ that divides an angle into ____________ angles of equal ________________
is called the ___________________________________.
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10
Geometry G
Worksheet 3.3
Name________________________
SHOW ALL YOUR WORK!!
1. Find m1 if mCUB = 78.
B
1
48
S
2. Find m2 if mWHI = 160.
E
U
I
T
C
42 104
2
H
W
3. mSOX = 160
m1 = x + 14
m2 = 3x – 10
Find m2
S
W
1
2
X
O
B
4. mBEA = 71. Find mREA.
2x
R
(5x + 8)
E
A
O
5. mWOV = 12x. Find mLOV.
W
76
(5x + 1)
V
L
11
6. mFIE = 3x, mRIE = 42, mFIR = 5x
Find mFIR.
E
F
R
I
7. mHAK = 4x – 2, mKAW = 2x – 5,
and mHAW = 77.
Find mHAK and mKAW.
H
K
W
A
8. US bisects BUL, mBUS = 2x + 10,
and mSUL = 3x – 18.
Find mBUL.
B
S
U
L
T
9. mTRI = 3x – 5, mIRB = x + 27,
and mTRB = 86.
Does RI bisect TRB?
I
R
B
N
10. Find the measure of each angle.
a. mNEO = _______
b. mDES = _______
c. mDEO = _______
d. mSEO = _______
D
O
27
18
S
E
C
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Section 3.4 – Adjacent Angles and Linear Pairs
In the diagrams below, 1 and 2 are …
T
1
1
2
2
H
O
L
2
1
E
Not Adjacent Angles
Adjacent Angles
Not Adjacent Angles
1 2
K
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.
1
1 2
2
1
2
In the diagrams below, 1 and 2 are …
1
1
2
2
1
a linear pair
a linear pair
2
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
_________________ ________________.
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14
3-5 – Complementary and Supplementary Angles
F
A
E
L
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 _____________________ ___________.
B
50°
40°
C
R
T
D
In the diagram above, ______ and ______ are _____________________ ___________.
A
Use the figure on the right to name each of the following.
L
1. Name a pair of complementary angles.
N
2. Name a pair of supplementary angles.
Q
M
3. Name a different pair of supplementary angles.
P
O
4. Name a linear pair.
Find the measure of each angle
5.
6.
56°
7.
56°
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Find the measure of each angle in the diagram.
 DAB is a right angle
 ADE is a right angle
 1 = 53
m  1 = m  12
 3 = 55
 5 = 88
m4 = m9
 ABE = 100
 DEB = 80
D
3
4
12
11
7
E
8
C 5
6
2
A
1
9
10
B
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Geometry G
Worksheet 3.5
Name_________________________
Supplementary and Complementary Angles
Find the measures of angles 1 through 22. Mark them in your diagram.
57
6
7
14
15
1
75
62
8
73
16
3
71 2
17
4
42
11
10
9
18 91
20 19
25
104
5
122
13
12
21
70
22
17
D
23) Find mDBC.
x
8x
B
A
C
C
24) Find mDBC.
D
(4x – 20)
x
A
B
25) 1 and 2 are complementary. m1 = 2x + 7 and m2 = 4x – 19. Find the measure of each angle.
26) 3 and 4 are supplementary. m3 = 5x + 22 and m4 = 7x + 2. Find the measure of each
angle.
27) Use the diagram on the right to name:
C
a) two complementary angles
B

D
b) a linear pair


G
E
c) two adjacent angles

A
F
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Name_________________________________________ Date________ Hour____
3.6 – Vertical Angles
Geometry G
Vertical Angles:
3
4
1
2
THEOREM:
Examples:
1) Find x, y, and z
x
51
Y
z
2) Given: m4 = (2x + 5)
m5 = (x +30)
Find: m6
4
5
6
3) Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or linear pair.
a) 1 and 2
b) 3 and 4
2
3
1
c) 5 and 4
4
5
d) 3 and 5
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4) Find x and y if CBD is congruent to FDG.
5) Find each of the following:
a)
x
b) mLAT
c)
mTAO
d) mPAO
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Vocabulary Words:
Complementary Angles
Angle Bisector
Adjacent Angles.
Supplementary Angles
Linear Pair
Right Angles
Vertical Angles
ST bisects RSW
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? ___________________________________
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In the figure, GA and GD, and GB and GE are opposite rays.
10]
Which angle forms a linear pair with DGC ? ________
11]
Do BGC and EGD form a linear pair?
12]
Name two angles that are adjacent to CGD . ________
13]
Name two angles that form a linear pair with BGD . ________
________
14]
Name three angles adjacent to AGB . ________
________
15]
Do CGE and CGB form a linear pair?
16]
Name the vertical angle to EGD . _______________
17]
Name another pair of vertical angles. ____________ and _______________
________
________
________
________
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Name:
E
D
A
G
F
C
B
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
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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 = ______
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5.
6.
Equation: _______________________
Equation: _______________________
x = ______
x = ______
7.
8.
Equation: _______________________
Equation: _______________________
x = ______
x = ______
mACB  ________
mABC  _________
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3.7 – Perpendicularity
Name____________________________
Date___________________ Hour_____
Geometry G
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:
DE EF
ab
E
a
D
F
b
What conclusions would I be able to make if given the following:
AB  BC
1)
A
2)
B
C
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Example 1: True or False?
1. PRN is acute.
2. 4  8
3. m5 + m6 = 90
4. QR  PR
5. 7 is obtuse
Example 2:
Find x.
Example 3:
Find mDBC.
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Geometry G
Section 2.5. Worksheet 3
Name____________________________
Warm – Up:
1.
2.
ST bisects RSW , mRST  27
mFOH  ________
mTSW  _________ mWSR  _________
3.
4.
mJPK  _________
mCBF  70 & BD bisects CBF.
mJPL  _________
mCBD  ________
5.
AB  CD
CE bisects DCB
mABC  ________
mDCE  ________
mACD  ________
mDCB  ________
6.
mDBC  ________
mCBE  ________
BC bisects DBE
mDBE  ________
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7.
mABE  _________
mDBC  _________
8.
BD bisects ABC and
mABD  32
mDBC  _________
mABC  _________
9.
10.
Given l  p
m1  ______
m2  ______
m4  ______
m5  ______
m6  ______
m7  ______
AB  CD
HE bisects CHB
mBHG  32
m3  ______
m1  ________
m2  ________
m3  ________
m4  ________
m5  ________
m6  ________
m7  ________
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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 = ______
mABC  _______
mABC  _______
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Geometry G
Section 2.5. Worksheet 6
Name____________________________
5.
6.
Equation: _______________________
Equation: _______________________
x = ______
x = ______
mABC  _______
mABC  _______
Note: Picture is not drawn to scale.
7.
BD bisects ABC
Equation: _______________________
8.
Equation: _______________________
x = ______
x = ______
mABC  _______
mEBD  _______
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9.
Equation: _______________________
x = ________
mAFD  ________
mAFB  ________
mCFD  ________
10.
11.
Equation:_________________________
Equation:_________________________
x = ________
x = ________
mABC  ________
mPQR  ________
mABE  ________
mRQT  ________
33
Geometry G
Section 2.5. Worksheet 7
Name____________________________
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.
8x
(6x + 12)
(7x + 10)
Equation: _______________________
(16x + 4)
(18x + 4)
Equation: _______________________
x = ______
x = ______
3.
4.
(7x - 12)
(16x + 4)
18x
(5x + 18)
Equation: _______________________
Equation: _______________________
x = ______
x = ______
mABC  _______
mABC  _______
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