Download BLoCK 1 ~ LInes And AngLes

BLoCK 1 ~ LInes And AngLes
angLe pairs
L esson 1
L esson 2
MeasUring and naMing a ngLes -------------------------------------cLassiFYing a ngLes -------------------------------------------------Explore! Classify an Angle
coMPLeMenTarY and sUPPLeMenTarY a ngLes ------------------------Explore! Complementary vs Supplementary
verTicaL a ngLes ----------------------------------------------------Explore! The Vertical Angle Relationship
a LTernaTe exTerior and inTerior a ngLes ---------------------------Explore! Alternate Exterior and Alternate Interior
corresPonding and saMe-side inTerior a ngLes --------------------Explore! More Angle Pairs
BLock 1 ~ a ngLe Pairs -----------------------------------------------
L esson 3
L esson 4
L esson 5
L esson 6
r eview
oBtuse
A lt
word wAll
A ngle
Ate
ern
le s
r
e x te
A ng
ior
A lte
su PPl e m e n t
A ry
A cu t e A
r nA
te
tr A
l ine
d
PAir
r io
r
i n te r i
or
A ngl
m
A
e nt
r ight A ngle
18
23
29
35
ngle s
s
r Ay
es
VerticAl A n
gl e s
e nt
Vertex
str Aight
le s
A ng
ry
A ngle
13
ngle
e Gr e e
Pl e
ide
congru
sAl
com
Ar
i n te
A ngles
A ngle
er
nsV
sA me- s
A dJAcent A
3
8
Protr
A ngle
acto
r
cor
g
ndin
r e s Po
A ngle
s
Block 1 ~ Lines And Angles ~ Angle Pairs
1
BLoCK 1 ~ AngLe PAIrs
tic - tac - tOe
PuZZling A ngles
Bisecting A ngles
FliP BooK
Find angle measures in a
complex diagram.
Use two types of
constructions to
bisect angles.
Create a flip book
which describes
special angle pairs.
Special Angle
Pairs
2
See page  for details.
See page  for details.
See page  for details.
croSSword
ProtrActor guide
trAnsVersAl collAge
Make a crossword
using vocabulary
from this block.
Write a guide for
using a protractor.
Find or take pictures
of transversals and
display them.
See page  for details.
See page  for details.
See page  for details.
A ngle A rt
BooK oF Poetry
duPlicAting A ngles
Create an original piece
of artwork with
lines and angles.
Write poems about special
angle pairs. Make an
illustrated poetry booklet.
Use a compass and
straightedge to
duplicate angles.
See page  for details.
See page  for details.
See page  for details.
Block 1 ~ Angle Pairs ~ Tic - Tac - Toe
measuring and naming angLes
Lesson 1
A
ngles are used in construction, architecture, graphic design, aerospace, art, machining and manufacturing,
as well as many other fields. An angle is formed by two rays with a common endpoint. A ray has one endpoint
and extends forever in one direction.
___›
__
(ra NA _›
yN
A)
The vertex of an angle is the common point of both rays. N is the vertex of
this angle.
N
When three points are used to name an angle, the vertex is written in the middle
of the name. The vertex can be written as the name of an angle when it is the
vertex for only one angle.
__
(ra NG _›
yN
G)
___›
G
exampLe 1
___›
Ray NA is written NA. Ray NG is written NG. The first point in the name of a
ray is the endpoint.
A
___›
Three ways to name the angle formed by NA and NG are ∠ANG, ∠GNA and ∠N.
give 4 different names for the given angle.
P
1
A
L
solutions
exampLe 2
1. ∠PAL
2. ∠LAP
3. ∠A
Is ∠W another name for ∠nWe? explain.
E
N
solution
4. ∠1
S
W
No, it is not clear whether ∠W refers to ∠NWE, ∠SWE or ∠NWS.
___›
Adjacent angles are two angles that share a ray. In Example 2, ∠NWE and ∠SWE share WE.
This means ∠NWE and ∠SWE are adjacent angles.
Lesson 1 ~ Measuring And Naming Angles
3
Identify at least one additional name for each angle. Write using proper
angle notation.
T
a. ∠1
exampLe 3
I
b. ∠TGI
1
c. ∠RGE
solutions
E
2
3
G
R
a. ∠TGE or ∠EGT
b. ∠2 or ∠IGT
c. ∠EGR or ∠3
A protractor is a tool used to measure angles. Angles are measured in units called degrees.
The “m” in front of an angle measure is notation for the word “measure”.
The statement in Figure 1 below reads, “The measure of ∠ABC is equal to sixty degrees.”
Figure 1
Figure 2
Figure 3
Z
A
P
60°
B
C
m∠ABC = 60°
4
Lesson 1 ~ Measuring And Naming Angles
Q
142°
m∠PQR = 142°
R
X
74°
Y
m∠ZYX = 74°
use a protractor to measure each angle.
C
a.
exampLe 4
b.
5
B
A
solutions
C
a.
b.
B
A
5
m∠ABC = 53°
m∠5 = 132°
exercises
give two different names for each angle.
1.
2.
Q
3.
L
B
P
U
J
K
A
sketch a diagram to represent each angle.
4. ∠DOG
5. ∠CUB also called ∠4
6. ∠PAL also called ∠2
7. ∠1 and ∠2 are adjacent angles
8. ∠XYZ and ∠XYU are adjacent angles
9. ∠HOT is approximately 90°
T
use each protractor to determine the measure of the angle.
10.
11.
L
G
M
S
B
I
Lesson 1 ~ Measuring And Naming Angles
5
use a protractor to measure each angle to the nearest degree.
12.
13.
D
G
14.
M
R
E
A
F
15.
N
16.
3
P
17.
1
I
V
use a protractor to draw each angle. Label the angle(s).
18. m∠SAM = 34°
19. m∠YAK = 115°
20. m∠CAT = 167°
21. an 80° angle with PQ and PR
22. two 35° angles with the same vertex
23. two adjacent angles that are 50° and 100°
___›
___›
use the diagram below to name an angle with the specified measure.
24. 90°
25. 50°
26. 17°
27. 145°
28. 163°
29. 130°
D
E
C
M
B
6
Lesson 1 ~ Measuring And Naming Angles
A
G
U
t ic -t Ac -t oe ~ d u P l ic At i ng A ngl e s
Constructions are part of geometry. A geometric construction is
made by using a compass and straightedge. Follow the steps to
duplicate ∠ABC on a piece of notebook paper.
A
C
B
step 1: Trace ∠ABC on your paper.
step 2: Use a straightedge to draw a ray.
This will be one side of the duplicate angle.
step 3: Place the stylus or sharp point of a compass
on the vertex of the traced angle.
Draw an arc on the angle.
Place stylus here.
step 4: Without changing the setting on the compass,
place the stylus on the endpoint of the duplicate
ray and draw an arc.
step 5: Use the compass to measure the width of the arc
drawn on the original angle. Place the stylus on
the intersection of a side and the arc. Adjust the
compass so the pencil is touching the other
intersection point.
Place stylus here.
step 6: Without changing the setting on the compass,
place the stylus on the intersection of the arc
and duplicate ray. Make a small arc intersecting
the larger arc.
step 7: Use a straightedge to connect the endpoint
of the duplicate ray. This is the second ray
needed to complete the duplication of ∠ABC.
1. Use a protractor to draw a 60° angle.
2. Using only a compass and straightedge, duplicate the 60° angle.
3. Measure the duplicated angle with a protractor to check accuracy.
4. Repeat these steps on a 25°, 128° and a 160° angle.
5. Which step is the most difficult for you in this process? Why?
Lesson 1 ~ Measuring And Naming Angles
7
cLassiFying angLes
Lesson 2
In Lesson 1 you named and measured angles. Angles can be classified into groups by their degree measure.
expLOre!
cLassiFy an angLe
step 1: Use a protractor to measure the angles in each group. Record each measurement.
grouP A
grouP B
grouP C
step 2: Answer each question for each group.
a. How are the angles in each group alike?
b. What do you notice about the degree measures of the angles in each group?
step 3: Write at least two sentences describing each group of angles. Include information about their
degree measures.
step 4: Compare what you wrote in step 3 to the definitions given below. Identify which definition is
appropriate for each group.
Acute angle – an angle that measures more than 0° and less than 90°
Right angle – an angle that measures exactly 90°
Obtuse angle – an angle that measures more than 90° but less than 180°
8
Lesson 2 ~ Classifying Angles
There are four classifications of angles based on their degree measure. In the Explore!, you learned about
acute, obtuse and right angles. An angle that has a measure of 180° is called a straight angle.
right
Angle
r
st
g
Ai
ht
gl
n
A
e
Right angles are often identified by drawing a small square in the vertex of the angle. If a square is present in
the vertex of an angle, the angle measures 90⁰.
Classify each angle by measuring it with a protractor.
C
a.
b.
exampLe 1
J
A
I
T
c.
F
O
L
d.
D
O
G
X
solutions
a. m∠CAT = 64° so the angle is acute.
b. m∠JIL = 110° so the angle is obtuse.
c. The box drawn at the vertex shows that m∠FOX = 90°.
d. m∠DOG = 180° so it is a straight angle.
Angles with equal measures are congruent. The symbol for congruent is ≅. Congruent angles are identified
in diagrams with congruence marks. The two marks on the arc inside each angle are congruence marks.
They show ∠RED ≅ ∠BLU. This is read, “Angle RED is congruent to angle BLU.”
D
∠X is congruent to ∠Y because each measures 90°. Each angle is
a right angle as indicated by the square in each vertex.
E
X
R
B
L
U
Y
Lesson 2 ~ Classifying Angles
9
exampLe 2
sketch a diagram of congruent and adjacent angles.
solution
The angles share OC which makes them adjacent.
The congruence marks indicate the angles are
congruent.
___›
C
D
B
O
exampLe 3
use the information in the diagram to write an equation. solve for x.
P
S
I
solution
(5x −
11)°
E
39°
L
M
Congruence marks show the angles are congruent.
Write the equation.
Add 11 to both sides of the equation.
Divide both sides of the equation by 5.
∠SIM ≅ ∠PLE
5x − 11 = 39
+11 +11
5x = __
50
__
5
5
x = 10
☑ Check the solution by substituting 10 for x.
5x − 11 = 39
5(10) − 11 =? 39
50 − 11 =? 39
39 = 39
exampLe 4
∠JAK is congruent to ∠hIL. The measure of ∠JAK = (12 − 3x)° and the measure
of ∠hIL = (44 − x)°. solve for x. Then find the degree measure of each angle.
solution
Write an equation showing the angles
have equal measures.
Add x to each side of the equation.
Subtract 12 from each side of the equation.
Divide each side of the equation by −2.
Write the given expression
for each angle.
Substitute −16 for x.
Simplify.
Add.
∠JAK ≅ ∠HIL
12 − 3x = 44 − x
+x
+x
12 − 2x = 44
−12
−12
−2x
32
___ = __
−2 −2
x = −16
m∠JAK = 12 − 3x
= 12 − 3(−16)
= 12 + 48
= 60
m∠HIL = 44 − x
= 44 − (−16)
= 44 + 16
= 60
m∠JAK = 60° and m∠HIL = 60°. Both angles are equal which verifies that they
are congruent.
10
Lesson 2 ~ Classifying Angles
exercises
estimate the degree measure for each angle. Classify each angle as acute, obtuse, right or straight.
1.
2.
3.
4.
5.
6.
sketch a diagram for each description. Label each angle.
7. two congruent angles
8. ∠GUY is obtuse
9. ∠GAL is acute
10. ∠JKL ≅ ∠POM
11. a right angle that can be identified using 3 names
12. two adjacent, right angles
13. two congruent angles that share a vertex
14. ∠PQR is straight
use each diagram to solve for x.
15.
16. m∠PQR = 124°
||
||
60°
R
(3x − 2)°
(x + 12)°
R
||
30°
O
(7x +
Y
3)°
M
x+
(4x + 6)°
20. m∠XYZ = m∠ABC
13
)°
19. ∠ROY ≅ ∠MAN
||
(2x − 10)°
Q
P
(5
18.
17.
N
m∠XYZ = (5 + 2x)°
m∠ABC = (3x + 1)°
A
21. ∠JAM and ∠GEM are congruent angles. The measure of ∠JAM is (8x + 5)° and m∠GEM = (x + 75)°.
a. Solve for x.
b. Find the measure of each angle.
22. ∠SML is an acute angle. The measure of ∠SML = (x − 7)°. What must x be less than?
Lesson 2 ~ Classifying Angles
11
23. m∠LRG = (4x + 22)°
a. If ∠LRG is a right angle, what must x equal?
b. If ∠LRG is an acute angle, what must x be less than?
c. If ∠LRG is an obtuse angle, what must x be greater than?
review
sketch a diagram to represent each figure.
24. ∠RMP
25. ∠PIE also called ∠3
26. ∠5 and ∠6 which are
adjacent
use a protractor to measure each angle.
27.
28.
29.
use a protractor to draw an angle with the given measure.
30. 67°
31. 135°
32. 180°
33. 11°
t ic -t Ac -t oe ~ P ro t r Ac t or g u i de
step 1: Create a user’s guide describing how to measure and draw
angles with a protractor. Your teacher may use the guide
to refresh a substitute teacher or for students who are absent
the day of the lesson.
step 2: Create a worksheet that can be completed using the guide.
step 3: Make an answer key for the worksheet.
12
Lesson 2 ~ Classifying Angles
cOmpLementary and suppLementary
angLes
Lesson 3
I
ndividual angles are classified as acute, right, obtuse or straight. Special pairs of angles can also be classified.
In Lessons 3-6, special pairs of angles and their relationships will be examined.
expLOre!
cOmpLementary vs suppLementary
step 1: Look at the angles in the chart.
a. What similarities do you notice about the pairs of angles called supplementary angles?
b. What similarities do you notice about the pairs of angles called complementary angles?
suPPLeMentArY AngLes
2
∠1 and ∠2 are
supplementary.
1
CoMPLeMentArY AngLes
R
M
E
O
C 40° T
A
D
140°
O
G
L
E
R
F
T
G
m∠6 = 80° and m∠7 = 100°
∠CAT and ∠DOG
are supplementary.
R
C
70°
A
∠LEF and ∠RGT are
supplementary.
∠6 and ∠7 are
supplementary.
∠ROM and ∠MOE
are complementary.
W
20°
K
L
∠CAR and ∠WLK
are complementary.
∠1 and ∠2 are
complementary.
1
2
m∠8 = 45° and m∠9 = 45°
∠8 and ∠9 are
complementary.
step 2: Write a definition for supplementary angles.
step 3: Write a definition for complementary angles.
step 4: Give at least two examples of angle measures for
each type of angle pair listed below.
a. Supplementary angles
b. Complementary angles
Lesson 3 ~ Complementary And Supplementary Angles
13
Complementary and supplementary angles are special pairs of angles. Complementary angles are two angles
with a sum of 90°. Two angles with a sum of 180° are called supplementary angles. These special pairs of
angles may or may not be adjacent.
exampLe 1
use the diagram to find m∠PAr.
r
45°
solution
P
m∠PAR and m∠TAR are supplementary.
t
m∠PAR + 45° = 180°
−45° −45°
m∠PAR = 135°
Supplementary angles have a sum of 180°.
Subtract 45 from both sides of the equation.
m∠PAR is 135°
☑ Check the solution.
exampLe 2
A
135° + 45° =? 180°
180° = 180°
∠grA and ∠Ins are supplementary.
a. Write an equation to solve for x.
b. determine the measure of each angle.
G
x+
(2
°
4)
A
I
R
(3x + 1)°
N
solutions
a. Supplementary angles have a sum of 180°.
Write an equation.
m∠GRA + m∠INS = 180°
(2x + 4) + (3x + 1) = 180
Combine like terms.
Subtract 5 from each side.
Divide each side by 5.
b. Write the given
expression for each angle.
Substitute 35 for x.
Multiply.
Add.
5x + 5 = 180
−5 −5
5x = ___
175
__
5
5
x = 35
m∠GRA = (2x + 4)°
= (2(35) + 4)°
= (70 + 4)°
= 74°
☑ m∠GRA + m∠INS = 180°
?
74° + 106° = 180°
180° = 180°
The measure of ∠GRA is 74°.
The measure of ∠INS is 106°.
14
Lesson 3 ~ Complementary And Supplementary Angles
S
m∠INS = (3x + 1)°
= (3(35) + 1)°
= (105 + 1)°
= 106°
exampLe 3
use the diagram to write an equation. solve for x.
e
o
62°
(x + 5)°
M
h
solution
Complementary angles have a sum of 90°.
Substitute the degreee measures.
Combine the like terms.
Subtract 67 from each side of the equation.
m∠HOM + m∠MOE = 90°
62 + (x + 5) = 90
x + 67 = 90
−67 −67
x = 23°
☑ Check the solution.
62 + (23 + 5) =? 90
62 + 28 =? 90
90 = 90
The value of x is 23.
exampLe 4
∠1 and ∠2 are complementary angles. The measure of ∠1 = (3x + 4)°
and m∠2 = (x + 6)°.
a. draw a diagram.
b. Write an equation and solve for x.
c. Find ∠1 and ∠2.
solutions
a.
(3x + 4)°
(x + 6)°
or
b. Complementary angles have a sum of 90°.
Substitute the degree measures.
Combine like terms on the same side
of the equation.
Subtract 10 from each side of the equation.
Divide by 4 on each side of the equation.
c. Write the given expression
for each angle.
Substitute 20 for x.
Multiply.
Add.
☑ Check the solution.
m∠1 = 64° and m∠2 = 26°.
(3
x+
°
4)
(x + 6)°
m∠1 + m∠2 = 90°
(3x + 4) + (x + 6) = 90
4x + 10 = 90
−10 −10
4x = __
80
__
4
4
x = 20
m∠1 = (3x + 4)°
= (3(20) + 4)
= (60 + 4)
= 64°
m∠2 = (x + 6)°
= (20 + 6)
= 26°
64° + 26° =? 90°
90 = 90
Lesson 3 ~ Complementary And Supplementary Angles
15
exercises
Identify each pair of angles as complementary, supplementary or neither.
1.
56°
2.
3.
110°
70°
34°
4.
5.
52°
38°
6.
122°
58°
122°
20°
7. m∠1 and m∠2 sum to 181°.
8. ∠A and ∠M have a sum of 90°.
Write an equation for each description. solve for x. Check your solution.
9. ∠A and ∠B are complementary.
42°
B
A
10.
41°
2x°
x°
x°
11.
12.
x°
70° (4 + 2x)°
13.
x°
)°
5
x−
7
(
(x + 3)°
x°
14.
15. ∠MAN and ∠MAP are supplementary. The measure of ∠MAN is 57°.
What is the measure of ∠MAP?
16. ∠5 and ∠7 are complementary angles. Find the measure of ∠7 if m∠5 = 47°.
17. The complement of ∠Q is 31°. Find m∠Q.
18. The supplement of ∠U is 62°. What is m∠U?
16
Lesson 3 ~ Complementary And Supplementary Angles
Find the measure of each angle in exercises 19-23. Check your solution.
19. ∠1 and ∠2 are supplementary; m∠1 = 3x° and m∠2 = 3x°
20. ∠W And ∠C are complementary; m∠W = (47 + 3x)° and m∠C = (10 + 8x)°
21. ∠G and ∠H are supplementary; m∠G = (x + 4)° and m∠H = (4x + 11)°
22. ∠V and ∠W are supplementary; ∠V is (12 + 3x)° and m∠W = (33 + 2x)°
23. ∠1 and ∠2 are complementary; m∠1 = 4x° and m∠2 = (x + 8)°
3 5
24. Use the figure on the right.
a. Name an obtuse angle.
b. Name three pairs of supplementary angles.
c. Name a pair of complementary angles.
d. Give possible measures for ∠4 and ∠1.
e. Determine m∠11.
10
4
1
2
11
7
6
8
9
review
draw and label a diagram to represent each statement.
25. ∠PAM is acute
26. ∠BIG is obtuse
27. ∠RHT is right
use a protractor to measure each angle to the nearest degree.
28.
29.
30.
t ic -t Ac -t oe ~ A ngl e A rt
step 1: Research how angles and lines are used in art. Write at
least a one page summary of your findings.
step 2: Create a work of art using the different types of angles in Block 1.
Use an 11 inch by 18 inch piece of paper for your work of art.
Give your creation a title and sign it.
Lesson 3 ~ Complementary And Supplementary Angles
17
verticaL angLes
Lesson 4
C
omplementary and supplementary angles are types of special angles. Another pair of special angles
are vertical angles. Vertical angles are formed by two intersecting lines. They have a common vertex but
are not adjacent.
1
3
4
2
Two adjacent angles whose non-common sides are opposite rays are a linear pair. In the diagram above,
there are four sets of linear pairs. For example, ∠1 and ∠3 form a linear pair. If two angles form a linear pair,
they are supplementary.
expLOre!
the verticaL angLe reLatiOnship
step 1: Trace ∠3 above onto a sheet of paper.
step 2: Place the traced ∠3 on top of ∠4. What do you notice?
step 3: Repeat steps 1 and 2 with ∠1 and ∠2. What do you find?
step 4: Draw two intersecting lines on a piece of paper.
step 5: Label the angles that are formed with the numbers 5, 6, 7 and 8. Identify the vertical angles
in your drawing.
step 6: Measure each angle in your drawing with a protractor.
step 7: What can you conclude about the measure of any pair of vertical angles?
22°
||
Lesson 4 ~ Vertical Angles
22°
|
140° 40°
40° 140°
|
|
18
158°
||
||
|
||
158°
Sierra and King used different methods to find the solution to the question below. Look at Sierra’s and
King’s work.
Question
If m∠2 = 36° and m∠3 = 144°,
what is the measure of ∠4?
4
3
sierra’s Work
King’s Work
Sierra knows that ∠4 and ∠3 are
supplementary because they are a linear
pair. She subtracted 144 from 180.
King knows that ∠2 and ∠4 are vertical
angles. They have the same degree
measure.
2
m∠4 = 36°
180° − 144° = 36°
m∠4 = 36°
There is often more than one way to arrive at a correct answer. Both Sierra and King answered the question
correctly but used different methods.
exampLe 1
solutions
Find the measure of each missing angle.
a. m∠3
b. m∠1
54°
c. m∠4
a. Vertical angles are congruent.
2
1
3
4
m∠2 = m∠3
54° = m∠3
b. ∠1 and ∠2 are a linear pair.
Substitute 54° for m∠2.
Subtract 54° from each side of the equation.
m∠1 + m∠2 = 180°
m∠1 + 54° = 180°
m∠1= 126°
c. Vertical angles are congruent.
m∠1 = m∠4
126° = m∠4
In geometry, sketches are used as a visual representation for the
information given. Sketches are not always accurate in terms of
actual length or degree measure. Information given in a diagram
should be used to solve a problem rather than measuring the
actual lengths and degree measures with a ruler and protractor.
For example, the sketch below shows that the angle is 45° and
the length of the segment is 5 feet. The angle in the sketch may
not measure exactly 45° and the line does not measure 5 feet.
You should still use this information to solve any problem asked
about the diagram.
45°
5 feet
Lesson 4 ~ Vertical Angles
19
exampLe 2
use the diagram at right.
a. solve for x.
b. Find the measure of each angle.
(3x + 7)°
(x + 30)°
solutions
a. Vertical angles have equal measures.
Subtract x from each side of the equation.
Subtract 7 from each side of the equation.
Divide each side of the equation by 2.
b. Write the given expression
for each angle.
Substitute the solution for x.
Multiply.
Add.
3x + 7 = x + 30
2x + 7 = 30
−7 −7
2x = 23
x = 11.5
(3x + 7)°
(x + 30)°
(3(11.5) + 7)°
34.5 + 7
41.5°
(11.5 + 30)°
41.5°
The measure of each angle is 41.5°.
Each special angle pair has properties that are important to remember. The sum of complementary angles
is 90°. The sum of supplementary angles or linear pairs is 180°. Vertical angles are congruent.
exercises
use the diagram at right. determine the measure of each unknown angle.
1. If m∠1 = 50°, find the following:
a. m∠2 = ?
b. m∠3 = ?
c. m∠4 = ?
3. If m∠3 = 126°, find the following:
a. m∠1 = ?
b. m∠2 = ?
c. m∠4 = ?
2. If m∠4 = 153°, find the following:
a. m∠1 = ?
b. m∠2 = ?
c. m∠3 = ?
4. If m∠2 = 16°, find the following:
a. m∠1 = ?
b. m∠3 = ?
c. m∠4 = ?
sketch a diagram to represent each situation. Label each diagram.
5. ∠7 and ∠8 are vertical angles.
6. ∠9 and ∠10 are obtuse vertical angles.
7. ∠1 and ∠2 are a linear pair.
8. ∠ABC and ∠DBE are vertical
∠1 is an acute angle.
20
Lesson 4 ~ Vertical Angles
and complementary angles.
1
4 3
2
Identify each special angle pair as vertical angles or a linear pair. solve for x. Check your solution.
9.
10.
114°
(5x + 4)°
11.
86°
)°
+2
(8x
42°
(4x + 30)°
13.
12.
14.
(x + 11)°
(6x + 4)° (4x − 14)°
15.
(3x + 7)°
16.
(5x + 9)°
(3x + 10)°
62°
89°
17.
(8x + 3)°
(7x + 12)°
C
18. Use the diagram at the right.
a. Solve for x.
b. Find m∠ABC.
c. Find m∠CBD.
(4x + 7)°
(2x − 1)°
B
A
19. Use the diagram at the right.
a. Solve for x.
b. Find m∠WX Y.
c. Find m∠TXY.
6x°
D
T
(6x + 13)°
U
Y
X
(10x − 7)°
W
Lesson 4 ~ Vertical Angles
21
review
Match each diagram to a description from the word bank. some diagrams may match more than one
description.
Acute angle
Obtuse angle
Right angle
Congruent angles
Straight angle
Vertical angles
Supplementary angles
Linear pair
Complementary angles
K
20.
50°
21.
22.
I
S
40°
26.
27.
m∠ABC = 70°
142°
25.
||
R
28.
50°
130°
t ic -t Ac -t oe ~ B ooK
||
|
24.
|
23. m∠PQR = 110°
oF
Q
P
P oe t r y
step 1: Write a separate poem about each special angle pair:
linear pair, supplementary, complementary, vertical,
alternate interior, alternate exterior, corresponding and
same-side interior angles. Title each poem with the name
of the special angle pair.
step 2: Put the eight poems in a booklet. Include diagrams and illustrations.
22
Lesson 4 ~ Vertical Angles
aLternate exteriOr and interiOr angLes
Lesson 5
The words “interior” and “exterior” are used in many non-math situations. The word “interior” is used to
describe things that are inside. Things that are exterior are on the outside. The two special angle pairs in this
lesson are called alternate interior angles and alternate exterior angles. Look at the diagrams in the table below.
What does it mean for a pair of angles to be alternate exterior angles? What about alternate interior angles?
Alternate exterior Angles
5
6
8
7
18
19
16
>>
>>
17
Alternate Interior Angles
∠6 and ∠8 are alternate
exterior angles.
∠1 and ∠2 are alternate
interior angles.
>>
∠5 and ∠7 are alternate
exterior angles.
∠3 and ∠4 are alternate
interior angles.
>>
∠17 and ∠18 are alternate
exterior angles.
∠10 and ∠13 are alternate
interior angles.
∠16 and ∠19 are alternate
exterior angles.
∠12 and ∠14 are alternate
interior angles.
3 1
2
10
14
4
12
13
A transversal is a line that intersects two or more lines.
Alternate exterior angles are two angles on the outside of two lines and on opposite sides of a transversal.
Alternate interior angles are two angles on the inside of two lines and on opposite sides of a transversal.
X
>>
B
4
1
2
3
A
‹___›
In this diagram, XY is the transversal.
‹___›
‹___›
AB || PQ reads, “Line AB is parallel to
line PQ.”
>>
P
8
5
6
7
Q
Y
Lesson 5 ~ Alternate Exterior And Interior Angles
23
expLOre!
aLternate exteriOr and aLternate interiOr
step 1: Draw two parallel lines intersected by a transversal. Make the diagram large enough to easily
measure the angles with a protractor. Label your angles 1 through 8.
step 2: Use a protractor to measure the eight angles on the drawing from step 1. Record their measures.
step 3: Which pairs of angles in your diagram are alternate exterior angles? What do you notice about the
degree measure of each pair of alternate exterior angles?
step 4: Which pairs of angles are alternate interior angles? What do you notice about the degree measure
of each pair of alternate interior angles?
step 5: Trace the figure from the bottom of page 23 onto a piece of paper.
step 6: Slide your tracing until angles 1, 2, 3 and 4 are on top of angles 5, 6, 7 and 8.
a. What do you observe? Explain.
b. Does this confirm your conclusion in steps 3 and 4?
step 7: Draw two lines that are not parallel. Draw a transversal that intersects both lines.
step 8: Use a protractor to measure the eight angles created in step 7.
step 9: Write a conclusion about the alternate interior angles and alternate exterior angles formed by
parallel lines compared to those formed by non-parallel lines.
name the angle relationship between ∠1 and ∠2. determine whether ∠1 and ∠2
are congruent.
b.
c.
a.
1
1
< 1
solutions
24
2
2
<<
<
<<
exampLe 1
2
a. Alternate interior. The angles are congruent because the lines are parallel.
b. Alternate exterior. The angles are congruent because the lines are parallel.
c. Alternate exterior. The lines are not parallel, so the angles are not congruent.
Lesson 5 ~ Alternate Exterior And Interior Angles
exampLe 2
Identify the special angle pair relationship. solve for x.
<<
<<
86°
(8x + 4)°
l
solution
m
The angles are alternate exterior angles.
Lines l and m are parallel so the alternate
exterior angles are congruent.
Subtract 4 from each side of the equation.
Divide each side of the equation by 8.
8x + 4 = 86
−4 −4
8x
82
__ = __
8
8
x = 10.25
☑Check the solution.
Substitute 10.25 for x.
Add.
exampLe 3
solutions
use the figure at right.
a. solve for x.
b. Find the measure of the angles.
a. The lines are parallel so alternate
interior angles are congruent.
Subtract 2x from each side of the equation.
Add 58 to each side of the equation.
Divide each side of the equation by 7.
(8x + 4)° = 86°
(8(10.25) + 4)° =? 86°
82 + 4° =? 86°
86° = 86°
(9x − 58)°
>>
(2x + 5)°
>>
9x − 58 = 2x + 5
−2x
−2x
7x − 58 = 5
+58 +58
63
7x = __
__
7
7
x=9
b. Write the given expression
for each angle.
Substitute 9 for x found in part a.
Multiply.
Simplify.
(9x − 58)°
(9(9) − 58)°
(81 − 58)°
23°
(2x + 5)°
(2(9) + 5)°
(18 + 5)°
23°
Each angle measures 23°. Alternate interior angles are congruent so part b verifies
the solution for x.
Lesson 5 ~ Alternate Exterior And Interior Angles
25
exercises
use one of the following special angle pairs to identify the relationship of the angles shown.
Alternate exterior
1.
Alternate Interior
Vertical
2.
Linear Pair
3.
57°
65°
65°
130° 50°
57°
4.
5.
6.
1
4
140°
40°
2
5
8.
>>
3
9.
90°
4
y°
>>
y°
>>
7.
>>
90°
name the special angle pair relationship between ∠1 and ∠2. explain whether ∠1 ≅ ∠2.
10.
1
11.
>>
12.
1
2
>>
2
1
2
(3x + 10)°
26
14.
>
Lesson 5 ~ Alternate Exterior And Interior Angles
(4x + 3)°
103°
>
>
15.
>>
70°
>
>>
name the special angle pair relationship. solve for x.
13.
>>
120°
8x°
>>
>>
(6x
)°
17.
+ 20
18.
>>
2)°
(2x + 1
>> (x
°
0)
+ 10
>>
19.
>>
16.
(6x
)°
(3x − 22
20. ∠1 and ∠2 are vertical angles
m∠1 = (5x + 7)°
m∠2 = (3x + 15)°
− 7)
>>
solve for x. Then find the measure of each identified angle. Check your solution.
°
(5x
+ 10
)°
21. ∠5 and ∠8 are
supplementary
m∠5 = (3x − 40)°
m∠8 = (7x − 120)°
2x°
56°
22. Explain how to distinguish between alternate exterior angles and alternate interior angles.
review
Write two possible names for each angle.
23.
24.
P
25.
A
W
B
5
M
3
K
o
sketch and label a diagram for each description.
26. an acute angle
27. vertical angles with each angle measuring 40°
28. adjacent supplementary angles
29. complementary angles that are not adjacent
t ic -t Ac -t oe ~ t r A n s V e r s A l c ol l Age
step 1: Find at least 10 photographs or pictures of transversals. You can
take photographs, locate and print pictures from the internet or
cut pictures from magazines or newspapers.
step 2: Identify the transversal and special angle pairs in each picture.
step 3: Make a collage to display the pictures.
Lesson 5 ~ Alternate Exterior And Interior Angles
27
t ic -t Ac -t oe ~ B i s e c t i ng A ngl e s
To bisect an angle means to cut it in half. You can bisect an
angle using two different methods. One construction method
to bisect an angle uses patty paper. Another method uses a
compass and straightedge.
Bisecting an Angle using Patty Paper or tracing Paper
step 1: Draw or trace an angle onto a piece of patty paper
or tracing paper. Record the measure of the angle.
step 2: Fold one ray of the angle onto the other
ray of the angle.
step 3: Trace the crease and measure each angle.
Bisecting an Angle using A Compass and straightedge
step 1: Draw or trace an angle. Record the measure of
the angle.
step 2: Place the stylus or sharp point of a compass
on the vertex. Use the compass to draw an arc
through the angle.
step 3: Place the stylus on one of the points of intersection
between a ray of the angle and the arc from step 2.
Draw an arc as shown.
Place stylus here.
step 4: Repeat step 3 at the other point of intersection.
step 5: Use a straightedge to draw a ray from the
vertex to the intersection of the two arcs drawn
in steps 3 and 4.
step 6: Measure each angle.
1. Use a protractor and patty/tracing paper to draw 90°, 24°, 115° and 160° angles.
Each angle should be on a separate sheet of patty/tracing paper.
2. Bisect each angle by folding.
3. Use a protractor to draw another set of angles with the measures listed in # 1 on regular paper.
4. Bisect each angle using a compass and straightedge.
5. Lay your matching patty/tracing paper constructions on top of each compass construction.
6. Summarize each method of construction. Include which method you prefer and explain why.
Discuss the pros and cons of each method.
28
Lesson 5 ~ Alternate Exterior And Interior Angles
cOrrespOnding and same - side interiOr
angLes
Lesson 6
T
wo additional special pairs of angles formed by two lines and a transversal are corresponding angles and
same-side interior angles.
Corresponding angles are two angles on the same side of a transversal. One angle is an exterior angle and the
other angle an interior angle. They are not adjacent angles.
Same-side interior angles are between the two lines on the same side of a transversal.
1 2
3 4
6
5
8
7
expLOre!
mOre angLe pairs
step 1: If two parallel lines are cut by a transversal, predict:
a. What is the relationship between a pair of same-side interior angles?
b. What is the relationship between a pair of corresponding angles?
step 2: Use a straight edge to draw a pair of parallel lines with a transversal. Make the figure large enough
that you can measure the angles with a protractor.
step 3: Use a protractor to measure each of the eight angles. Record each measure.
step 4: What do you observe about the corresponding angle measures?
step 5: Look at the measures of the same-side interior angles. What do you observe about their measures?
step 6: Must the lines be parallel for these relationships to occur? Draw a figure without parallel lines to
test your answer.
Lesson 6 ~ Corresponding And Same - Side Interior Angles
29
name the special angle pair relationship between ∠1 and ∠2.
a.
b.
c.
exampLe 1
1
2
2
1
1
2
a. corresponding angles
b. same-side interior angles
c. corresponding angles
solutions
1 2
3 4
>>
>>
6
5
8
7
Claudia and Ping come to different conclusions when finding the missing angles in the diagram below.
Determine who is correct and why.
Claudia’s Work
1 2
108° 7
4
3
6 5
m∠1 = 72°
m∠2 = 108°
m∠3 = 108°
m∠4 = 72°
m∠5 = 72°
m∠6 = 108°
m∠7 = 72°
Ping’s Work
m∠1 = 72°
m∠2 = 108°
m∠3 = cannot determine
m∠4 = cannot determine
m∠5 = cannot determine
m∠6 = cannot determine
m∠7 = 72°
Two lines in the diagram are intersected by a transversal. It is not known
whether the lines are parallel. Unless lines are marked parallel it cannot be
assumed they are parallel. Ping is correct. The measures of ∠3, ∠4, ∠5
and ∠6 cannot be determined.
30
Lesson 6 ~ Corresponding And Same - Side Interior Angles
exampLe 2
Write an equation and solve for x.
>>
>>
solution
(4x + 9)°
117°
Corresponding angles have equal measures.
Subtract 9 from each side of the equation.
Divide each side of the equation by 4.
4x + 9 = 117
−9 −9
108
4x = ___
__
4
4
x = 27
☑ Check the solution.
4(27) + 9 =? 117
108 + 9 =? 117
117 = 117
Substitute 27 for x in the equation.
exampLe 3
Write an equation and solve for x. Then find the measure of each identified angle.
<<
+
(3x
+
(3x
1)°
)°
44
<<
solution
The lines are parallel so the same side
interior angles sum to 180°.
Combine like terms.
Subtract 45 from each side of the equation.
Divide each side of the equation by 6.
(3x + 1) + (3x + 44) = 180
6x + 45 = 180
−45 −45
6x = ___
135
__
6
6
x = 22.5
Find the measure of each angle.
Substitute 22.5 for x.
☑ 68.5° + 111.5° =? 180°
(3x + 1)°
(3(22.5) + 1)°
(67.5 + 1)°
68.5°
(3x + 44)°
(3(22.5) + 44)°
(67.5 + 44)°
111.5°
180° = 180°
Lesson 6 ~ Corresponding And Same - Side Interior Angles
31
exercises
name the special angle pair relationship between ∠1 and ∠2 in each diagram.
2
<<
2
4.
5.
1
1
7.
1
2
8.
1
<<
<<
1
2
6.
2
<<
<<
2
3.
1
<<
>
2.
1
>
1.
2
9.
<<
<<
1
1
2
2
10. Which diagrams in exercises 1-9 have parallel lines?
44°
>>
12.
>>
(47 + 2x)°
154°
129°
>>
4x°
>>
13.
>
11.
>
solve for x.
(5x + 34)°
solve for x. Then find the measure of each identified angle. Check your solution.
15.
°
16.
>>
>>
32
Lesson 6 ~ Corresponding And Same - Side Interior Angles
(90 − 5x)°
46°
(62 − 2x)°
>>
110
(4x
°
)
+ 40
>>
6)°
>>
(x +
>>
14.
Find the measure of each numbered angle. Check your solution.
>>
>>
>>
7
5
1 (10x + 5)°
2
3
4
19.
(8x + 25)°

6
4
18.
1 81°
2 3

17.
2
1
4
5
3
6
)°
(4x + 2
)°
(6x − 7
>>
sketch a diagram to represent each special pair of angles. Label the angles in each pair as ∠1 and ∠2.
20. congruent same-side interior angles
21. alternate interior angles that are not congruent
22. acute corresponding angles
23. right alternate exterior angles
24. obtuse same-side interior angles
25. acute vertical angles
review
use a protractor and/or straightedge to draw each diagram. Label each diagram.
26. m∠GUM = 162°
27. m∠RPQ = 20°
‹___›
28. 3 parallel lines with transversal AB
29. right angle named ∠2
name each special angle pair. solve for x.
30.
31.
x°
119°
32.
3x°
2x°
(5x + 1)°
(3x + 44)°
t ic -t Ac -t oe ~ F l i P B ooK
Special Angle
Pairs
step 1: Create a flip book showing all of the special angle pairs
in Block 1.
◆ Supplementary angles
◆ Vertical angles
◆ Corresponding angles
◆ Alternate interior angles
◆ Same-side interior angles
◆ Alternate exterior angles
◆ Complementary angles
◆ Linear pairs
step 2: Include a definition, a diagram and an example for each special angle
pair.
Lesson 6 ~ Corresponding And Same - Side Interior Angles
33
t ic -t Ac -t oe ~ P u Z Z l i ng A ngl e s
Find the measure of the numbered angles in each diagram.You will need to
research about the sum of the angles in a triangle before completing the puzzle.
>>
>>
2
3
1
5 6
68° 7
4
14 120°
13 12
9 8
11 10
38
41 40
39
15
18 16
17
19
20
>>
37
35
36
21
22
28 27 26 23
30 29 25 24
34 33
31 32
t ic -t Ac -t oe ~ c ro s s wor d
Create a crossword puzzle using all of the vocabulary from Block 1.
Make a blank master copy and an answer key.
34
Lesson 6 ~ Corresponding And Same - Side Interior Angles
review
BLoCK 1
vocabulary
acute angle
adjacent angles
alternate exterior angles
alternate interior angles
angle
complementary angles
congruent
corresponding angles
degree
linear pair
obtuse angle
protractor
ray
right angle
same-side interior angles
straight angle
supplementary angles
transversal
vertex
vertical angles
Lesson 1 ~ Measuring and Naming Angles
use a protractor to measure each angle to the nearest degree.
1.
2.
3.
List the four names for each angle.
4.
5.
O
X
5
M
5
P
Y
Z
sketch and label a diagram to represent each angle. use a protractor, as needed.
6. m∠YOU = 125°
7. m∠BAT = 40°
8. ∠HAM and ∠HAP
are adjacent
Block 1 ~ Review
35
Lesson 2 ~ Classifying Angles
Classify each angle as acute, right, obtuse or straight.
9.
10.
11.
134°
51°
sketch and label a diagram.
12. ∠CAT ≅ ∠DOG
13. a right angle that can be named three different ways
set up an equation and solve for x. Check your solution.
14.
15.
(35 + 2x )°
31°
(2x + 16 )°
16.
(3x – 12)°
17. m∠BIG = (x + 14)°
a. What must x be equal to if ∠BIG is a right angle?
b. What must x be less than if ∠BIG is acute?
c. What must x be greater than if ∠BIG is obtuse?
Lesson 3 ~ Complementary and Supplementary Angles
Identify each pair of angles as complementary, supplementary or neither.
18.
19.
60°
120°
20.
40°
51°
39°
135°
solve for x. Check your solution.
21.
22. m∠A = (25 + 2x)°
42°
(x + 7)°
36
Block 1 ~ Review
m∠P = (10 + 3x)°
∠A and ∠P are
complementary.
23.
142°
x°
sketch a diagram for each situation. Label it and solve for x.
24. ∠PAR and ∠TYE are supplementary; m∠PAR is 83° and m∠TYE is (x + 5)°
25. ∠F and ∠G are supplementary; m∠F is 46° and m∠G is (3x – 25)°
26. ∠1 and ∠2 are complementary; m∠1 is 2x° and m∠2 is 3x°
Lesson 4 ~ Vertical Angles
determine the measure of the angles labeled a, b and c.
27.
28.
b
a
c
45°
a
b
123°
c
name the special angle pair. solve for x. Check your solution.
29.
30.
(x + 8)°
31.
(2x + 12)°
141°
(5x + 27)°
33.
32.
(4x + 10)°
(6x + 7)°
34.
(5x)°
(5 + 3x)°
(10 + 7x)°
(x + 76)°
(5x)°
Lesson 5 ~ Alternate Exterior and Interior Angles
name the special angle pair relationship between ∠1 and ∠2. explain whether or not the angles
are congruent.
>>
>>
2
1
2
37.
1
2
>>
36.
1
>>
35.
Block 1 ~ Review
37
solve for x. Then find the measure of each angle. Check your solution.
(x + 11)°
>>
40.
>>
(60 −
2x)°
>>
)°
x
(81 +
>>
39.
>>
38.
>>
(5x +
(3x − 35)°
(3x +
2)°
40)°
Lesson 6 ~ Corresponding and Same-Side Interior Angles
name the special angle pair relationship between ∠1 and ∠2.
41.
42.
43.
1
1
>>
2
2
>>
>>
1
2
>>
44. Copy the table below. List each special angle pair from the blue box in one category. Assume lines are
parallel for *angle pairs.
Linear Pairs
*Alternate exterior Angles
*Alternate Interior Angles
*Corresponding Angles
have a sum of 90°
Vertical Angles
Complementary Angles
*same-side Interior Angles
supplementary Angles
have a sum of 180°
Are equal in measure
Write an equation to solve for x. Then find the measure of each angle. Check your solution.
>>
>>
38
(2x + 10)°
(x + 5)°
Block 1 ~ Review
>>
46.
>>
45.
(3x + 70)°
(7x + 30)°
>>
47.
(8x
)°
− 12
(48
)°
− 2x
>>
Jerry
nursery mAnAger
turner, oregon
CAreer
FoCus
I am a nursery manager. I work with a staff planning and planting
25 different tree species. Some of the seedlings I plant are very
recognizable. I plant Douglas Fir, Noble Fir, Ponderosa Pine and Giant
Sequoia. Some of the trees I grow are for planting around homes. Others
go into the woods for reforestation. When loggers cut down trees, the trees I grow
are planted in their place. This helps make sure that there will be trees to harvest in the
future. I help grow 6.5 to 7.5 million trees for about 150 different customers. I work at a
small nursery, but some nurseries in the Northwest grow over 25 million trees a year.
I use math every day in my job. When preparing to plant, I have to know how many trees will fit on
each acre. Some seedlings are planted at 75 per foot and some are planted at 18 per foot. Math helps me
to determine how many seeds I will need to make sure I will have enough trees to fill a customer’s order.
Fertilizers are another area where I use math. I take soil samples and calculate at what rate I should apply
fertilizer to make sure that the seedlings grow well. When I prepare to ship trees to a customer, I also
have to use math to make sure that they are counted correctly and packaged right for each customer.
Most nursery managers have a Bachelor of Science degree in Agriculture or a related horticultural field.
These degrees require high-level math classes like calculus. About 10% of nursery managers do not have
a college degree. Those managers usually work their way up to a manager position with many years of
work and experience. Nursery manager salaries start at around $35,000 to $40,000 per year and can get
as high as $65,000 or more per year. Often the pay is related to the size of the nursery and how many
trees the nursery sells.
I feel lucky to be doing something I enjoy and making a positive contribution to the world I live in by
growing trees. I have helped grow over 180,000,000 trees in my career. I enjoy going to the woods and
seeing stands of trees I grew. I know that some of those trees will still be living when my grandchildren’s
grandchildren are old.
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