Download GETE0304

Parallel Lines and the
Triangle Angle-Sum Theorem
3-4
3-4
1. Plan
What You’ll Learn
Check Skills You’ll Need
• To classify triangles and find
Classify each angle as acute, right, or obtuse.
the measures of their
angles
GO for Help
1.
2.
Lesson 1-6
Objectives
1
2
3.
• To use exterior angles of
triangles
acute
right
. . . And Why
acute
Examples
1
Solve each equation.
To find the reclining angle
of a lounge chair, as in
Example 4
4. 30 + 90 + x = 180 60
5. 55 + x + 105 = 180 20
6. x + 58 = 90 32
7. 32 + x = 90 58
2
3
New Vocabulary • acute triangle • right triangle • obtuse triangle
• equiangular triangle • equilateral triangle
• isosceles triangle • scalene triangle
• exterior angle of a polygon • remote interior angles
1
4
The diagrams at the top of the Activity Lab on page 146 suggest the Triangle-Angle
Sum Theorem.
Theorem 3-12
Triangle Angle-Sum Theorem
C
A
When alterations of Euclid’s
Parallel Postulate lead to different
geometries, the Triangle AngleSum Theorem appears strikingly
different. In a hyperbolic
geometry, the sum of a triangle’s
angle measures is less than 180;
in an elliptic geometry, the sum
is greater than 180.
More Math Background: p. 124C
The sum of the measures of the angles of a triangle is 180.
m&A + m&B + m&C = 180
Applying the Triangle
Angle-Sum Theorem
Classifying a Triangle
Using the Exterior Angle
Theorem
Real-World Connection
Math Background
Finding Angle Measures in Triangles
Key Concepts
To classify triangles and find
the measures of their angles
To use exterior angles of
triangles
B
Lesson Planning and
Resources
The following proof of Theorem 3-12 relies on the idea that through a point not on
a given line you can draw a line parallel to the given line.
Proof
Proof of Theorem 3-12
See p. 124E for a list of the
resources that support this lesson.
PowerPoint
Given: #ABC
P
C
Bell Ringer Practice
13 2
Prove: m&A + m&B + m&3 = 180
Proof: By the Protractor Postulate, you can draw
)
CP so that m&1 = m&A. Then, &1 and &A are
* )
congruent alternate interior angles, so CP 6 AB.
A
&2 and &B are also alternate interior angles, so by
the Alternate Interior Angles Theorem, m&2 = m&B.
By substitution, m&A + m&B + m&3 = m&1 + m&2 + m&3,
which is equal to 180 by the Angle Addition Postulate.
Check Skills You’ll Need
For intervention, direct students to:
B
Finding Angle Measures
Lesson 1-6; Example 2
Extra Skills, Word Problems, Proof
Practice, Ch. 1
Solving Linear Equations
Algebra 1 Review, page 30
Lesson 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
Special Needs
Below Level
L1
Tell students that acute, right, and obtuse angles can
help them identify acute, right, and obtuse triangles.
Have students prepare a chart that shows each
triangle with its defining characteristic in color.
learning style: visual
147
L2
Using geometry software to draw, measure, and
manipulate the seven triangles on page 148 will help
students discover results such as the Isosceles Triangle
Theorem.
learning style: visual
147
2. Teach
1
Applying the Triangle Angle-Sum Theorem
EXAMPLE
Algebra Find the values of x and y.
Guided Instruction
39 + 65 + x = 180
104 + x = 180
Each new lesson requires
students to keep track of
the names of more and more
theorems, so shorthand ways
to write their names can help
students remember them. Point
out that the symbol S is used in
mathematics to indicate a sum.
So, they could abbreviate the
Triangle Angle-Sum Theorem as:
x = 76
EXAMPLE
Simplify.
Subtract 104 from each side.
m&GJF + m&GJH = 180
65 x y z
H
J
F
Angle Addition Postulate
x + y = 180
76 + y = 180
y = 104
Quick Check
Substitute.
Substitute 76 for x.
Subtract 76 from each side.
1 Find the value of z in two different ways, each way using the Triangle Angle-Sum
Theorem. 55
Alternative Method
Before finding any values, ask:
How many triangles are in the
diagram? 3 Use the question to
highlight the alternate method
of first finding the value of z
using GFH.
2
39
Triangle Angle-Sum Theorem
To find the value of y, look at &FJH. It is a straight angle.
&S
EXAMPLE
21
To find the value of x, use #GFJ.
Teaching Tip
1
G
In Chapter 1, you classified an angle by its measure. You can also classify a triangle
by its angles and sides.
nline
Equiangular
all angles congruent
Math Tip
Students have not yet learned
that a triangle is equilateral if
and only if it is equiangular.
As students see more triangles,
encourage them to develop
hypotheses about triangles,
including what is impossible,
such as an equilateral obtuse
triangle.
Acute
all angles acute
Right
one right angle
Obtuse
one obtuse angle
Visit: PHSchool.com
Web Code: aue-0775
Equilateral
all sides congruent
2
EXAMPLE
Isosceles
at least two sides congruent
Scalene
no sides congruent
Classifying a Triangle
PowerPoint
Classify the triangle by its sides and its angles.
Additional Examples
1 In triangle ABC, &ACB is a
right angle, and CD # AB. Find
the values of a, b, and c.
C
c°
A
At least two sides are congruent, so the triangle is isosceles. All the angles are
acute, so the triangle is acute.
70°
b°
a°
D
The triangle is an acute isosceles triangle.
B
Quick Check
a = 70, b = 20, c = 20
2 Classify the triangle by its
sides and its angles.
5
4
148
2 Draw and mark a triangle to fit each description. If no triangle can be drawn, write
not possible and explain why. a–c. See margin.
a. acute scalene
b. isosceles right
c. obtuse equiangular
Chapter 3 Parallel and Perpendicular Lines
2
obtuse scalene
Advanced Learners
English Language Learners ELL
L4
After Example 1, help students discover the Triangle
Exterior Angle Theorem by finding x and y as sums of
angles in the triangles. x21°55°, y65°39°
148
learning style: visual
To help students remember the meaning of remote
interior angle, use the word remote in other contexts,
such as a remote island or a television remote control.
learning style: verbal
Guided Instruction
2
1
Using Exterior Angles of Triangles
Tactile Learners
An exterior angle of a polygon is an
angle formed by a side and an extension of
an adjacent side. For each exterior angle of a
triangle, the two nonadjacent interior angles
are its remote interior angles.
Interior angle of a
triangle means the same
as “angle of a triangle.”
3
Encourage students to copy
the triangle diagram shown
immediately above the Triangle
Exterior Angle Theorem. Have
them cut out the exterior angle
and two remote interior angles
and then superimpose the interior
angles over the exterior angle to
demonstrate the Triangle Exterior
Angle Theorem.
2
Remote interior angles
The diagram at the right suggests a relationship
between an exterior angle and its two remote
interior angles. Theorem 3-13 states this relationship.
You will prove this theorem in Exercise 35.
1
3
3
2
Vocabulary Tip
Exterior
angle
1
2
4
Key Concepts
Theorem 3-13
Triangle Exterior Angle Theorem
2
The measure of each exterior angle of a triangle equals
the sum of the measures of its two remote interior angles.
1
3
m&1 = m&2 + m&3
Connection
to Algebra
EXAMPLE
Suggest that students let x
represent the measure of the
angle to help them associate the
equations with more familiar
algebraic equations.
PowerPoint
3
EXAMPLE
Using the Exterior Angle Theorem
Algebra Find each missing angle measure.
a.
b.
40
1
30
m&1 = 40 + 30
For: Triangle Theorems Activity
Use: Interactive Textbook, 3-4
Quick Check
5
A
B
B
E
1
43 = m&2
35
3 Two angles of a triangle measure 45. Find the measure of an exterior angle at each
vertex. 135, 135, 90
EXAMPLE
E
D
D
C
2
Real-World
Connection
4 Explain what happens to
the angle formed by the back of
the chair and the armrest as you
make a lounge chair recline more.
The angle increases in measure.
Resources
• Daily Notetaking Guide 3-4 L3
• Daily Notetaking Guide 3-4—
L1
Adapted Instruction
E
D
C
125°
70
Multiple Choice The lounge chair has
different settings that change the angles
formed by its parts. Suppose m&2 is 32
and m&3 is 81. Find m&1, the angle formed
by the back of the chair and the arm rest.
E
D
C
B
A
D
C
B
A
4
C
B
A
3
C
B
A
2
3 Find m&1.
113
113 = 70 + m&2
m&1 = 70
4
1
Additional Examples
D
E
E
67
113
Test-Taking Tip
To check exterior
angle-measure
answers, remember
that an exterior angle
and the adjacent
interior angle must be
supplementary.
81
180
m&1 = m&2 + m&3
Exterior Angle
Theorem
m&1 = 32 + 81
Substitute.
m&1 = 113
Simplify.
1
Closure
2
3
Explain what is wrong with this
diagram.
B
The angle formed is a 1138 angle.
The correct choice is C.
Lesson 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
149
C
Quick Check
b.
10.
2. a.
c. Not possible; an
equilateral k has all
acute '.
11. Not possible; a right k will always
have one longest side opp. the
right l.
85°
80°
A
D
m&BAD must be greater
than m&BCA.
149
3. Practice
Quick Check
Assignment Guide
1 A B 1-15, 23-25, 27-32, 34,
EXERCISES
37, 38
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
2 A B
16-22, 26, 33, 35, 36
C Challenge
39-41
Test Prep
Mixed Review
4 a. Change the setting on the lounge chair so that m&2 = 33 and m&3 = 97. Find
the new measure of &1. 130
b. Explain how you can find m&1 without using the Exterior Angle Theorem.
Answers may vary. Sample: Find the measure of the third l of the triangle.
Subtract this from 180.
A
42-44
45-48
Practice by Example
Find ml1.
Example 1
GO for
Help
Homework Quick Check
1.
(page 148)
2.
117
1
33
30
3.
52.2
83.1
44.7
To check students’ understanding
of key skills and concepts, go over
Exercises 4, 18, 25, 28, 34.
33
57
1
90
1
x 2 Algebra Find the value of each variable.
4.
Exercise 11 Discuss why this
figure is impossible to draw.
When students say that the side
opposite the right angle is always
longer than either of the other
sides, point out that their
observation is a theorem that
the longest side of a triangle is
always opposite the angle with
the greatest measure.
x 70;
y 110;
z 30
5.
30
70
6.
60 30
30
40
x
80 x y
y
z
x 80; y 80
c
Use a protractor and a centimeter ruler to measure the angles and the sides of each
triangle. Classify each triangle by its angles and sides.
Example 2
(page 148)
7.
8.
9.
obtuse,
isosceles
acute,
equiangular,
equilateral
right,
scalene
If possible, draw a triangle to fit each description. Mark the triangle to show known
information. If no triangle can be drawn, write not possible and explain why.
10. acute equilateral 10–15. See margin 11. equilateral right
12. obtuse scalene pp. 150–151.
13. obtuse isosceles
14. isosceles right
GPS Guided Problem Solving
(page 149)
L2
Reteaching
b. l1 and l3 for l5
l1 and l2 for l6
l1 and l2 for l8
L1
Adapted Practice
Name
Class
Practice 3-3
L3
Date
16. a. Which of the numbered angles at the
right are exterior angles? l5, l6, l8
b. Name the remote interior angles for each.
c. How are exterior angles 6 and 8 related?
They are O vert. '.
17. a. How many exterior angles at the right
are at each vertex of the triangle? 2
b. How many exterior angles does a triangle
have in all? 6
Example 3
L4
Enrichment
Practice
Parallel Lines and the Triangle Angle-Sum Theorem
Find the value of each variable.
1.
2.
3
y
65
x
30
60
n
39
75 68
4.
5.
93
6.
61
8.
10
55
a b c
44
25
53
y
w
11.
1
12.
2
32
© Pearson Education, Inc. All rights reserved.
140
14.
2
69.7
18.
62
v t
4
60
115.5
3
15.
2
116 1
1
2
45
13
3 4
38
31
20.
128.5
5
120
46
126.8
3
19.
123
70
72
86
ml3 92;
ml4 88
56
63
16. The sides of a triangle are 10 cm, 8 cm, and 10 cm. Classify the triangle.
17. The angles of a triangle are 44°, 110°, and 26°. Classify the triangle.
150
Use a protractor and a centimeter ruler to measure the angles and the sides
of each triangle. Classify each triangle by its angles and sides.
18.
19.
Chapter 3 Parallel and Perpendicular Lines
20.
12.
150
4
5
3
8 6
7
x 2 Algebra Find each missing angle measure.
9.
28
z
x
p
x
Find the measure of each numbered angle.
13.
2
46
79
10.
1
m
36
7.
15. scalene acute
L3
13.
14.
15.
47
(page 149)
a
B
Connection to Music
21. Music The lid of a grand piano
is held open by a prop stick
whose length can vary,
depending upon the effect
desired. The longest prop stick
makes angles as shown. What
are the values of x and y?
x 147, y 33
22. A short prop stick makes the
angles shown below. What are
the values of a and b?
a 162,
b 18
Example 4
Exercise 21 Ask students what
y
x
other instruments can modify
volume by blocking the escape
of sound waves. For example,
trumpets and trombones have
“mutes” that block sound when
inserted into the bells, and French
horn players place their fists in
the bells of their instruments to
reduce the volume.
57
Exercises 23–26 Done together
as a class, these exercises provide
a good opportunity to review
the cumulative knowledge of
students. Ask students to justify
their solutions step by step for
the rest of the class, giving the
names or descriptions of the
theorems and postulates they
use to find the angle measures.
72
b
Apply Your Skills x 2 Algebra Find the values of the variables and then the measures of the
angles. Classify each triangle by its angles. Note that some figures have
more than one triangle.
x 7; 55,
(8x 1) 35, 90;
right
23.
24.
(2x 4)
(2x 9)
(4x 7)
x 37; 37,
65, 78; acute
Exercises 25, 26 Remind students
to begin each problem by asking:
How many triangles are in the
diagram?
x
Exercise 30 Students can work
25.
GPS
B
26.
a 67, b 58, c 125,
d 23, e 90; kFGH: 58, 67, 55;
acute; kFEH: 125, 32, 23; obtuse;
54
A
kEFG: 67, 23, 90; right
y x
z
D
x 38, y 36,
26. See left.
z 90; kABD:
E
d
36, 90, 54; right;
kBCD: 90, 52, 38;
right; kABC: 74,
52
52, 54; acute
C
e
32
c
55
H
F
b
a
G
27. Reasoning What is the measure of each angle of an equiangular triangle?
Explain. 60; 180 3 60
28. Writing Is every equilateral triangle isosceles? Is every isosceles triangle
equilateral? Explain. See margin.
29. Visualization The diagram shows a triangle on a 3-by-3 geoboard.
How many triangles with different shapes can be made on this
geoboard? Classify each triangle by its sides and angles.
See margin.
30. Multiple Choice The measure of one angle of a triangle is 115. The other two
angles are congruent. What is the measure of each? A
32.5
65
57.5
115
Problem Solving Hint
x 2 31. Algebra A right triangle has acute angles whose measures are in the ratio 1 : 2.
In Exercise 31, use x
and 2x for the angle
measures. In Exercise
32, use 2x, 3x, and 4x.
x 32. a. Algebra The ratio of the angle measures in #BCR is 2 : 3 : 4. Find the
backward by multiplying each
answer choice by 2 and adding
their result to 115. Alternatively,
they can subtract 115 from 180,
and divide their result by 2.
Exercise 32 To help students
understand why 2x, 3x, and 4x are
good representations of the angle
measures, have students choose
values for x and calculate the
ratio of the angle measures.
Diversity
Exercise 37 The textile art of
some countries features patterns
of triangles. Students may be able
to describe, show photographs
of, or bring in examples of such
textiles.
Find the measures of these angles. 30 and 60
2
angle measures. 40, 60, 80
b. What type of triangle is #BCR?
acute
Lesson 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
28. Yes, an equilateral k is
isosc. because if three
sides of a k are O, then
two sides are O. No, the
third side of an isosc. k
does not need to be O
to the other two.
29. eight
151
Acute isosceles
Right isosceles
Right scalene
Obtuse scalene
151
4. Assess & Reteach
GO
PowerPoint
Homework Help
Visit: PHSchool.com
Web Code: aue-0304
Lesson Quiz
1. A triangle with a 90° angle has
sides that are 3 cm, 4 cm, and 5
cm long. Classify the triangle by
its sides and angles. scalene
right triangle
Use the diagram for
Exercises 2–6.
33. Draw any triangle. Label it #ABC. Extend both sides of the triangle to form
two exterior angles at vertex A. Use the two exterior angles to explain why it
does not matter which side of a triangle is extended to form an exterior angle.
nline
33. Check students’
work. Answers may
vary. Sample: The
two exterior '
formed at vertex A
are vertical ' and
thus have the same
measure.
Proof
34. Prove the following theorem. 34-35. See margin.
The acute angles of a right triangle are complementary.
B
Given: #ABC with right angle C
Prove: &A and &B are complementary.
Proof
A
C
35. Prove the Triangle Exterior Angle Theorem.
2
Given: &1 is an exterior angle of the triangle.
Prove: m&1 = m&2 + m&3
1 4
3
3
1 2
4
36. Reasoning Two angles of a triangle measure 64 and 48. Find the measure of
the largest exterior angle. Explain. See margin.
5
Real-World
2. Find m&3 if m&2 = 70
and m&4 = 42. 68
3. Find m&5 if m&2 = 76
and m&3 = 90. 166
Connection
Patricia Watson Tsinnie often
uses isosceles triangles in her
rug designs.
37. Open-Ended Study the design in the Navajo weaving below. Make a design
of your own that makes repeated use of isosceles triangles. Check students’
work.
4. Find x if m&1 = 4x,
m&3 = 2x + 28, and
m&4 = 32. 30
5. Find x if m&2 = 10x,
m&3 = 5x + 40, and
m&4 = 3x - 4. 8
6. Find m&3 if m&1 = 125 and
m&5 = 160. 105
Alternative Assessment
Have students draw and label a
triangle with an exterior angle at
each vertex. They should measure
and label each angle, classify the
triangle, and then explain how
the measurements illustrate the
Triangle Angle-Sum Theorem
and the Triangle Exterior Angle
Theorem.
38. The measures of the angles of #RST are 5!x, 7 !x, and 8!x.
a. Find the value of x. 81
b. Give the measure of each angle. 45, 63, 72
c. What type of triangle is #RST? acute
C
Challenge
Test Prep
Resources
CD bisects &ACB. Find m&DBF. 115
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 193
• Test-Taking Strategies, p.188
• Test-Taking Strategies with
Transparencies
41. What can you conclude about the bisector
of an exterior angle of a triangle if the
remote interior angles are congruent?
Justify your response. See margin.
152
34. By the definition of right
angle, mlC ≠ 90. By the
Triangle Angle-Sum
Theorem, mlA ± mlB ±
mlC ≠ 180.
152
39. Reasoning Sketch a triangle and two exterior angles that have a side of the
triangle in common. For what type of triangle, if any, is each statement true?
Justify each answer. See back of book.
a. The bisectors of the two exterior angles are parallel.
b. The bisectors of the two exterior angles are perpendicular.
c. The bisectors of the two exterior angles and the common side of the given
triangle form an isosceles triangle.
A
40. In the figure at the right, CD ' AB and
D
F B
(3x 2)
C
(5x 20)
Chapter 3 Parallel and Perpendicular Lines
Subtracting 90 from each side
gives mlA ± mlB ≠ 90, so
lA and lB are
complementary by the
definition of comp. angles.
35. ml1 ± ml4 ≠ 180 by
the l Add. Postulate.
ml2 ± ml3 ± ml4 ≠
180 by the k l-Sum
Theorem. ml1 ± ml4
± ml2 ± ml3 ± ml4
by the Trans. Property of
Equality. ml1 ± ml2 ±
ml3 by the Subtr.
Property of Equality.
Test Prep
Multiple Choice
Use the diagram at the right for Exercises 42–44.
Use this Checkpoint Quiz to check
students’ understanding of the
skills and concepts of Lessons 3-1
through 3-4.
M
42. m&M = 25 and m&L = 43. What is m&JKM? G
F. 18
G. 68
H. 117
J. 162
Resources
43. m&M = 4x, m&L = 5x, and m&MKL = 6x.
What is m&JKM? B
A. 72
B. 108
C. 120
D. 132
J
K
L
Grab & Go
• Checkpoint Quiz 1
44. m&JKM = 15x - 48, m&L = 5x + 12, and m&M = 40. What is m&MKL? H
F. 9
G. 57
H. 78
J. 97
Mixed Review
GO for
Help
Lesson 3-3
Use the diagram at the right for Exercises 45–46.
1
a
45. If &1 and &2 are supplementary, what can you
conclude about lines a and c? Justify your answer.
b
46. If a 6 c, what can you conclude about
lines a and b? Justify your answer.
a n b; two lines n to the same line are n to each other.
Lesson 1-6 x 2 47. Algebra In the figure at the right,
A
m&AOB = 3x + 20, m&BOC = x + 32,
and m&AOC = 80. Find the value of x. 7
c
45. a n c by the Conv. of the
Same-Side Ext. ' Thm.
Lesson 1-1
O
Draw the next figure in each sequence.
48.
B
C
49.
Checkpoint Quiz 1
7. Converse of Alternate
Exterior Angles Theorem
2
Lessons 3-1 through 3-4
Use the diagram at the right for Exercises 1–9. State the theorem or postulate that
justifies each statement.
Corr. ' Postulate
Conv. of Corr. ' Post.
a
b
c
1. &1 > &3
2. If &5 > &9, then d 6 e.
3. Same-Side Int. ' Thm.
1
2 3
3. m&1 + m&2 = 180
4. If &4 > &7, then d 6 e.
d
4. Conv. of the Alt. Int. ' Thm.
4
5 6
5. &1 > &4
6. &7 > &9
Vertical ' Theorem
Alt. Int. ' Thm.
7
8
7. If &3 > &9, then d 6 e. 8. &4 > &5
e
9
Corr. ' Postulate
9. If e ' b, then e ' c.
If a line is ' to 1 of 2 parallel lines, it is ' to both.
(2x 23)
10. Find the measures of the angles
of each triangle. Classify each
triangle by its angles.
w
x
y
38, 55, 87; acute
38
55, 26, 99; obtuse
(4w 5)
lesson quiz, PHSchool.com, Web Code: aua-0304
36. 132; since the l is 68, the
largest l is 180 – 48 ≠
132.
41. Answers may vary.
Sample: The measure of
the ext. l is ≠ to the sum
of the measures of the
Lesson 3-4 The Triangle Angle-Sum Theorem
two remote int. '. Since
these ' are O, the '
formed by the bisector of
the ext. l are O to each of
them. Therefore, the
bisector is n to the
included side of the
153
remote ' by the Conv. of
the Alt. Int. ' Thm.
153