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2-5
2-5
Proving Angles Congruent
1. Plan
Objectives
1
To prove and apply theorems
about angles
Examples
1
2
Using the Vertical Angles
Theorem
Proving Theorem 2-2
What You’ll Learn
• To prove and apply
theorems about angles
To find the measures of
angles formed by the legs of
a director’s chair, as in
Exercise 11
Lesson 1-6
x Algebra Find the value of each variable.
1. 50
. . . And Why
GO for Help
Check Skills You’ll Need
2
2. 90
130 x
3. 35
z
y
55
Fill in each blank.
4. Perpendicular lines are two lines that intersect to form 9. right angles
Math Background
Inductive reasoning can lead to
conjectures, but only deductive
reasoning can establish truth
conclusively. Proving a theorem
by logically progressing from a
given condition to a necessary
conclusion is a fundamental
mathematical activity.
5. An angle is formed by two rays with the same endpoint.
The endpoint is called the 9 of the angle. vertex
New Vocabulary • theorem
1
• paragraph proof
Theorems About Angles
More Math Background: p. 78D
Hands-On Activity: Vertical Angles
• Draw two intersecting lines.
Number the angles as shown.
Lesson Planning and
Resources
1
3
• Fold &1 onto &2.
See p. 78E for a list of the
resources that support this lesson.
4
2
• Fold &3 onto &4.
• Make a conjecture about vertical angles.
Vertical angles are congruent.
PowerPoint
Bell Ringer Practice
You can use deductive reasoning to show that a conjecture is true. The set of steps
you take is called a proof. The statement that you prove true is a theorem. The
Investigation above leads to a conjecture that becomes the following theorem.
Check Skills You’ll Need
For intervention, direct students to:
Identifying Angle Pairs
Lesson 1-6: Examples 4, 5, 6
Extra Skills, Word Problems, Proof
Practice, Ch. 1
Key Concepts
Theorem 2-1
Vertical Angles Theorem
Vertical angles are congruent.
3
&1 > &2 and &3 > &4
1
2
4
In the proof of a theorem, a “Given” list shows you what you know from the
hypothesis of the theorem. You prove the conclusion of the theorem. A diagram
records the given information visually.
110
Chapter 2 Reasoning and Proof
Special Needs
Below Level
L1
Have students perform the lesson Activity using lines
that are not close to being perpendicular. Then have
them cut out the angles and match them to reinforce
the fact that vertical angles are congruent.
110
learning style: tactile
L2
Have students use protractors to verify the Vertical
Angles Theorem. Actually measuring the angles also
will reinforce the underlying algebraic argument in the
paragraph proof on p. 111.
learning style: tactile
2. Teach
Here is what the start of many proofs will look like.
what you
know
diagram that
shows what
you know
Given:
Prove:
what you
must show
Guided Instruction
Hands-On Activity
There are many forms of proofs. A paragraph proof is written as sentences in a
paragraph. Here is a paragraph proof of Theorem 2-1.
Proof Given: &1 and &2 are
d what you know S
Prove: &1 > &2
1
2
3
vertical angles.
d what you show
Technology Tip
Paragraph Proof: By the Angle Addition Postulate, m&1 + m&3 = 180 and
m&2 + m&3 = 180. By substitution, m&1 + m&3 = m&2 + m&3. Subtract
m&3 from each side. You get m&1 = m&2, or &1 > & 2.
Students can explore the Vertical
Angles Theorem with geometry
software.
Tactile Learners
You can use the Vertical Angles Theorem to solve for variables and find the
measures of angles.
1
4
5
A
A
B
B
E
D
E
D
D
C
C
D
E
E
Test-Taking Tip
Be sure you shade the
correct ovals in the
grid. Grading software
reads only the shaded
ovals, not the answer
across the top.
(4x)
4x = 3x + 35
x = 35
Quick Check
Have students copy the diagrams
for the Vertical Angles and
Congruent Supplements Theorems
to reinforce each Given and the
conclusions that follow.
Using the Vertical Angles Theorem
Gridded Response Find the value of x.
E
D
C
B
E
D
C
B
A
3
C
B
A
2
C
B
A
1
EXAMPLE
(3x 35)
Provide patty paper (which is
also used to separate frozen
hamburger patties) for students
to fold for this activity.
/
3
/
5
.
.
.
.
Alternative Method
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9
0
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9
0
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9
0
1
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8
9
Work as a class to write the
paragraph proof preceding
Example 1 in two columns,
justifying each step in the right
column as in Lesson 2-4.
1
Vertical angles are congruent.
Subtract 3x from each side.
1 a. Find the measures of the labeled pair of vertical angles in the diagram above.
140; 140
b. Find the measures of the other pair of vertical angles. 40; 40
c. Check to see that adjacent angles are supplementary. 140 + 40 = 180
The Vertical Angles Theorem is actually a special case of the following theorem.
A proof of this theorem is shown on the next page. You can write a proof of
another form of this theorem in Exercise 27.
EXAMPLE
Teaching Tip
Ask: How is this solution method
like writing a paragraph proof?
How is it different? Sample: You
work step-by-step, building on
what you know; you can quickly
see the list of steps.
PowerPoint
Additional Examples
1 Find the value of x. 52
Key Concepts
Theorem 2-2
Congruent Supplements Theorem
If two angles are supplements of the same angle (or of congruent angles),
then the two angles are congruent.
(2x + 3)
(4x – 101)
Lesson 2-5 Proving Angles Congruent
Advanced Learners
111
English Language Learners ELL
L4
Have students prove the statement: “If two angles are
congruent, then their supplements are congruent.”
This is the converse of the Congruent Supplements
Theorem.
learning style: verbal
Help students recognize that a proof is a set of
steps to show a conjecture is true. Proofs use given
information, definitions, postulates, and theorems as
reasons that justify a step.
learning style: verbal
111
PowerPoint
Proof
Additional Examples
Proving Theorem 2-2
EXAMPLE
Study what you are given, what you are to prove, and the diagram. Write a
paragraph proof.
2 Write a paragraph proof using
the given, what you are to prove,
and the diagram.
W X
2
Given: &1 and &2 are supplementary.
&3 and &2 are supplementary.
1
2
Prove: &1 > &3
Y Z
3
Given: WX = YZ
Prove: WY = XZ
Proof: It is given that
WX ≠ YZ. By adding the middle
segment, XY, to each smaller
segment gives WX XY ≠
YZ XY. By the Segment
Addition Postulate, WX XY ≠
WY and XY YZ ≠ XZ. Using
substitution, you get WY ≠ XZ.
Proof: By the definition of supplementary angles, m&1 + m&2 = 180 and
m&3 + m&2 = 180. By substitution, m&1 + m&2 = m&3 + m&2. Subtract
m&2 from each side. You get m&1 = m&3, or &1 > &3.
Quick Check
2 In the proof above, which Property of Equality allows you to subtract m&2 from
each side of the equation? Subtraction Property of Equality
Theorem 2-3 is like the Congruent Supplements Theorem. You can demonstrate its
proof in Exercises 7 and 28.
Resources
• Daily Notetaking Guide 2-5 L3
• Daily Notetaking Guide 2-5—
L1
Adapted Instruction
Key Concepts
Theorem 2-3
Congruent Complements Theorem
If two angles are complements of the same angle (or of congruent angles),
then the two angles are congruent.
Theorem 2-4
Closure
All right angles are congruent.
Point out two things that are
incorrect in the diagram. Explain
your reasoning.
Theorem 2-5
If two angles are congruent and supplementary, then each is a right angle.
60°
120°
You can complete proofs of Theorems 2-4 and 2-5 in Exercises 14 and 21, respectively.
130°
50°
Pairs of vertical angles do not
have the same measure; the
Vertical Angles Theorem says
they are congruent. One pair of
supplementary angles has a sum
of 170, and another pair has a
sum of 190; supplementary
angles have a sum of 180.
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
A
Practice by Example
Example 1
GO for
Help
Find the value of each variable.
1.
2.
3.
(page 111)
3x
(80 - x )
y
3x
75
x 5 25, y 5 105
20
Find the measures of the labeled angles in each exercise.
4. Exercise 1
60, 60
112
112
Chapter 2 Reasoning and Proof
5. Exercise 2
75, 105
(x + 90) 4x
30
6. Exercise 3
120, 120
Example 2
(page 112)
7. Developing Proof Complete this proof of one form of Theorem 2-3 by filling in
the blanks.
If two angles are complements of the same angle, then the two angles
are congruent.
Given: &1 and &2 are complementary.
&3 and &2 are complementary.
Assignment Guide
1 A B 1-28
1
Prove: &1 > &3
2
Proof: By the definition of complementary angles,
90
m&1 + m&2 = a. 9 and m&3 + m&2 = b. 9. 90
3
Then m&1 + m&2 = m&3 + m&2 by c. 9. substitution
Apply Your Skills
C Challenge
29-32
Test Prep
Mixed Review
33-39
40-47
Homework Quick Check
Subtract m&2 from each side. You get m&1 = d. 9, or &1 > &3. ml3
B
3. Practice
8. Writing How is a theorem different from a postulate? Answers may vary.
Sample: A theorem is proven and a postulate is assumed to be true.
9. Open-Ended Give an example of vertical angles in your home.
Answers may vary. Sample: scissors
10. Reasoning Explain why this statement is true:
If m&1 + m&2 = 180 and m&3 + m&2 = 180, then &1 > &3. See margin.
11. Design The two back legs of the director’s chair pictured at the left meet in a
728 angle. Find the measure of each angle formed by the two back legs. See margin
To check students’ understanding
of key skills and concepts, go over
Exercises 4, 7, 8, 20, 21.
Exercise 7 When they finish the
proof, have students compare
their justifications to those of the
Vertical Angles Theorem proof.
x 2 Algebra Find the value of each variable and the measure of each labeled angle.
x 5 14, y 5 15; 50, 50, 130
13.
(5x 20)
(3x 8)
12.
(4x 35)
(x 10)
(5x 4y)
15; 25, 25
14. Developing Proof Complete this proof of Theorem 2-4 by filling in the blanks.
All right angles are congruent.
Given: &X and &Y are right angles.
L3
Guided Problem
L4
L2
Reteaching
Prove: &X > &Y
Exercise 11
GPS
Enrichment
X
Y
L1
Adapted Practice
right angle
Proof: By the definition of a. 9, m&X = 90 and m&Y = 90.
By the Substitution Property, m&X = b. 9, or &X > &Y. mlY
Practice
Name
Class
Proving Angles Congruent
Find the values of the variables.
1.
2.
(2x 10)
(3x 40)
(6y 10) (6y 10)
Main St.
t.
Elm
kS
St.
Oa
15. Multiple Choice What is the measure of the
angle formed by Park St. and Oak St.? C
35°
45°
55°
90°
L3
Date
Practice 2-5
3.
4.
32
(4z 10)
(9x 4)
z
5.
6.
(7x 3)
(4x 1)
(4y)
65
(6y)
Park St.
35
Write true or false.
7. &1 and &2 are vertical angles.
GO
nline
Homework Help
© Pearson Education, Inc. All rights reserved.
8. &2 and &3 are supplementary angles.
Name two pairs of congruent angles in each figure. Justify your answers. 16–18.
See margin.
16.
17.
18.
B
K
D
E
F
9. m&1 = m&3
10. m&3 + m&4 = 180
I
O
A
C
G
H
P
J
L
2
3
4
12. &4 and &2 are adjacent angles.
Write three conclusions that can be drawn from each figure.
13.
14.
15.
O
P
B
C
B
D
125
M
N
Q
Visit: PHSchool.com
Web Code: aue-0205
1
11. m&1 + m&3 = 180
A
O
E
A
W
C
D
M
19. Coordinate Geometry &DOE contains points
) D(2, 3), O(0, 0), and E(5, 1).
Find the coordinates of a point F so that OF is a side of an angle that is
adjacent and supplementary to &DOE. Answers may vary. Sample: (–5, –1)
Lesson 2-5 Proving Angles Congruent
113
113
Exercise 8 When students are
GPS 20. Coordinate Geometry &AOX contains points A(1, 3), O(0, 0), and X(4, 0).
a. Find the coordinates of a point B so that &BOA and &AOX are adjacent
complementary angles. a–b. See margin. )
b. Find the coordinates of a point C so that OC is a side of a different angle
that is adjacent and complementary to &AOX.
finished, ask: How is a theorem
similar to a postulate? Sample:
Both are true statements about
geometric figures. Point out that
students will use both postulates
and theorems that are already
proved to prove each new
theorem.
21. Developing Proof Complete this proof of Theorem 2-5 by filling in the blanks.
If two angles are congruent and supplementary,
then each is a right angle.
Given: &W and &V are congruent
and supplementary.
Connection to Architecture
Exercise 9 Ask: What angles are
used frequently in house and
building design? Sample: right
angle, straight angle Have
students discuss where these
angles appear.
Prove: &W and &V are right angles.
Proof: &W and &V are congruent, so m&W = m& a. 9. V
&W and &V are supplementary so m&W + m&V = b. 9. 180
Substituting m&W for m&V, you get m&W + m&W = 180, or 2m&W = 180.
By the c. 9 Property of Equality, m&W = 90. Division
Exercise 15 Have students
identify what they are trying to
find in the diagram. Then ask:
What angle information are you
given? one right angle and a
35° angle
V
W
Since &W > &V, m&V = 90, too. Then both angles are d. 9 angles. right
3
1
2
22. Sports In the photograph, the wheels of the racing wheelchair are tilted so
that &1 > &2. What theorem can you use to justify the statement &3 > &4?
Supplements of O ' are O.
x 2 Algebra Find the measure of each angle.
4
Exercise 22
Exercises 16–18 Have partners
23. &A is twice as large as its complement, &B.
explain their reasoning to each
other. Point out that being able
to explain your reasoning
carefully is like writing a good
proof in geometry.
24. &A is half as large as its complement, &B. mlA ≠ 30, mlB ≠ 60
25. &A is twice as large as its supplement, &B. mlA ≠ 120, mlB ≠ 60
26. &A is half as large as twice its supplement, &B. mlA ≠ 90, mlB ≠ 90
Proof
Exercises 27, 28 Do these exercises
27. Write a proof for this form of Theorem 2-2. See margin.
If two angles are supplements of congruent
angles, then the two angles are congruent.
as a class activity. Students may
refer back to the proofs in the
lesson for ideas, if necessary.
Given: &1 and &2 are supplementary.
&3 and &4 are supplementary.
&2 > &4
Connection to Algebra
Exercises 30–32 Students must
Proof
Given: &1 and &2 are complementary.
&3 and &4 are complementary.
&2 > &4
10. If ml1 ± ml2 ≠ 180,
and ml2 ± ml3 ≠ 180,
then ml1 ± ml2 ≠
ml2 ± ml3 by subst.
Subtr. ml2 from each
side ml1 ≠ ml3 or
l1 O l3.
Prove: &1 > &3
C
Challenge
11. The two acute ' have
measure 72. The two
obtuse ' have measure
108.
114
2
3
4
28. Write a proof for this form of Theorem 2-3. See margin.
If two angles are complements of congruent
angles, then the two angles are congruent.
17. lEIG O lFIH since all
rt. ' are O; lEIF O
lHIG since they are
compl. of the same l.
1
Prove: &1 > &3
set up and solve a system of two
equations.
16. lDOB O lAOC and
lDOA O lBOC since
they are vert. '.
mlA ≠ 60, mlB ≠ 30
114
1
2
3
4
29. Paper Folding After you’ve done the Activity on page 110, answer these
questions.
a–b. It is the bisector of both angles.
a. How is the first fold line you make related to angles 3 and 4?
b. How is the second fold line you make related to angles 1 and 2?
c. How are the two fold lines related to each other? Give a convincing
argument to support your answer. Sample: perpendicular; bisectors of
two adjacent supplementary angles form two adjacent angles whose
measures add to 1
2 (180) , or 90.
Chapter 2 Reasoning and Proof
18. lKPJ O lMPJ since
they are marked O;
lKPL O lMPL since
they are suppl. of O '.
19. Answers may vary.
Sample: (–5, –1)
20. a. Answers may vary. B
can be any point on
the positive y-axis.
Sample: (0, 5).
b. Answers may vary.
Sample: (3, –1)
x 2 Algebra Find the value of each variable and the measure of each labeled angle.
30.
(y + x)
(y - x)
2x
31.
(x + y + 5)
y
32.
PowerPoint
2x
2x
4y
x ≠ 30, y ≠ 90;
60, 120, 60
4. Assess & Reteach
x ≠ 35, y ≠ 70;
70, 110, 70
Lesson Quiz
(x y 10)
x ≠ 50, y ≠ 20;
80, 100, 80
Use the diagram and mlABS ≠
3x ± 6 and mlRBC ≠ 5x – 20 for
Exercises 1-4.
R
Test Prep
A
S
Gridded Response
Find the measure of each angle.
B
C
1. Find x. 13
33. an angle with measure 8 less than the measure of its complement 41
2. Find m&ABS. 45°
34. one angle of a pair of complementary vertical angles 45
3. Find m&SBC. 135°.
35. an angle with measure three times the measure of its supplement 135
4. Without using the Vertical
Angle Theorem, what theorem
can you use to prove that
&ABR &SBC? Congruent
Supplements Theorem
Use the diagram at the right to find the measure
of each of the following angles.
36. &1 20
37. &2 90
38. &3 70
39. &4 110
70
4
3 2
1
Alternative Assessment
Mixed Review
GO for
Help
Lesson 2-4
Write this statement on the board.
If two angles are congruent and
complementary, then each has
a measure of 45.
Use the given property to complete each statement.
40. Subtraction Property of Equality
If 3x + 7 = 19, then 3x = 9. 12
Have partners draw and label a
diagram that fits the theorem and
write a paragraph proof of
the theorem.
41. Reflexive Property of Congruence
AB > 9 AB
42. Substitution Property
If MN = 3 and MN + NP = 15, then 9. 3 1 NP 5 15
Lesson 2-3
Use deductive reasoning to draw a conclusion. If not possible, write not possible.
43. If two lines intersect, then they are coplanar.
Lines m and n are coplanar. not possible
44. If two angles are vertical angles, then they are congruent.
&1 and &2 are vertical angles. l1 and l2 are congruent.
Lesson 2-2
45. If y ≠ 25, then
y + 7 ≠ 32.
y + 7 ≠ 32 if and
only if y ≠ 25.
46. If you live south of
the equator, then
you live in Australia.
Each conditional statement below is true. Write its converse. If the converse is also
true, combine the statements as a biconditional.
45. If y + 7 = 32, then y = 25.
46. If you live in Australia, then you live south of the equator.
47. If n . 0, then n2 . 0. If n2 S 0, then n S 0.
Test Prep
A sheet of blank grids is available
in the Test-Taking Strategies with
Transparencies booklet. Give this
sheet to students for practice with
filling in the grids.
Resources
lesson quiz, PHSchool.com, Web Code: aua-0205
27. By the def. of suppl. ',
ml1 ± ml2 ≠ 180 and
ml3 ± ml4 ≠ 180. By
the Subst. Prop. ml1 ±
ml2 ≠ ml3 ± ml4. It
is given that l2 O l4,
so ml2 ≠ ml4. Then by
Alternatively, if students are not
yet ready to write a paragraph
proof, ask them to write a
full explanation of why the
theorem makes sense. Writing an
explanation is similar to writing
a proof, but the language may
be less intimidating to students.
Lesson 2-5 Proving Angles Congruent
the Subtr. Prop. of ≠, ml1
≠ ml3, or l1 O l3.
28. By the def. of compl. ',
ml1 ± ml2 ≠ 90 and
ml3 ± ml4 ≠ 90. By
the Subst. Prop. of ≠,
115
ml1 ± ml2 ≠ ml3 ±
ml4. It is given that l2
O l4, so ml2 ≠ ml4.
Then by the Subtr. Prop.
of ≠, ml1 ≠ ml3 or
l1 O l3.
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 121
• Test-Taking Strategies, p. 116
• Test-Taking Strategies with
Transparencies
115