2.4
What's Your Proof?
Angle Postulates and Theorems
2
LEARNING GOALS
In this lesson, you will:
•
•
•
•
Use the Corresponding Angle Postulate.
Prove the Alternate Interior Angle Theorem.
Prove the Alternate Exterior Angle Theorem.
Prove the Same-Side Interior
Angle Theorem.
• Prove the Same-Side Exterior
Angle Theorem.
KEY TERMS
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•
•
•
•
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Corresponding Angle Postulate
conjecture
Alternate Interior Angle Theorem
Alternate Exterior Angle Theorem
Same-Side Interior Angle Theorem
Same-Side Exterior Angle Theorem
Y
ou are constantly bombarded with information through magazines, newspapers,
television, and the Internet. However, not all “facts” that you read about are
actually true! If you want to be an educated consumer of information, you should
always be looking for the argument, or proof, to back up a statement. If you can’t find
such information then you should be skeptical.
© Carnegie Learning
Sometimes you need to carefully examine the evidence. For example, say someone
claims that 4 out of 5 dentists recommend a certain toothpaste. Sounds pretty
impressive, right? However, what if you learned that only five dentists were asked
their opinions? You might start to question the claim. What if you also learned that the
dentists were paid by the toothpaste company for their opinions? As you can see,
sometimes the “truth” isn’t always what it appears to be.
2.4
Angle Postulates and Theorems
191
Problem 1
The Corresponding Angle
Postulate is stated. Students
use the postulate to form
conjectures regarding pairs
of alternate interior angles,
alternate exterior angles, sameside interior angles, and sameside exterior angles. These
conjectures will then be proven
and renamed as theorems.
2
PROBLEM 1
The Corresponding Angle Postulate
The Corresponding Angle Postulate states: “If two parallel lines are intersected by a
transversal, then corresponding angles are congruent.”
3
y
4
7 8
1 2
5 6
w
x
1. Name all pairs of angles that are congruent using the Corresponding Angle Postulate.
I can use the Corresponding Angle Postulate to state /1 > /3, /2 > /4,
/5 > /7, and /6 > /8.
Grouping
A conjecture is a hypothesis that something is true. The hypothesis can later be proved
or disproved.
Have students complete
Questions 1 through 3 with
a partner. Then have
students share their responses
as a class.
2. Write a conjecture about each pair of angles formed by parallel lines cut by a
transversal. Explain how you made each conjecture.
a. alternate interior angles.
The conjecture is that if two parallel lines are intersected by a transversal, then
alternate interior angles are congruent.
In the figure, /2 and /7 are alternate interior angles. By the Corresponding Angle
Postulate, /2 > /4. By the Vertical Angle Theorem, /4 > /7. Therefore,
by the Transitive Property, /2 > /7.
Guiding Questions
for Share Phase,
Questions 1 through 3
• Is the Corresponding
Similarly, /3 and /6 are alternate interior angles. By the Corresponding Angle
Postulate, /3 > /1. By the Vertical Angle Theorem, /1 > /6. Therefore,
by the Transitive Property, /3 > /6.
Angle Postulate a
conditional statement?
•
192
In the figure, /4 and /5 are alternate exterior angles. By the Corresponding
Angle Postulate, /4 > /2. By the Vertical Angle Theorem, /2 > /5. Therefore,
by the Transitive Property, /4 > /5.
What is the conclusion
of the Corresponding
Angle Postulate?
•
How would you describe
the location of a pair of
corresponding angles?
•
How many pairs of congruent
corresponding angles are
formed when two parallel
lines are intersected by
a transversal?
•
The conjecture is that if two parallel lines are intersected by a transversal, then
alternate exterior angles are congruent.
What is the hypothesis
of the Corresponding
Angle Postulate?
How would you describe the
location of a pair of alternate
interior angles?
Chapter 2 Introduction to Proof
Similarly, /1 and /8 are alternate exterior angles. By the Corresponding Angle
Postulate, /1 > /3. By the Vertical Angle Theorem, /3 > /8. Therefore,
by the Transitive Property, /1 > /8.
•
How many pairs of congruent alternate interior angles are formed when two
parallel lines are intersected by a transversal?
•
•
How would you describe the location of a pair of alternate exterior angles?
•
How would you describe the location of a pair of same-side interior angles?
How many pairs of congruent alternate exterior angles are formed when two
parallel lines are intersected by a transversal?
© Carnegie Learning
•
b. alternate exterior angles.
•
How many pairs of sameside interior angles are
formed when two parallel
lines are intersected by
a transversal?
•
How would you describe the
location of a pair of sameside exterior angles?
•
How many pairs of sameside exterior angles are
formed when two parallel
lines are intersected by
a transversal?
c. same-side interior angles
The conjecture is that if two parallel lines are intersected
by a transversal, then same-side interior angles are
supplementary.
Test it
out! Conjecture is
all about testing.
In the figure, /2 and /3 are same-side interior angles.
By the Corresponding Angle Postulate, /2 > /4.
By the Linear Pair Postulate, /3 and /4 are
supplementary. Therefore, by the Substitution
Property, /2 and /3 are supplementary.
2
Similarly, /6 and /7 are same-side interior angles.
By the Corresponding Angle Postulate, /6 > /8.
By the Linear Pair Postulate, /7 and /8 are supplementary. Therefore, by the
Substitution Property, /6 and /7 are supplementary.
Note
Emphasize the difference
between a conjecture and
a theorem.
d. same-side exterior angles
The conjecture is that if two parallel lines are intersected by a transversal, then
same-side exterior angles are supplementary.
In the figure, /1 and /4 are same-side exterior angles. By the Corresponding
Angle Postulate, /1 > /3. By the Linear Pair Postulate, /3 and /4 are
supplementary. Therefore, by the Substitution Property, /1 and /4 are
supplementary.
Similarly, /5 and /8 are same-side exterior angles. By the Corresponding Angle
Postulate, /5 > /7. By the Linear Pair Postulate, /7 and /8 are supplementary.
Therefore, by the Substitution Property, /5 and /8 are supplementary.
3. Did you use inductive or deductive reasoning to make each conjecture?
© Carnegie Learning
I used inductive reasoning because I used postulates and properties to make these
conjectures.
2.4
Angle Postulates and Theorems
193
Problem 2
Students use a variety of
methods of proof to prove
the Alternate Interior Angle
Theorem, the Alternate Exterior
Angle Theorem, the Same-Side
Interior Angle Theorem, and
the Same-Side Exterior
Angle Theorem.
2
PROBLEM 2
Conjecture or Theorem?
If you can prove that a conjecture is true, then it becomes a theorem.
1. The Alternate Interior Angle Conjecture states: “If two parallel lines are intersected by a
transversal, then alternate interior angles are congruent.”
1 2
3 4
w
Grouping
5 6
7 8
x
Have students complete
Question 1 with a partner.
Then have students share their
responses as a class.
z
a. Use the diagram to write the “Given” and “Prove” statements for the Alternate
Interior Angle Conjecture.
Given: w i x, z is a transversal
Prove: /3 > /6
Guiding Questions
for Share Phase,
Question 1
• How is the Corresponding
•
Which pair of alternate
interior angles did you
prove congruent?
•
Can you prove a different
pair of alternate interior
angles congruent using the
same strategy?
•
Do you need to prove both
pairs of alternate interior
angles congruent to prove
this theorem?
•
Why is it necessary to prove
a second pair of alternate
interior angles congruent to
prove this theorem?
•
194
Did you use the flow
chart proof to create the
two-column proof? Was
it helpful?
Chapter 2 Introduction to Proof
•
•
•
How is the flow chart proof similar to the two-column proof?
How is the flow chart proof different than the two-column proof?
Which type of proof is easier to create? Why?
© Carnegie Learning
Angle Postulate used to help
prove the Alternate Interior
Angle Theorem?
b. Complete the flow chart proof of the Alternate Interior Angle Conjecture by writing
the reason for each statement in the boxes provided.
w x
Given
/2 > /6
/2 > /3
Corresponding Angle Postulate
Vertical angles are congruent
2
/3 > /6
Transitive Property
c. Create a two-column proof of the Alternate Interior Angle Theorem.
Statements
Reasons
1. w i x
1. Given
2. /1 > /5
2. Corresponding Angle Postulate
3. /1 > /4
3. Vertical angles are congruent
4. /4 > /5
4. Transitive Property
Congratulations!
You can now use this
theorem as a valid reason in
proofs.
© Carnegie Learning
You have just proven the Alternate Interior Angle Conjecture.
It is now known as the Alternate Interior Angle Theorem.
2.4
Angle Postulates and Theorems
195
Grouping
Have students complete
Question 2 with a partner.
Then have students share their
responses as a class.
2. The Alternate Exterior Angle Conjecture states: “If two parallel lines are intersected by a
transversal, then alternate exterior angles are congruent.”
a. Draw and label a diagram illustrating the Alternate Exterior Angle Conjecture. Then,
write the given and prove statements.
Given: w i x, z is a transversal
Prove: /1 > /8 or /2 > /7
Guiding Questions
for Share Phase,
Question 2
• How is the Corresponding
2
Which pair of alternate
exterior angles did you
prove congruent?
•
Can you prove a different
pair of alternate exterior
angles congruent using the
same strategy?
•
Do you need to prove both
pairs of alternate exterior
angles congruent to prove
this theorem?
5 6
7 8
x
Angle Postulate used to help
prove the Alternate Exterior
Angle Theorem?
•
1 2
3 4
w
z
b. Prove the Alternate Exterior Angle Conjecture.
Students may use a flow chart, two-column, or paragraph proof. The flow chart
proof is provided. Students only need to use one method.
Remember, you
can write proofs as
flow charts, two columns, or
paragraphs. Choose the form of
proof that you are most
comfortable with.
w x
Given
/1 > /5
Corresponding Angle Postulate
/5 > /8
Vertical angles are congruent
/1 > /8
Transitive Property
© Carnegie Learning
You have just proven the Alternate Exterior Angle Conjecture. It is now known as the
Alternate Exterior Angle Theorem.
196
Chapter 2 Introduction to Proof
Grouping
Have students complete
Question 3 with a partner.
Then have students share their
responses as a class.
3. The Same-Side Interior Angle Conjecture states: “If two parallel lines are intersected by a
transversal, then interior angles on the same side of the transversal are supplementary.”
a. Draw and label a diagram illustrating the Same-Side Interior Angle Conjecture.
Then, write the given and prove statements.
Given: w i x, z is a transversal
Guiding Questions
for Share Phase,
Question 3
• How is the Corresponding
•
Which pair of Same-Side
interior angles did you
prove supplementary?
•
Can you prove a different
pair of Same-Side interior
angles supplementary using
the same strategy?
•
Do you need to prove both
pairs of Same-Side interior
angles supplementary to
prove this theorem?
2
5 6
7 8
x
Angle Postulate used to help
prove the Same-Side Interior
Angle Theorem?
Prove: /3 and /5 are
supplementary angles or /4 and
/6 are supplementary angles
1 2
3 4
w
z
b. Prove the Same-Side Interior Angle Conjecture.
Students may use a flow chart, two-column, or paragraph proof. The two-column
proof is provided. Students only need to use one method.
Statements
Reasons
1. w i x
1. Given
2. /1 and /3 are a linear pair
2. Definition of linear pair
3. /1 and /3 are supplementary
3. Linear Pair Postulate
4. m/1 1 m/3 5 1808
4. Definition of supplementary angles
5. /1 > /5
5. Corresponding Angle Postulate
6. m/1 5 m/5
6. Definition of congruent angles
7. m/3 1 m/5 5 1808
7. Substitution Property
8. /1 and /3 are supplementary angles 8. Definition of supplementary angles
You’re on
a roll! Only one
more conjecture
to go.
© Carnegie Learning
You have just proven the Same-Side Interior Angle
Conjecture. It is now known as the Same-Side Interior
Angle Theorem.
2.4
Angle Postulates and Theorems
197
Grouping
Have students complete
Question 4 with a partner.
Then have students share their
responses as a class.
4. The Same-Side Exterior Angle Conjecture states: “If two parallel lines are
intersected by a transversal, then exterior angles on the same side of the transversal
are supplementary.”
a. Draw and label a diagram illustrating the Same-Side Exterior Angle Conjecture.
Then, write the given and prove statements.
Given: w i x, z is a transversal
Guiding Questions
for Share Phase,
Question 4
• How is the Corresponding
2
Angle Postulate used to help
prove the Same-Side Exterior
Angle Theorem?
•
•
•
Which pair of Same-Side
exterior angles did you prove
supplementary?
Can you prove a different
pair of Same-Side exterior
angles supplementary using
the same strategy?
Do you need to prove both
pairs of Same-Side exterior
angles supplementary to
prove this theorem?
Prove: /1 and /7 are
supplementary angles or /2 and
/8 are supplementary angles
1 2
3 4
w
5 6
7 8
x
z
b. Prove the Same-Side Exterior Angle Conjecture.
Students may use a flow chart, two-column, or paragraph proof. The flow chart
proof is provided. Students only need to use one method.
w!x
Given
/1 and /3 are a linear pair
Definition of linear pair
/3 > /7
Corresponding Angle Postulate
/1 and /3 are supplementary
Linear Pair Postulate
m/3 > m/7
Definition of congruent angles
m/1 1 m/3 180°
Definition of supplementary angles
m/1 1 m/7 180°
Substitution Property
Note:
Suggest that students use
abbreviations for the theorem
names such as;
/1 and /7 are supplementary angles
Definition of supplementary angles
Alternate Interior Angle
Theorem: AIA Theorem
Same-Side Interior Angle
Theorem: SSIA Theorem
Same-Side Exterior Angle
Theorem: SSEA Theorem
198
Chapter 2 Introduction to Proof
You have just proven the Same-Side Exterior Angle Conjecture. It is now known as the
Same-Side Exterior Angle Theorem.
© Carnegie Learning
Alternate Exterior Angle
Theorem: AEA Theorem
Talk the Talk
The theorems in this lesson
are summarized in this activity.
Encourage students to write
these theorems in their
notebooks. Suggest they sketch
a small diagram beside each
theorem highlighting at least
one pair of angles depicted in
each theorem.
Talk the Talk
?
P
T
R
5
1 3
2 4
7
6
8
2
Given: m/4 5 37°
1. Gail determined the measures of all eight angles labeled using the given information.
Stu said she could only calculate the measure of four angles with certainty.
Who is correct? Explain your reasoning.
© Carnegie Learning
Stu is correct. Gail cannot determine the measures of angles 5, 6, 7, and 8 unless
it is stated that line P is parallel to line R.
2.4
Angle Postulates and Theorems
199
Guiding Questions for
Discuss Phase
• How many pairs of
corresponding angles exist
when two parallel lines are
intersected by a transversal?
•
2
•
•
•
How many pairs of alternate
interior angles exist when
two parallel lines are
intersected by a transversal?
How many pairs of alternate
exterior angles exist when
two parallel lines are
intersected by a transversal?
How many pairs of sameside interior angles exist
when two parallel lines are
intersected by a transversal?
How many pairs of sameside exterior angles exist
when two parallel lines are
intersected by a transversal?
If two parallel lines are intersected by a transversal, then:
r corresponding angles are congruent.
r alternate interior angles are congruent.
r alternate exterior angles are congruent.
r same-side interior angles are supplementary.
r same-side exterior angles are supplementary.
Each of these relationships is represented by a postulate or theorem.
r Corresponding Angle Postulate: If two parallel lines are intersected by a
transversal, then corresponding angles are congruent.
r Alternate Interior Angle Theorem: If two parallel lines are intersected by a
transversal, then alternate interior angles are congruent.
r Alternate Exterior Angle Theorem: If two parallel lines are intersected by a
transversal, then alternate exterior angles are congruent.
r Same-Side Interior Angle Theorem: If two parallel lines are intersected
by a transversal, then interior angles on the same side of the transversal
are supplementary.
r Same-Side Exterior Angle Theorem: If two parallel lines are intersected
by a transversal, then exterior angles on the same side of the transversal
are supplementary.
2. Did you use inductive or deductive reasoning to prove each theorem?
I used deductive reasoning because I used general rules, definitions, and postulates
about geometric relationships to make conclusions.
© Carnegie Learning
Be prepared to share your solutions and methods.
200
Chapter 2 Introduction to Proof