Download 16.4 Reasoning and Proof

LESSON
16.4
Reasoning and Proof
Name
Class
Date
16.4 Reasoning and Proof
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PAGE 647
BEGINS HERE
Essential Question: How do you go about proving a statement?
Common Core Math Standards
The student is expected to:
Resource
Locker
G-CO.9
Explore
Prove theorems about lines and angles. Also A-REI.A.1
Exploring Inductive and Deductive
Reasoning
A conjecture is a statement that is believed to be true. You can use reasoning to investigate whether
a conjecture is true. Inductive reasoning is the process of reasoning that a rule or statement
may be true by looking at specific cases. Deductive reasoning is the process of using
logic to prove whether all cases are true.
Mathematical Practices
MP.8 Patterns
Language Objective
Complete the steps to make a conjecture about the sum of three consecutive
counting numbers.
Have students fill in a chart explaining the meaning of conditionals,
counterexample and conditional.
A
Write a sum to represent the first three consecutive
counting numbers, starting with 1.
B
Is the sum divisible by 3?
Essential Question: How do you go
about proving a statement?
C
Write the sum of the next three consecutive counting
Possible answer: You can make a conjecture, or
statement, that you believe is true. Then through
inductive or deductive reasoning, you can prove the
statement is true by showing specific cases are true
or by using logical steps.
PREVIEW: LESSON
PERFORMANCE TASK
View the online Engage. Discuss the photo. Discuss
why the figure seems to be impossible. Then preview
the Lesson Performance Task.
© Houghton Mifflin Harcourt Publishing Company
ENGAGE
1+2+3
Yes. 1 + 2 + 3 = 6 and 6 ÷ 3 = 2.
numbers, starting with 2.
2+ 3+4
D
Is the sum divisible by 3?
Yes. 2 + 3 + 4 = 9 and 9 ÷ 3 = 3.
E
Complete the conjecture:
The
sum
of three consecutive counting numbers is divisible by
3
.
Recall that postulates are statements you accept are true. A theorem is a statement that you can
prove is true using a series of logical steps. The steps of deductive reasoning involve using
appropriate undefined words, defined words, mathematical relationships, postulates,
or other previously-proven theorems to prove that the theorem is true.
Use deductive reasoning to prove that the sum of three consecutive counting numbers
is divisible by 3.
F
Let the three consecutive counting numbers be represented by n, n + 1, and
G
The sum of the three consecutive counting numbers can be written as 3n +
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(
n+1
).
H
The expression 3n + 3 can be factored as 3
I
The expression 3(n + 1) is divisible by
J
Recall the conjecture in Step E: The sum of three consecutive counting numbers
is divisible by 3.
EXPLORE
for all values of n.
3
Look at the steps in your deductive reasoning. Is the conjecture true or false?
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True
QUESTIONING STRATEGIES
Reflect
1.
When a detective solves a case, is the detective
more likely to use inductive or deductive
reasoning? Explain. deductive reasoning, because
the solution is likely based on logical conclusions
drawn from the evidence
Discussion A counterexample is an example that shows a conjecture to be false. Do you think that
counterexamples are used mainly in inductive reasoning or in deductive reasoning?
Possible answer: A counterexample would be used in inductive reasoning to show that at
least one specific case makes the conjecture false.
2.
Exploring Inductive and Deductive
Reasoning
Suppose you use deductive reasoning to show that an angle is not acute. Can you conclude that the angle is
obtuse? Explain.
No; if the angle is not acute, I can conclude that it is right, obtuse, or straight.
Explain 1
When might you want to make a conjecture
about a set of numbers? If the numbers seem
to form a pattern, you might want to make a
conjecture based on the number pattern.
Is one counterexample enough to prove that a
conjecture is false? Explain. Yes, the
conjecture must be true for every case. So, if even
one counterexample exists, the conjecture is false.
Introducing Proofs
A conditional statement is a statement that can be written in the form “If p, then q” where p is the hypothesis
and q is the conclusion. For example, in the conditional statement “If 3x - 5 = 13, then x = 6,” the hypothesis is
“3x - 5 = 13” and the conclusion is “x = 6 .”
Properties of Equality
Addition Property of Equality
If a = b, then a + c = b + c.
Subtraction Property of Equality
If a = b, then a - c = b - c.
Multiplication Property of Equality
If a = b, then ac = bc.
Division Property of Equality
If a = b and c ≠ 0, then __ac = _bc .
Reflexive Property of Equality
a=a
Symmetric Property of Equality
If a = b, then b = a.
Transitive Property of Equality
If a = b and b = c, then a = c.
Substitution Property of Equality
If a = b, then b can be substituted for a in any expression.
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© Houghton Mifflin Harcourt Publishing Company
Most of the Properties of Equality can be written as conditional statements. You can use these properties to solve an
equation like “3x - 5 = 13” to prove that “x = 6 .”
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Discuss why a conjecture is like a hypothesis
in the scientific method. Elicit that a conjecture, like a
hypothesis, is often based on inductive reasoning.
EXPLAIN 1
Introducing Proofs
Lesson 4
AVOID COMMON ERRORS
PROFESSIONAL DEVELOPMENT
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Integrate Mathematical Practices
This lesson provides an opportunity to address Mathematical Practice MP.3,
which calls for students to “construct viable arguments.” Students use deductive
reasoning, and explain steps logically from definite premises to a definite general
conclusion. They use inductive reasoning to make a conjecture about what is true
in general by examining several cases, and they justify the falsehood of a
conclusion by citing a counterexample.
4/19/14 12:06 PM
Students may have difficulty identifying the correct
property of equality to justify a step when solving an
algebraic equation. Students may need to include
steps where they show the property of equality to
help them recognize how it is applied. For example,
they may need to show the step where the value is
added to both sides of the equation to apply the
Addition Property of Equality.
Reasoning and Proof 816
Will changing the order of the hypothesis and
the conclusion in a true conditional statement
change whether or not the statement is true?
Explain. Yes, the statement may still be true but
you would have to prove that it is.
Do you always use deductive reasoning when
you solve an equation algebraically?
Explain. Yes, you should always be able to support
each step in the solution using a property.
Use deductive reasoning to solve the equation. Use the Properties of
Equality to justify each step.
Example 1
QUESTIONING STRATEGIES
14 = 3x - 4

14 = 3x - 4
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18 = 3x
Addition Property of Equality
6=x
Division Property of Equality
x= 6
Symmetric Property of Equality
9 = 17 - 4x

BEGINS HERE
9 = 17 - 4x
Subtraction Property of Equality
9 - 17 = -4x
-8
= -4x
Division
=x
2
x=
2
Property of Equality
Symmetric Property of Equality
Your Turn
Write each statement as a conditional.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Digital
Vision/Getty Images
3.
4.
All zebras belong to the genus Equus.
If an animal is a zebra, then it belongs to the genus Equus.
The bill will pass if it gets two-thirds of the vote in the Senate.
If the bill gets two-thirds of the vote in the Senate,
then it will pass.
5.
Use deductive reasoning to solve the equation 3 - 4x = -5.
3 - 4x = -5
-4x = -8
x=2
6.
Subtraction Property of Equality
Division Property of Equality
Identify the Property of Equality that is used in each statement.
If x = 2, then 2x = 4.
Multiplication Property of Equality
5 = 3a; therefore, 3a = 5 .
Symmetric Property of Equality
If T = 4, then 5T + 7 equals 27.
Substitution Property of Equality
If 9 = 4x and 4x = m, then 9 = m.
Transitive Property of Equality
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Lesson 4
COLLABORATIVE LEARNING
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Small Group Activity
Have students work in small groups. The first student writes a number or draws a
figure. The next student writes or draws a second item, beginning a pattern. Have
them continue until each student has contributed to the pattern. Then ask the first
student to describe a rule for the pattern. Have the groups repeat this activity until
each student has gone first.
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Explain 2
Using Postulates about Segments and Angles
EXPLAIN 2
Recall that two angles whose measures add up to 180° are called supplementary angles. The following theorem shows
one type of supplementary angle pair, called a linear pair. A linear pair is a pair of adjacent angles whose noncommon sides are opposite rays. You will prove this theorem in an exercise in this lesson.
Using Postulates about Segments and
Angles
The Linear Pair Theorem
If two angles form a linear pair, then they are supplementary.
4
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Have students identify the property of
3
m∠3 + m∠4 = 180°
You can use the Linear Pair Theorem, as well as the Segment Addition Postulate and Angle Addition Postulate, to find
missing values in expressions for segment lengths and angle measures.
Example 2

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equality used as they complete each step to solve for
the variable using the postulates about segments and
angles.
Use a postulate or theorem to find the value of x in each figure.
Given: RT = 5x - 12
x+ 2
R
QUESTIONING STRATEGIES
3x - 8
S
5x - 12
How are the segment and angle addition
postulates applied to solve for a variable? Set
the sum of the non-overlapping segments or angles
equal to the measure of the whole segment or angle.
T
Use the Segment Addition Postulate.
RS + ST = RT
(x + 2) + (3x - 8) = 5x - 12
6=x
x=6
Module 16
AVOID COMMON ERRORS
© Houghton Mifflin Harcourt Publishing Company
4x - 6 = 5x - 12
818
When solving equations using the Segment or Angle
Addition Postulates, students may forget to combine
like terms or use inverse operations to solve. Review
how to combine like terms and use inverse operations
as needed.
Lesson 4
DIFFERENTIATE INSTRUCTION
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Kinesthetic Experience
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Have students act out the Reflexive, Symmetric, and Transitive Properties. For the
Reflexive Property, have students look in a mirror. For the Symmetric Property,
have two students stand next to each other and then change places. For the
Transitive Property, have one student give a second student a sheet of paper, and
have the second student give the paper to a third. The result is the same as if the
first student had given the paper directly to the third.
Reasoning and Proof 818
Given: m∠RST = (15x - 10)°
B
R
P
(x + 25)°
(5x + 10)°
S
T
Use the Angle Addition Postulate.
m∠RST = m∠ RSP
+ m∠ PST
(15x − 10)° = (x + 25) ° + (5x + 10)°
15x − 10 = (6x + 35)
9
x=
45
x=
5
Reflect
7.
Discussion The Linear Pair Theorem uses the terms opposite rays as well as adjacent angles. Write a
definition for each of these terms. Compare your definitions with your classmates.
Possible answers: Opposite rays are rays that share a common endpoint and form a line.
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Adjacent angles are two angles in the same plane with a common vertex and a common
side, but no common interior points.
Your Turn
8.
Two angles LMN and NMP form a linear pair. The measure of ∠LMN is twice the measure of ∠NMP.
Find m∠LMN.
Use the Linear Pair Theorem. Substitute for m∠LMN.
m∠LMN + m∠NMP = 180°
(2 ⋅ m∠NMP) + m∠NMP = 180°
3 ⋅ m∠NMP = 180°
m∠NMP = 60°
m∠LMN = 2 ⋅ m∠NMP = 2 ⋅ 60° = 120°
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LANGUAGE SUPPORT
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Connect Vocabulary
For the properties of equality based on the operations, have students highlight the
operation to connect to the corresponding property. For the Reflexive, Symmetric,
and Transitive Properties, discuss what these words bring to mind. For example,
reflexive might remind students of a reflection in a mirror. You see the same thing
on both sides of a mirror, so, a = a.
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Explain 3
Using Postulates about Lines and Planes
EXPLAIN 3
Postulates about points, lines, and planes help describe geometric figures.
Postulates about Points, Lines, and Planes
Using Postulates about Lines and
Planes
Through any two points, there is exactly one line.
COOPERATIVE LEARNING
Through any three noncollinear points, there is exactly one plane
containing them.
To help students understand the rationale behind the
postulates about points, lines, and planes, ask them to
draw additional examples with the points and lines in
different locations to demonstrate each postulate.
Share the drawings with the class.
If two points lie in a plane, then the line containing those points lies in the plane.
If two lines intersect, then they intersect in exactly one point.
If two planes intersect, then they intersect in exactly one line.
© Houghton Mifflin Harcourt Publishing Company
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Lesson 4
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Reasoning and Proof 820
QUESTIONING STRATEGIES
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Example 3
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BEGINS HERE
Use each figure to name the results described.

Must any two planes intersect? Why or why
not? Name planes in the classroom that
support your answer. No, if the planes are parallel
they will never intersect. Possible example: opposite
walls in the classroom
A
B
E
C
D


If a line lies in a plane, how many points of
intersection do the line and the plane
have? an infinite number: every point that lies on
the line
Description
VISUAL CUES
Students may have difficulty interpreting the
diagrams showing intersecting planes. Make a slit in
the side of one piece of paper and hold the paper in a
horizontal plane. Slide a second sheet in a vertical
plane perpendicular to the horizontal plane to
provide students with a visual demonstration of two
intersecting planes. Locate points and lines as needed.
Example from the figure
the line of intersection of two planes
Possible answer: The two planes intersect
in line BD.
the point of intersection of two lines
The line through point A and the line through
point B intersect at point C.
three coplanar points
Possible answer: The points B, D, and E are
coplanar.
three collinear points
The points B, C, and D are collinear.

m

F
H
© Houghton Mifflin Harcourt Publishing Company
J

Description
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Lesson 16.4
ℓ
G
Example from the figure
the line of intersection of two planes
Possible answer: The two planes
intersect in line JF.
the point of intersection of two lines
The line through point F and the line
through point H intersect at point J.
three coplanar points
Possible answer: The points F, J, and H
are coplanar.
three collinear points
The points F, J, and G are collinear.
821
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Reflect
9.
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BEGINS HERE
Find examples in your classroom that illustrate the postulates of lines, planes, and points.
Possible answers: walls, floors, corners, desktops, blackboard
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Point out that students are starting to build a
10. Draw a diagram of a plane with three collinear points and three points that are noncollinear.
Possible answer: in the figure, B, C, and A are collinear; D, C, and A are noncollinear.
B
D
C

ELABORATE
A
catalog of definitions, postulates, and theorems about
geometric relationships that they will use throughout
the course. Discuss why it is important to become
familiar with using them as tools in deductive
reasoning.
Elaborate
11. What is the difference between a postulate and a definition? Give an example of each.
Possible answers: A postulate is a statement that is self-evident or is generally accepted to
be a true statement.
A definition is a statement that explains the meaning of a word in terms of previously
accepted words or statements.
QUESTIONING STRATEGIES
Possible examples:
How can you use a Property of Equality to
write the equation 4x = 8 as a conditional
statement? Using the Division Property of Equality,
x = 2. The conditional statement is “If 4x = 8,
then x = 2.”
Postulate: x = x is called the Reflexive Property of Equality.
Definition: An even number is a number that is divisible by 2.
12. Give an example of a diagram illustrating the Segment Addition Postulate. Write the
Segment Addition Postulate as a conditional statement.
R
S
T
If S is between R and T, then RS + ST = RT.
CONNECT VOCABULARY
© Houghton Mifflin Harcourt Publishing Company
13. Explain why photographers often use a tripod when taking pictures.
Through any three noncollinear points,
there is only one plane, so the feet of the
tripod are all always flat against the plane
of the ground, which steadies the camera.
14. Essential Question Check-In What are some of the reasons you can
give in proving a statement using deductive reasoning?
You can use given facts, definitions, postulates or properties, and
Connect conditionals in geometry to conditional
statements in everyday life. For example: “If I walk in
the rain without an umbrella, then I will get wet.”
Have students express their own conditional
statements tied to reality so that they can see this
connection.
previously-proven theorems.
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SUMMARIZE THE LESSON
822
Lesson 4
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How can you use deductive reasoning to
establish a conclusion? Provide a logical
sequence of statements supported by postulates
and established theorems in which each statement
logically follows from the preceding statement up
to the conclusion.
Reasoning and Proof 822
Evaluate: Homework and Practice
EVALUATE
• Online Homework
• Hints and Help
• Extra Practice
Explain why the given conclusion uses inductive reasoning.
1.
Find the next term in the pattern: 3, 6, 9.
The next term is 12 because the previous terms are multiples of 3.
The conclusion is based on observing three numbers.
2.
ASSIGNMENT GUIDE
3 + 5 = 8 and 13 + 5 = 18, therefore the sum of two odd numbers is an
even number.
The conclusion is based on two examples.
Concepts and Skills
Practice
Explore
Exploring Inductive and Deductive
Reasoning
Exercises 1–12
Example 1
Introducing Proofs
Exercises 13–16
Example 2
Using Postulates about Segments
and Angles
Exercises 17–20
Example 3
Using Postulates about Lines and
Planes
Exercises 21–26
3.
My neighbor has two cats and both cats have yellow eyes.
Therefore when two cats live together, they will both have yellow eyes.
The conclusion is based on two observations.
4.
The conclusion is based on a limited number of observations.
Give a counterexample for each conclusion.
5.
If x is a prime number, then x + 1 is not a prime number.
Counterexample: 2; if x = 2, then x + 1 = 3, which is a prime number.
6.
The difference between two even numbers is positive.
Counterexample: 6 − 10 = −4, which is negative.
7.
Points A, B, and C are noncollinear, so therefore they are noncoplanar.
When I draw three points that are noncollinear, I can draw a single plane
through all three points, so they are coplanar after all.
Have students compare counterexamples used to
demonstrate conjectures that are not true.
GRAPHIC ORGANIZERS
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© Houghton Mifflin Harcourt Publishing Company
COMMUNICATING MATH
Students work in pairs to complete a chart. The chart
has three columns and each column is labeled with
the highlighted vocabulary. Students must discuss in
depth the meaning of each word, and take notes on
their discussion. Then they write down their
agreed-upon ideas under the word in each column.
It always seems to rain the day after July 4th.
8.
The square of a number is always greater than the number.
1 __
1
The square of __
is 19 , which is less than __
.
3
3
In Exercises 9–12 use deductive reasoning to write a conclusion.
9.
If a number is divisible by 2, then it is even.
The number 14 is divisible by 2.
The number 14 is an even number.
Module 16
Exercise
IN1_MNLESE389762_U7M16L4 823
Lesson 16.4
Depth of Knowledge (D.O.K.)
Mathematical Practices
1–6
1 Recall of Information
MP.3 Logic
7–8
2 Skills/Concepts
MP.3 Logic
9–16
1 Recall of Information
MP.3 Logic
17–20
1 Recall of Information
MP.2 Reasoning
21–22
1 Recall of Information
MP.3 Logic
23–24
2 Skills/Concepts
MP.3 Logic
3 Strategic Thinking
MP.3 Logic
25
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Use deductive reasoning to write a conclusion.
AVOID COMMON ERRORS
10. If two planes intersect, then they intersect in exactly one line.
Planes ℜ and ℑ intersect.
Students may have difficulty identifying the
hypothesis and the conclusion given a conditional
statement when then is missing from the statement.
Suggest that students first rewrite the statement in the
“if, . . . then” form before they identify the hypothesis
and conclusion.
Planes ℜ and ℑ intersect in exactly one line.
11. Through any three noncollinear points, there is exactly one plane containing them.
Points W, X, and Y are noncollinear.
There is exactly one plane containing points W, X, and Y.
12. If the sum of the digits of an integer is divisible by 3, then the number is divisible by 3.
The sum of the digits of 46,125 is 18, which is divisible by 3.
The number 46,125 is divisible by 3.
AVOID COMMON ERRORS
Identify the hypothesis and conclusion of each statement.
Remind students to combine like terms when they
solve equations by applying the Segment and Angle
Addition Postulates.
13. If the ball is red, then it will bounce higher.
Hypothesis: the ball is red
Conclusion: it will bounce higher
14. If a plane contains two lines, then they are coplanar.
Hypothesis: a plane contains two lines Conclusion: the lines are coplanar
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 After students solve for the variable using the
15. If the light does not come on, then the circuit is broken.
Hypothesis: the light does not come on
Conclusion: the circuit is broken
16. You must wear your jacket if it is cold outside.
Hypothesis: it is cold outside
Conclusion: you must wear your jacket
Segment and Angle Addition Postulates, have them
use the value to find the actual lengths or angle
measures represented by the expressions. They can
also use this method to check their work.
Use a definition, postulate, or theorem to find the value of x in the figure described.
17. Point E is between points D and F. If DE = x - 4, EF = 2x + 5, and DF = 4x − 8, find x.
© Houghton Mifflin Harcourt Publishing Company
Use the Segment Addition Postulate; DE + EF = DF; (x − 4) + (2x + 5) = 4x − 8;
3x + 1 = 4x − 8; 9 = x
_
18. Y is the midpoint of XZ. If XZ = 8x − 2 and YZ = 2x + 1, find x.
―
Because Y is the midpoint of XZ, XY = YZ. Use this fact and the Segment
Addition Postulate; XY + YZ = XZ; (2x + 1) + (2x + 1) = 8x − 2;
4x + 2 = 8x − 2; 4 = 4x; 1 = x
→
‾ is an angle bisector of ∠RST. If m∠RSV = (3x + 5)° and m∠RST = (8x − 14)°, find x.
19. SV
→
‾ is an angle bisector of ∠RST, m∠RSV = m∠VST. Use this fact and
Because SV
the Angle Addition Postulate; m∠RSV + m∠VST = m∠RST;
(3x + 5) + (3x + 5) = 8x - 14; 6x + 10 = 8x - 14; 24 = 2 x; 12 = x
20. ∠ABC and ∠CBD are a linear pair. If m∠ABC = m∠CBD = 3x - 6, find x.
Use the Linear Pair Theorem.; m∠ABC + m∠CBD = 180°;
(3x - 6)°+ (3x - 6)° = 180°; 6x - 12 = 180; 6x = 192; x = 32
Module 16
Exercise
IN1_MNLESE389762_U7M16L4 824
Lesson 4
824
Depth of Knowledge (D.O.K.)
Mathematical Practices
26
2 Skills/Concepts
MP.3 Logic
27
3 Strategic Thinking
MP.3 Logic
28
2 Skills/Concepts
MP.6 Precision
15/09/14 11:59 PM
Reasoning and Proof 824
Use the figure for Exercises 21 and 22.
PEERTOPEER DISCUSSION
21. Name three collinear points.
Have students use a ruler to draw a line segment
made from two non-overlapping segments. Have
them label the lengths of one section and the total
segment. Then have them exchange papers and
explain how to use deductive reasoning to find the
missing length.
S
Possible answer: P, R, and T
P
R
22. Name two linear pairs.
Possible answers: ∠PRQ and ∠QRT, ∠STR and ∠UTR
Q
T
U
Explain the error in each statement.
23. Two planes can intersect in a single point.
When two planes cross, they intersect each other at an infinite number
of points, i.e., in a line.
JOURNAL
24. Three points have to be collinear.
Have students write examples of the following: a
conjecture, a counterexample to a conjecture,
inductive reasoning, and deductive reasoning.
The three points could be the vertices of a triangle.
25. A line is contained in exactly one plane
A line can be in more than one plane.
26. If x 2 = 25, then x = 5.
The value of x could also be -5.
HARDBOUND SE
H.O.T. Focus on Higher Order Thinking
PAGE 655
© Houghton Mifflin Harcourt Publishing Company
BEGINS HERE
27. Analyze Relationships What is the greatest number of intersection points 4 coplanar lines can have?
What is the greatest number of planes determined by 4 noncollinear points?
Draw diagrams to illustrate your answers.
Four coplanar lines can intersect in up to 6 points.
Module 16
IN1_MNLESE389762_U7M16L4 825
825
Lesson 16.4
825
Up to four planes can be determined by
4 noncollinear points.
Lesson 4
4/19/14 12:06 PM
28. Justify Reasoning Prove the Linear Pair Theorem.
Given: ∠MJK and ∠MJL are a linear pair of angles.
Prove: ∠MJK and ∠MJL are supplementary.
LANGUAGE SUPPORT
M
K
Complete the proof by writing the missing reasons.
Choose from the following reasons.
J
Angle Addition Postulate
Definition of linear pair
Substitution Property of Equality
Given
Statements
A conjecture is an opinion or proposition that is
supported by evidence but has not been proven. It
begins with an observation, such as: “I added twenty
pairs of odd numbers, and the sums were all even.”
The conjecture based on this might be, “The sum of
two odd numbers is always even.” A conjecture
doesn’t have to be true. For example: “I saw eleven
kids with apples at lunch today, and all the apples
were red. Conjecture: All apples are red.”
L
Reasons
1. ∠MJK and ∠MJL are a linear pair.
→
→
‾ are opposite rays.
‾ and JK
2. JL
→
→
‾ and JK
‾ form a straight line.
3. JL
1. Given
4. m∠LJK = 180°
4. Definition of straight angle
5. m∠MJK + m∠MJL = m∠LJK
5. Angle Addition Postulate
6. m∠MJK + m∠MJL = 180°
6. Substitution Property of Equality
7. ∠MJK and ∠MJL are supplementary.
7. Definition of supplementary angles
2. Definition of linear pair
3. Definition of opposite rays
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Ask students to state a possible conjecture for
Lesson Performance Task
HARDBOUND SE
PAGE 656
BEGINS HERE
If two planes intersect, then they intersect in exactly one line.
Find a real-world example that illustrates the postulate above. Then formulate a conjecture by completing the
following statement:
.
Justify your conjecture with real-world examples or a drawing.
Possible example: a wall and the ceiling
Students may make any of the following conjectures:
If three planes intersect, then they intersect in a point.
If three planes intersect, then they intersect in a line.
If three planes intersect, then they intersect in either a point or a line.
Obviously the third conjecture is the most complete. Possible examples: two
walls and the ceiling intersect at a point; the pages of a book are planes that
intersect in a line, the spine of the book.
Module 16
826
“I’ve seen thousands of creatures with wings and all
of them could fly.” Conjecture: If a creature has
wings, then it can fly. Refute: Penguins and ostriches
have wings but cannot fly.
“I’ve seen hundreds of bicycles and all of them had
two wheels.” Conjecture: If a vehicle is a bicycle,
then it has two wheels. Support: The prefix bimeans two, and cycle refers to a wheel. So, the
definition of bicycle includes the requirement that
the vehicle have two wheels.
© Houghton Mifflin Harcourt Publishing Company
If three planes intersect, then
the statement, then support or refute it:
Lesson 4
EXTENSION ACTIVITY
IN1_MNLESE389762_U7M16L4 826
The “impossible” triangle in the photo in the online Engage activity is an example
of an optical illusion. Have students research optical illusions and either sketch or
print ones that especially intrigue them. For each illusion, students should
describe what’s intriguing about it, and then explain—if they can!—how the
illusion is accomplished.
4/19/14 12:06 PM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Reasoning and Proof 826