Download Proof, Parallel and Perpendicular Lines

Unit
Proof, Parallel and 1
Perpendicular Lines
Unit Overview
In this unit you will begin the study of an axiomatic
system, Geometry. You will investigate the concept
of proof and discover the importance of proof in
mathematics. You will extend your knowledge of the
characteristics of angles and parallel and perpendicular
lines and explore practical applications involving
angles and lines.
Essential Questions
?
Why are properties,
postulates, and theorems
important in mathematics?
?
How are angles and parallel
and perpendicular lines used
in real-world settings?
© 2010 College Board. All rights reserved.
Academic Vocabulary
Add these words and others you encounter in this unit
to your Math Notebook.
angle bisector
midpoint of a segment
complementary angles
parallel
conditional statement
perpendicular
congruent
postulate
conjecture
proof
counterexample
supplementary angles
deductive reasoning
theorem
inductive reasoning
EMBEDDED
ASSESSMENTS
These assessments, following
Activities 1.4, 1.7, and 1.9, will give
you an opportunity to demonstrate
what you have learned about
reasoning, proof, and some basic
geometric figures.
Embedded Assessment 1
Conditional Statements and Logic
p. 37
Embedded Assessment 2
Angles and Parallel Lines
p. 65
Embedded Assessment 3
Slope, Distance, and Midpoint
p. 81
1
UNIT 1
Getting
Ready
1. Solve each equation.
a. 5x - 2 = 8
b. 4x - 3 = 2x + 9
c. 6x + 3 = 2x + 8
6. Give the characteristics of a rectangular prism.
7. Draw a right triangle and label the hypotenuse
and legs.
8. Use a protractor to find the measure of each
angle.
a.
2. Graph y = 3x - 3 and label the x- and
y-intercepts.
3. Tell the slope of a line that contains the points
(5, -3) and (7, 3).
1
4. Write the equation of a line that has slope __
3
and y-intercept 4.
b.
5. Write the equation of the line graphed below.
10
y
8
c.
6
4
2
–10 –8 –6 –4 –2
–2
2
4
6
8
10
x
–4
–6
–8
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–10
2
SpringBoard® Mathematics with Meaning™ Geometry
Geometric Figures
What’s My Name?
ACTIVITY
1.1
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Interactive Word Wall, Activating Prior Knowledge,
Group Presentation
Below are some types of figures you have seen in earlier mathematics
courses. Describe each figure. Using geometric terms and symbols, list as
many names as possible for each figure.
1. Q
2. F
3. X Y
5.
G
4. D
Z
m
6.
E
J
K
© 2010 College Board. All rights reserved.
7.
H
χ
P
T
N
A
D
B
C
TRY THESE A
Identify each geometric figure. Then use symbols to write two different
names for each.
a. Q
P
b.
N
C
Unit 1 • Proof, Parallel and Perpendicular Lines
3
ACTIVITY 1.1
Geometric Figures
continued
What’s My Name?
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Interactive Word Wall, Activating Prior Knowledge,
Discussion Group, Self/Peer Revision
My Notes
8. Draw four angles with different characteristics. Describe each angle.
Name the angles using numbers and letters.
9. Compare and contrast each pair of angles.
When you compare and contrast
two figures, you describe how
they are alike or different.
4
a.
b.
1
SpringBoard® Mathematics with Meaning™ Geometry
A
2
D
B
C
© 2010 College Board. All rights reserved.
MATH TERMS
Geometric Figures
ACTIVITY 1.1
continued
What’s My Name?
SUGGESTED LEARNING STRATEGIES: Discussion Group,
Peer/Self Revision
My Notes
c.
R
D
60°
30°
E
d.
F
D
T
150°
X
30°
E
S
F
Y
Z
10. a. The figure below shows two intersecting lines. Name two angles
that are supplementary to ∠4.
b. Explain why the angles you named in part a must have the same
measure.
3
4
5
© 2010 College Board. All rights reserved.
6
Unit 1 • Proof, Parallel and Perpendicular Lines
5
ACTIVITY 1.1
Geometric Figures
continued
What’s My Name?
SUGGESTED LEARNING STRATEGIES: Discussion Group,
Create Representations, Activating Prior Knowledge
My Notes
TRY THESE B
Complete the chart by naming all the listed angle types in each figure.
C
8 7
10 9
A
D
B
E
F
Acute angles
Obtuse angles
Angles with the
same measure
Supplementary
angles
11. In the circle below, draw and label each geometric term.
a. Radius OA
b. Chord BA
c. Diameter CA
12. Refer to your drawings in circle above. What is the geometric term for
point O?
6
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Complementary
angles
Geometric Figures
ACTIVITY 1.1
continued
What’s My Name?
SUGGESTED LEARNING STRATEGIES: Discussion Group,
Create Representations, Activating Prior Knowledge
My Notes
13. In the space below, draw a circle with center P and radius
1 in. Locate a point B
PQ = 1 in. Locate a point A so that PA = 1__
2
3 in.
so that PB = __
4
14. Use your diagram to complete these statements.
a. A lies _________________ the circle
because ___________________________________.
b. B lies _________________ the circle
because ___________________________________.
© 2010 College Board. All rights reserved.
15. In your diagram above, draw circles with radii PA and PB.
These three circles are called _______________________.
Unit 1 • Proof, Parallel and Perpendicular Lines
7
ACTIVITY 1.1
Geometric Figures
continued
What’s My Name?
My Notes
TRY THESE C
Classify each segment in circle O. Use all terms that apply.
___
AF : ________________________
___
A
BO : ________________________
___
B
C
CO : ________________________
___
DO : ________________________
O
___
EO : ________________________
___
CE : ________________________
F
E
D
CHECK YOUR UNDERSTANDING
Writeyour
youranswers
answersononnotebook
notebook
paper.
Show
Write
paper.
Show
youryour work.
6. Compare and contrast the terms acute angle,
work.
obtuse angle, right angle, and straight angle.
2. Describe all acceptable ways to name a plane.
3. Compare and contrast collinear and coplanar
points.
4. What are some acceptable ways to name an
angle?
5. a. Why is there a problem with using ∠B to
name the angle below?
b. How many angles are in the figure?
A
C
D
B
8
SpringBoard® Mathematics with Meaning™ Geometry
7. The measure of ∠A is 42°.
a. What is the measure of an angle that is
complementary to ∠A?
b. What is the measure of an angle that is
supplementary to ∠A?
8. Draw a circle P.
a. Draw a segment that has one endpoint on
the circle but is not a chord.
b. Draw a segment that intersects the circle in
two points, contains the center, but is not a
radius, diameter, or chord.
9. MATHEMATICAL How can you use the
R E F L E C T I O N figures in this activity to
describe real-world objects and situations?
Give examples.
© 2010 College Board. All rights reserved.
1. Describe all acceptable ways to name a line.
Logical Reasoning
Riddle Me This
ACTIVITY
1.2
SUGGESTED LEARNING STRATEGIES: Interactive Word Wall,
Vocabulary Organizer, Think/Pair/Share, Look for a Pattern
My Notes
The ability to recognize patterns is an important aspect of mathematics.
But when is an observed pattern actually a real pattern that continues
beyond just the observed cases? In this activity, you will explore patterns
and check to see if the patterns hold true beyond the observed cases.
Inductive reasoning is the process of observing data, recognizing
patterns, and making a generalization. This generalization is a conjecture.
1. Five students attended a party and ate a variety of foods. Something
caused some of them to become ill. JT ate a hamburger, pasta salad,
and coleslaw. She became ill. Guy ate coleslaw and pasta salad but not
a hamburger. He became ill. Dean ate only a hamburger and felt fine.
Judy didn’t eat anything and also felt fine. Cheryl ate a hamburger
and pasta salad but no coleslaw, and she became ill. Use inductive
reasoning to make a conjecture about which food probably caused
the illness.
ACADEMIC VOCABULARY
conjecture
inductive reasoning
2. Use inductive reasoning to make a conjecture about the next two
terms in each sequence. Explain the pattern you used to determine
the terms.
© 2010 College Board. All rights reserved.
a. A, 4, C, 8, E, 12, G, 16,
b. 3, 9, 27, 81, 243,
,
,
CONNECT TO HISTORY
c. 3, 8, 15, 24, 35, 48,
d. 1, 1, 2, 3, 5, 8, 13,
,
,
The sequence of numbers in
Item 2d is named after Leonardo
of Pisa, who was known as
Fibonacci. Fibonacci’s 1202
book Liber Abaci introduced the
sequence to Western European
mathematics.
Unit 1 • Proof, Parallel and Perpendicular Lines
9
ACTIVITY 1.2
Logical Reasoning
continued
Riddle Me This
SUGGESTED LEARNING STRATEGIES: Look for a Pattern,
Group Presentation
My Notes
3. Use the four circles on the next page. From each of the given points,
draw all possible chords. These chords will form a number of
non-overlapping regions in the interior of each circle. For each circle,
count the number of these regions. Then enter this number in the
appropriate place in the table below.
Number of Points
on the Circle
2
3
4
5
Number of Non-Overlapping
Regions Formed
4. Look for a pattern in the table above.
a. Describe, in words, any patterns you see for the numbers in the
column labeled Number of Non-Overlapping Regions Formed.
© 2010 College Board. All rights reserved.
b. Use the pattern that you described to predict the number of
non-overlapping regions that will be formed if you draw all
possible line segments that connect six points on a circle.
10
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Logical Reasoning
ACTIVITY 1.2
continued
Riddle Me This
SUGGESTED LEARNING STRATEGIES: Visualization, Look
for a Pattern
© 2010 College Board. All rights reserved.
My Notes
Unit 1 • Proof, Parallel and Perpendicular Lines
11
ACTIVITY 1.2
Logical Reasoning
continued
Riddle Me This
SUGGESTED LEARNING STRATEGIES: Discussion Group
My Notes
5. Use the circle on the next page. Draw all possible chords connecting
any two of the six points.
a. What is the number of non-overlapping regions formed by chords
connecting the points on this circle?
b. Is the number you obtained above the same number you
predicted from the pattern in the table in Item 4b?
d. Try to find a new pattern for predicting the number of regions
formed by chords joining points on a circle when two, three, four,
five, and six points are placed on a circle. Describe the pattern
algebraically or in words.
12
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c. Describe what you would do to further investigate the pattern in
the number of regions formed by chords joining n points on a
circle, where n represents any number of points placed on a circle.
Logical Reasoning
ACTIVITY 1.2
continued
Riddle Me This
SUGGESTED LEARNING STRATEGIES: Visualization, Look
for a Pattern
© 2010 College Board. All rights reserved.
My Notes
Unit 1 • Proof, Parallel and Perpendicular Lines
13
ACTIVITY 1.2
Logical Reasoning
continued
Riddle Me This
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share
My Notes
6. Write each sum.
1+3=
1+3+5=
1+3+5+7=
1+3+5+7+9=
1 + 3 + 5 + 7 + 9 + 11 =
7. Look for a pattern in the sums in Item 6.
a. Describe algebraically or in words the pattern you see for the
sums in Item 6.
c. Check the patterns in Items 7a and 7b to see if they predict the
next few sums beyond the list of sums given in Item 6. Record or
explain your findings below.
14
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b. Compare your description above with the descriptions that others
in your group or class have written. Below, record any descriptions
that were different from yours.
Logical Reasoning
ACTIVITY 1.2
continued
Riddle Me This
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Interactive Word Wall, Marking the Text, Predict
and Confirm, Look for a Pattern, Group Presentation,
Discussion Group
My Notes
8. Consider this array of squares.
A
B
C
D
E
F
B
B
C
D
E
F
C
C
C
D
E
F
D
D
D
D
E
F
E
E
E
E
E
F
F
F
F
F
F
F
© 2010 College Board. All rights reserved.
a. Round 1: Shade all the squares labeled A.
How many A squares did you shade?
How many total squares are shaded?
b. Round 2: Shade all the squares labeled B.
How many B squares did you shade?
How many total squares are shaded?
c. Complete the table for letters C, D, E, and F.
Letter
Round 1: A
Round 2: B
Round 3: C
Round 4: D
Round 5: E
Round 6: F
Squares Shaded in
this Round
1
3
Total Number of
Shaded Squares
1
4
Unit 1 • Proof, Parallel and Perpendicular Lines
15
ACTIVITY 1.2
Logical Reasoning
continued
Riddle Me This
My Notes
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Interactive Word Wall, Marking the Text, Predict
and Confirm, Look for a Pattern, Group Presentation,
Discussion Group
9. Think about the process you used to complete the table for Item 8c.
Use your findings in Item 8 to answer the following questions.
a. One method of describing the pattern in Item 6 would be to say,
“The new sum is always the next larger perfect square.” Analyze
the process you used in Item 8 and then write a convincing
argument that this pattern description will continue to hold true
as the array is expanded and more squares are shaded.
© 2010 College Board. All rights reserved.
b. Another method of describing the pattern in Item 6 would be
to say, “The new sum is always the result of adding the next odd
integer to the preceding sum.” Analyze the process you used
in Item 8 and write a convincing argument that this pattern
description will continue to hold true as the array is expanded and
more squares are shaded.
16
SpringBoard® Mathematics with Meaning™ Geometry
Logical Reasoning
ACTIVITY 1.2
continued
Riddle Me This
SUGGESTED LEARNING STRATEGIES: Interactive Word
Wall, Vocabulary Organizer, Marking the Text
My Notes
In this activity, you have described patterns and made conjectures about
how these patterns would extend beyond your observed cases, based
on collected data. Some of your conjectures were probably shown to be
untrue when additional data were collected. Other conjectures took on a
greater sense of certainty as more confirming data were collected.
In mathematics, there are certain methods and rules of argument that
mathematicians use to convince someone that a conjecture is true, even
for cases that extend beyond the observed data set. These rules are called
rules of logical reasoning or rules of deductive reasoning. An argument
that follows such rules is called a proof. A statement or conjecture that
has been proven, that is, established as true without a doubt, is called a
theorem. A proof transforms a conjecture into a theorem.
Below are some definitions from arithmetic.
Even integer: An integer that has a remainder of 0 when it is divided
by 2.
Odd integer: An integer that has a remainder of 1 when it is divided
by 2.
In the following items, you will make some conjectures about the
sums of even and odd integers.
ACADEMIC VOCABULARY
deductive reasoning
proof
theorem
© 2010 College Board. All rights reserved.
10. Calculate the sum of some pairs of even integers. Show the examples
you use and make a conjecture about the sum of two even integers.
11. Calculate the sum of some pairs of odd integers. Show the examples
you use and make a conjecture about the sum of two odd integers.
12. Calculate the sum of pairs of integers consisting of one even integer and one odd integer. Show the examples you use and make a
conjecture about the sum of an even integer and an odd integer.
Unit 1 • Proof, Parallel and Perpendicular Lines
17
ACTIVITY 1.2
Logical Reasoning
continued
Riddle Me This
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Look for a Pattern, Use Manipulatives
My Notes
The following items will help you write a convincing argument (a proof)
that supports each of the conjectures you made in Items 10–12.
13. Figures A, B, C, and D are puzzle pieces. Each figure represents an
integer determined by counting the square pieces in the figure. Use
these figures to answer the following questions.
Figure A
Figure B
Figure C
Figure D
a. Which of the figures can be used to model an even integer?
c. Compare and contrast the models of even and odd integers.
18
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© 2010 College Board. All rights reserved.
b. Which of the figures can be used to model an odd integer?
Logical Reasoning
ACTIVITY 1.2
continued
Riddle Me This
SUGGESTED LEARNING STRATEGIES: Use Manipulatives,
Discussion Group
My Notes
14. Which pairs of puzzle pieces can fit together to form rectangles? Make
sketches to show how they fit.
© 2010 College Board. All rights reserved.
15. Explain how the figures (when used as puzzle pieces) can be used to
show that each of the conjectures in Items 10 through 12 is true.
In Items 14 and 15, you proved the conjectures in Items 10 through
12 geometrically. In Items 16 through 18, you will prove the same
conjectures algebraically.
Unit 1 • Proof, Parallel and Perpendicular Lines
19
ACTIVITY 1.2
Logical Reasoning
continued
Riddle Me This
SUGGESTED LEARNING STRATEGIES: Discussion Group
My Notes
This is an algebraic definition of even integer: An integer is even if and
only if it can be written in the form 2p, where p is an integer. (You can
use other variables, such as 2m, to represent an even integer, where
m is an integer.)
CONNECT TO PROPERTIES
• A set has closure under an
operation if the result of the
operation on members of the
set is also in the same set.
16. Use the expressions 2p and 2m, where p and m are integers, to
confirm the conjecture that the sum of two even integers is an
even integer.
• In symbols, distributive
property of multiplication over
addition can be written as
a(b + c) = a(b) + a(c).
This is an algebraic definition of odd integer: An integer is odd if and only
if it can be written in the form 2t + 1, where t is an integer. (Again, you
do not have to use t as the variable.)
18. Use expressions for even and odd integers to confirm the conjecture
that the sum of an even integer and an odd integer is an odd integer.
20
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17. Use expressions for odd integers to confirm the conjecture that the
sum of two odd integers is an even integer.
Logical Reasoning
ACTIVITY 1.2
continued
Riddle Me This
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Summarize/Paraphrase/Retell, Create Representations,
Group Presentation
My Notes
In Items 14 and 15, you developed a geometric puzzle-piece argument to
confirm conjectures about the sums of even and odd integers. In Items
16–18, you developed an algebraic argument to confirm these same
conjectures. Even though one method is considered geometric and the
other algebraic, they are often seen as the same basic argument.
19. Explain the link between the geometric and the algebraic methods of
the proof.
© 2010 College Board. All rights reserved.
20. State three theorems that you have proved about the sums of even and
odd integers.
Unit 1 • Proof, Parallel and Perpendicular Lines
21
ACTIVITY 1.2
Logical Reasoning
continued
Riddle Me This
CHECK YOUR UNDERSTANDING
Writeyour
youranswers
answersonon
notebook
paper.
Show
Write
notebook
paper.
Show
youryour work.
4. a. Draw the next shape in the pattern,
work.
b. Write a sequence of numbers that could be
used to express the pattern,
1. Use inductive reasoning to determine the next
two terms in the sequence.
c. Verbally describe the pattern of the
1 , __
1 , __
1 , ___
1 ,…
sequence.
a. __
2 4 8 16
5. Use expressions for even and odd integers to
b. A, B, D, G, K, …
confirm the conjecture that the product of
2. Write the first five terms of two different
an even integer and an odd integer is an even
sequences that have 10 as the second term.
integer.
3. Generate a sequence using the description: the
6. MATHEMATICAL Suppose you are given the
first term in the sequence is 4 and each term is
R E F L E C T I O N first five terms of a
three more than twice the previous term.
numerical sequence. What are some
approaches you could use to determine the
rule for the sequence and then write the next
two terms of the sequence? You may use an
example to illustrate your answer.
© 2010 College Board. All rights reserved.
Use this picture pattern for Item 4 at the top of
the next column.
22
SpringBoard® Mathematics with Meaning™ Geometry
The Axiomatic System of Geometry
Back to the Beginning
ACTIVITY
1.3
SUGGESTED LEARNING STRATEGIES: Close Reading, Quickwrite,
Think/Pair/Share, Vocabulary Organizer, Interactive Word Wall
My Notes
Geometry is an axiomatic system. That means that from a small, basic
set of agreed-upon assumptions and premises, an entire structure of
logic is devised. Many interactive computer games are designed with
this kind of structure. A game may begin with basic set of scenarios.
From these scenarios, a gamer can devise tools and strategies to win
the game.
In geometry, it is necessary to agree on clear-cut meanings, or
definitions, for words used in a technical manner. For a definition to be
helpful, it must be expressed in words whose meanings are already known
and understood.
Compare the following definitions.
Fountain: a roundel that is barry wavy of six argent and azure.
Guige: a belt that is worn over the right shoulder and used to
support a shield.
© 2010 College Board. All rights reserved.
1. Which of the two definitions above is easier to understand? Why?
For a new vocabulary term to be helpful, it should be defined using words
that have already been defined. The first definitions used in building a
system, however, cannot be defined in terms of other vocabulary words
because no other vocabulary words have been defined yet. In geometry,
it is traditional to start with the simplest and most fundamental terms—
without trying to define them—and use these terms to define other terms
and develop the system of geometry. These fundamental undefined terms
are point, line, and plane.
MATH TERMS
The term, line segment, can be
defined in terms of undefined
terms: A line segment is part of
a line bounded by two points on
the line called endpoints.
2. Define each term using the undefined terms.
a. Ray
b. Collinear points
c. Coplanar points
Unit 1 • Proof, Parallel and Perpendicular Lines
23
ACTIVITY 1.3
The Axiomatic System of Geometry
continued
Back to the Beginning
SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer,
Interactive Word Wall, Activating Prior Knowledge
My Notes
After a term has been defined, it can be used to define other terms.
For example, an angle is defined as a figure formed by two rays with a
common endpoint.
3. Define each term using the already defined terms.
ACADEMIC VOCABULARY
a. Complementary angles
complementary angles
supplementary angles
postulate
b. Supplementary angles
Addition Property of Equality
If a = b and c = d,
then a + c = b + d.
Subtraction Property
of Equality
If a = b and c = d,
then a - c = b - d.
Multiplication Property
of Equality
The process of deductive reasoning, or deduction, must have a starting
point. A conclusion based on deduction cannot be made unless there is
an established assertion to work from. To provide a starting point for the
process of deduction, a number of assertions are accepted as true without
proof. These assertions are called axioms, or postulates.
When you solve algebraic equations, you are using deduction. The
properties you use are like postulates in geometry.
4. Using one operation or property per step, show how to solve the
equation 4x + 9 = 18 - _12 x. Name each operation or property used to
justify each step.
If a = b, then ca = cb.
Division Property of Equality
a = __
b
If a = b and c ≠ 0, then __
c c.
Distributive Property
a(b + c) = ab + ac
Reflexive Property
a=a
Symmetric Property
If a = b, then b = a.
Transitive Property
If a = b and b = c, then a = c.
Substitution Property
If a = b, then a can be substituted
for b in any equation or inequality.
24
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CONNECT TO ALGEBRA
The Axiomatic System of Geometry
ACTIVITY 1.3
continued
Back to the Beginning
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share
My Notes
You can organize the steps and the reasons used to justify the steps in two
columns with statements (steps) on the left and reasons (properties) on
the right. This format is called a two-column proof.
EXAMPLE 1
Given: 3(x + 2) - 1 = 5x + 11
Prove: x = -3
Statements
1. 3(x + 2) - 1 = 5x + 11
2.
3(x + 2) = 5x + 12
3.
3x + 6 = 5x + 12
4.
6 = 2x + 12
5.
-6 = 2x
6.
-3 = x
7.
x = -3
Reasons
1. Given equation
2. Addition Property of Equality
3. Distributive Property
4. Subtraction Property of Equality
5. Subtraction Property of Equality
6. Division Property of Equality
7. Symmetric Property of Equality
TRY THESE A
© 2010 College Board. All rights reserved.
a. Supply the reasons to justify each statement in the proof below.
6+x
x - 3 = _____
Given: _____
5
2
Statements
6+x
x - 3 = _____
_____
1.
5
2
6+x
x - 3 = 10 _____
2. 10 _____
5
2
(
)
(
Prove: x = 9
Reasons
1.
)
2.
3.
5(x - 3) = 2(6 + x)
3.
4.
5x - 15 = 12 + 2x
4.
5.
3x - 15 = 12
5.
6.
3x = 27
6.
7.
x=9
7.
Unit 1 • Proof, Parallel and Perpendicular Lines
25
ACTIVITY 1.3
The Axiomatic System of Geometry
continued
Back to the Beginning
My Notes
SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer,
Think/Pair/Share, Interactive Word Wall, Marking the Text,
Group Presentation
TRY THESE A (continued)
b. Complete the Prove statement and write a two-column proof for the
equation given in Item 4. Number each statement and corresponding
reason.
1
Given: 4x + 9 = 18 - __
2x
Prove:
ACADEMIC VOCABULARY
conditional statement
Rules of logical reasoning involve using a set of given statements along
with a valid argument to reach a conclusion. Statements to be proved are
often written in if-then form. An if-then statement is called a conditional
statement. In such statements, the if clause is the hypothesis, and the
then clause is the conclusion.
WRITING MATH
The letters p and q are
often used to represent the
hypothesis and conclusion,
respectively, in a conditional
statement. The basic form of an
if-then statement would then be,
“If p, then q.”
EXAMPLE 2
Conditional statement: If 3(x + 2) - 1 = 5x + 11, then x = -3.
Hypothesis
Conclusion
3(x + 2) - 1 = 5x + 11
x = -3
TRY THESE B
a. What is the hypothesis?
b. What is the conclusion?
c. State the property of equality that justifies the conclusion of the
statement.
26
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
Use the conditional statement: If x + 7 = 10, then x = 3.
The Axiomatic System of Geometry
ACTIVITY 1.3
continued
Back to the Beginning
SUGGESTED LEARNING STRATEGIES: Vocabulary
Organizer, Interactive Word Wall
My Notes
Conditional statements may not always be written in if-then form. You
can restate such conditional statements in if-then form.
5. Restate each conditional statement in if-then form.
a. I’ll go if you go.
READING MATH
Forms of conditional statements
include:
b. There is smoke only if there is fire.
• If p, then q.
• p only if q
• p implies q.
c. x = 4 implies x2 = 16.
• q if p
An if-then statement is false if an example can be found for which the
hypothesis is true and the conclusion is false. This type of example is a
counterexample.
ACADEMIC VOCABULARY
counterexample
6. This is a false conditional statement.
If two numbers are odd, then their sum is odd.
© 2010 College Board. All rights reserved.
a. Identify the hypothesis of the statement.
b. Identify the conclusion of the statement.
c. Give a counterexample for the conditional statement and justify
your choice for this example.
Unit 1 • Proof, Parallel and Perpendicular Lines
27
ACTIVITY 1.3
The Axiomatic System of Geometry
continued
Back to the Beginning
SUGGESTED LEARNING STRATEGIES: Vocabulary
Organizer, Interactive Word Wall, Think/Pair/Share
My Notes
converse
inverse
contrapositive
CONNECT TO AP
When both a statement and its
converse are true, you can connect
the hypothesis and conclusion
with the words “if and only if.”
Conditional:
If p, then q.
Converse:
If q, then p.
Inverse:
If not p, then not q.
Contrapositive:
If not q, then not p.
7. Given the conditional statement:
If a figure is a triangle, then it is a polygon.
Complete the table.
Form of the
statement
Conditional
statement
Converse of
the conditional
statement
Inverse of the
conditional
statement
Contrapositive
of the
conditional
statement
28
SpringBoard® Mathematics with Meaning™ Geometry
Write the
statement
True or
False?
If the statement
is false, give a
counterexample.
If a figure is a
triangle, then it is a
polygon.
© 2010 College Board. All rights reserved.
MATH TERMS
Every conditional statement has three related conditionals. These are
the converse, the inverse, and the contrapositive of the conditional
statement. The converse of a conditional is formed by interchanging
the hypothesis and conclusion of the statement. The inverse is formed
by negating both the hypothesis and the conclusion. Finally, the
contrapositive is formed by interchanging and negating both the
hypothesis and the conclusion.
The Axiomatic System of Geometry
ACTIVITY 1.3
continued
Back to the Beginning
SUGGESTED LEARNING STRATEGIES: Group Presentation
My Notes
If a given conditional statement is true, the converse and inverse are not
necessarily true. However, the contrapositive of a true conditional is always
true, and the contrapositive of a false conditional is always false. Likewise,
the converse and inverse of a conditional are either both true or both false.
Statements with the same truth values are logically equivalent.
MATH TERMS
The truth value of a statement
is the truth or falsity of that
statement
8. Write a true conditional statement whose inverse is false.
9. Write a true conditional statement that is logically equivalent to its
converse.
When a statement and its converse are both true, they can be combined
into one statement using the words “if and only if ”. All definitions you have
learned can be written as “if and only if ” statements.
10. Write the definition of perpendicular lines in if and only if form.
TRY THESE C
Use this statement: Numbers that do not end in 2 are not even.
© 2010 College Board. All rights reserved.
a. Rewrite the statement in if-then form and state whether it is true or false.
b. Write the converse and state whether it is true or false. If false, give a
counterexample.
c. Write the inverse and state whether it is true or false.
d. Write the contrapositive and state whether it is true or false. If false, give
a counterexample.
Unit 1 • Proof, Parallel and Perpendicular Lines
29
ACTIVITY 1.3
The Axiomatic System of Geometry
continued
Back to the Beginning
CHECK YOUR UNDERSTANDING
Writeyour
youranswers
answersononnotebook
notebook
paper.
Show your work.
Write
paper.
Show
9. Which of the following is a counterexample of
your work.
this statement?
2. Complete the prove statement and write a two
column proof for the equation:
Given: x - 2 = 3(x - 4)
Prove:
3. Write the statement in if-then form.
Two angles have measures that add up to 90°
only if they are complements of each other.
Use the following statement for Items 4–7.
If a vehicle has four wheels, then it is a car.
a. a 100° angle
b. a 70° angle
c. an 80° angle
d. a 90° angle
10. Give an example of a false statement that has a
true converse.
11. A certain conditional statement is true. Which
of the following must also be true?
a. converse
b. inverse
c. contrapositive
d. all of the above
12. Given: (1) If X is blue, then Y is gold.
(2) Y is not gold.
4. State the hypothesis.
Which of the following must be true?
5. Write the converse.
a. Y is blue.
b. Y is not blue.
6. Write the inverse.
c. X is not blue.
d. X is gold.
7. Write the contrapositive.
8. Given the false conditional statement:
If a vehicle has four wheels, then it is a car.
Write a counterexample.
30
If an angle is acute, then it measures 80°.
SpringBoard® Mathematics with Meaning™ Geometry
13. MATHEMATICAL Some test questions ask you
R E F L E C T I O N to tell if a statement is
sometimes true, always true, or never true.
How can counterexamples help you to answer
these types of questions?
© 2010 College Board. All rights reserved.
1. Identify the property that justifies the
statement: If 4x - 3 = 7, then 4x = 10.
Truth Tables
The Truth of the Matter
ACTIVITY
1.4
SUGGESTED LEARNING STRATEGIES: Close Reading, Activating
Prior Knowledge, Group Discussion, Interactive Word Wall,
Vocabulary Organizer, Think/Pair/Share
My Notes
Symbolic logic allows you to determine the validity of statements
without being distracted by a lot of text. You can use symbols, such as p
and q, to represent simple statements. A compound statement is formed
when two or more simple statements are connected as a conditional
(if-then) a biconditional (if and only if), a conjunction (and), or a
disjunction (or).
1. The symbol → represents a conditional. You read p → q as, “if p,
then q,” or “p implies q.” Let p represent the statement “you arrive
before 7 PM,” and let q represent the statement “you will get a good
seat.” Use the information in the play poster to the right to write the
statement that is represented by p → q.
© 2010 College Board. All rights reserved.
Truth tables are used to
determine the conditions under
which a statement is true or
false. This truth table displays
the truth values for p → q,
which are dependent on the
truth values for p and q.
Row 1
Row 2
Row 3
Row 4
p
q
T
T
F
F
T
F
T
F
p→q
T
F
T
T
2. Row 1 addresses the case when both p and q are true. Refer to the
play announcement that you completed in Item 1. Explain why p → q
would be true if both p and q are true.
3. Row 2 addresses the case when p is true and q is false. In terms of the
play announcement, explain why p → q would be false if p is true and
q is false.
4. Refer to Rows 3 and 4. In terms of the play announcement, explain
why p → q is true if p is false.
Unit 1 • Proof, Parallel and Perpendicular Lines
31
ACTIVITY 1.4
Truth Tables
continued
The Truth of the Matter
SUGGESTED LEARNING STRATEGIES: Close Reading, Create
Representations, Think/Pair/Share
My Notes
The symbol ~p is the negation of p and can be read as
“not p.” This truth table shows the conditions under
which ~p is true or false. When p is true, ~p is false,
and when p is false, ~p is true.
p
~p
T
F
F
T
5. Let p represent the simple statement “Triangles are convex,” which is a
true statement. Write the statement denoted by ~p and state whether
it is true or false.
6. Create a simple statement p that you know to be false. Write the
statement ~p and state whether is true or false.
EXAMPLE
Make a truth table for ~p → q.
Step 1
Write down all
possible T and F
combinations for
p and q.
Step 3
Add a column
for ~p → q and
evaluate ~p → q.
32
SpringBoard® Mathematics with Meaning™ Geometry
Step 2
Add a column
for ~p and
negate p.
p
q
T
T
F
F
T
F
T
F
p
q
~p
T
T
F
F
T
F
T
F
F
F
T
T
~p → q
T
T
T
F
p
q
~p
T
T
F
F
T
F
T
F
F
F
T
T
© 2010 College Board. All rights reserved.
7. Let p represent “you are not in the band” and let q represent “you can
go on the trip.” Write the compound statement represented by ~p → q.
Truth Tables
ACTIVITY 1.4
continued
The Truth of the Matter
SUGGESTED LEARNING STRATEGIES: Close Reading,
Think/Pair/Share, Create Representations, Identify a
Subtask, Activating Prior Knowledge, Interactive Word
Wall, Vocabulary Organizer
My Notes
8. Follow the steps and complete the truth table for ~(p → q).
Step 1
Find all possible T and F
combinations for p and q.
Step 1
Step 2
Step 3
p
p→q
~(p → q)
q
Step 2
Evaluate p → q.
Step 3
Negate p → q.
9. Write the steps and complete the truth table for ~q → ~p.
Step 1
© 2010 College Board. All rights reserved.
p
q
Step 2
~p
~q
Step 3
~q → ~p
10. The symbol ↔ represents a biconditional. You read it as “if and
only if.” Let p represent “A chord in a circle contains the center,” and
let q represent “The chord is a diameter.” Write the statement that is
represented by p ↔ q.
Unit 1 • Proof, Parallel and Perpendicular Lines
33
ACTIVITY 1.4
Truth Tables
continued
The Truth of the Matter
My Notes
Because biconditionals are true
only when both parts have the
same truth value, many definitions are written in the form of a
biconditional.
SUGGESTED LEARNING STRATEGIES: Close Reading, Think/
Pair/Share, Create Representations, Identify a Subtask,
Discussion Group, Group Presentation, Interactive Word
Wall, Vocabulary Organizer
The truth table for p ↔ q is
shown to the right. Notice that
p ↔ q is true only when
p and q are both true or both false.
p
q
T
T
F
F
T
F
T
F
p↔q
T
F
F
T
11. Construct a truth table for (p → q) ↔ ~q.
A rectangle is a parallelogram
with a right angle.
A rhombus is a parallelogram
with consecutive congruent
sides.
12. The symbol ∧ represents a conjunction. You read it as “and.” Let
p represent “the figure is a rectangle,” and let q represent “the figure
is a rhombus.”
a. Write the statement that is represented by p ∧ q.
b. Write the statement that is represented by p ∧ ~q.
The truth table for p ∧ q is shown to
the right. Notice that p ∧ q is only
true when both p and q are true.
34
SpringBoard® Mathematics with Meaning™ Geometry
p
q
T
T
F
F
T
F
T
F
p∧q
T
F
F
F
© 2010 College Board. All rights reserved.
MATH TERMS
Truth Tables
ACTIVITY 1.4
continued
The Truth of the Matter
SUGGESTED LEARNING STRATEGIES: Close Reading,
Create Representations, Identify a Subtask
My Notes
13. The symbol ∨ represents a disjunction. You read it as “or.” Let p
represent “the figure is not a rectangle,” and let q represent “the figure
is a rhombus.”
a. Write the statement that is represented by p ∨ q.
b. Write the statement that is represented by ~p ∨ q.
The truth table for p ∨ q is shown to the right.
Notice that p ∨ q is only false when both
p and q are false.
p
q
T
T
F
F
T
F
T
F
p∨q
T
T
T
F
© 2010 College Board. All rights reserved.
14. Construct a truth table for p ∨ ( p ∧ q).
Unit 1 • Proof, Parallel and Perpendicular Lines
35
ACTIVITY 1.4
Truth Tables
continued
The Truth of the Matter
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Identify a Subtask, Create Representations, Identify a
Subtask
My Notes
A tautology is a statement that is true for all cases of p and q. The
corresponding truth table will have only T’s in the last column.
15. Construct a truth table for (p → q) ↔ (p ∧ q). Is this a tautology?
CHECK YOUR UNDERSTANDING
Writeyour
youranswers
answersononnotebook
notebook
paper.
Show
Write
paper.
Show
youryour work.
4. (p → q) ∧ p
work.
5. q ∧ (p → ~q)
Construct a truth table for the compound statements.
6. (p → q) ↔ (~q → ~p)
Identify which statements, if any, are tautologies.
7. MATHEMATICAL How could a truth table
1. ~p ∧ q
R E F L E C T I O N help you to analyze a
campaign promise made by a person running
2. q → (p ∨ ~p)
for office? Think of a conditional statement
3. p ∨ (q → p)
starting, “If I am elected, …”
36
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
16. Construct a truth table for ~(p ∨ q) ↔ (~p ∧ ~q). Is this a tautology?
Conditional Statements and Logic
Embedded
Assessment 1
HEALTHY HABITS
Use after Activity 1.4.
Statements given in advertisements often involve conditional statements. Analyze
the advertisement below.
“If you want to feel your very best, take one SBV tablet every day. People who
take SBV vitamins care about their health. You owe it to those close to you to
take care of yourself. Begin taking SBV today!”
Write your answers on notebook paper. Show your work.
1. Consider the first statement in the advertisement: “If you want to feel your very
best, take one SBV tablet every day.”
a. Identify the hypothesis and conclusion of the statement.
b. Write the converse of the statement. Discuss the validity of the converse and
why it might be used to influence people to buy the vitamin.
2. Consider the second statement: “People who take SBV vitamins care about their
health.”
a. Rewrite the statement in if-then form.
b. Write the inverse of the statement. Discuss the validity of the inverse and
why it might be used to influence people to buy the vitamin.
© 2010 College Board. All rights reserved.
According to current guidelines, people are advised to maintain cholesterol levels
below 200 units. Four friends; Juana, Bea, Les, and Hugh, all have cholesterol levels
above the recommended limit. They have all heard cholesterol-lowering strategies
discussed on talk shows and in commercials. Those strategies included: taking two
1 hour
red yeast rice tablets daily; eating oatmeal daily; practicing meditation for __
2
each day; and getting at least twenty minutes of aerobic exercise daily.
3. For one month, they each used strategies daily. They had their cholesterol
rechecked at the end of the month. Juana meditated and ate oatmeal. Her
cholesterol level fell 5 points. Hugh exercised and meditated. His cholesterol
lowered 10 points. Les took red yeast rice tablets and meditated. His cholesterol
level stayed the same. Bea took red yeast rice tablets and ate oatmeal. Her
cholesterol level went down 7 points. Which technique(s) appear to be most
effective in lowering cholesterol levels? Explain your reasoning.
Unit 1 • Proof, Parallel and Perpendicular Lines
37
Embedded
Assessment 1
Conditional Statements and Logic
Use after Activity 1.4.
HEALTHY HABITS
Exemplary
Proficient
Emerging
The student:
The student:
• Correctly
• Correctly
identifies the
identifies the
hypothesis or
hypothesis and
conclusion but
conclusion. (1a)
not both.
• Writes the correct
• Writes a correct
converse of the
if-then form but
statement. (1b)
not the correct
• Writes a correct
inverse.
if-then form and
inverse of the
statement. (2a, b)
The student:
• Correctly
identifies neither
the hypothesis
nor the
conclusion.
• Writes an incorrect
converse of the
statement.
• Writes neither
a correct ifthen form nor a
correct inverse.
Problem Solving
#3
The student
correctly identifies
the seemingly
most effective
technique(s). (3)
The student
identifies a
technique that
is somewhat
effective.
The student
identifies a
technique that is
not effective.
Communication
#1b, 2b, 3
The student:
• Gives a complete
discussion of the
validity of the
converse. (1b)
• Gives a complete
discussion of the
validity of the
inverse. (2b)
• Gives logical
reasoning for his/
her choice(s) of
technique(s). (3).
The student:
• Gives an
incomplete
discussion of the
validity of the
converse.
• Gives an
incomplete
discussion of the
validity of the
inverse.
• Gives incomplete
reasoning for the
choice.
The student:
• Gives an irrelevant
discussion of the
validity of the
converse.
• Gives an irrelevant
discussion of the
validity of the
inverse.
• Gives illogical
reasoning for the
choice.
© 2010 College Board. All rights reserved.
Math Knowledge
#a, b; 2a, b
38
SpringBoard® Mathematics with Meaning™ Geometry
Segment and Angle Measurement
It All Adds Up
ACTIVITY
1.5
SUGGESTED LEARNING STRATEGIES: Close Reading
My Notes
If two points are no more than one foot apart, you can find the distance
between them by using an ordinary ruler. (The inch rulers below have
been reduced to fit on the page.)
A
B
inches
1
2
3
4
5
READING MATH
6
7
In the figure, the distance between point A and point B is 5 inches. Of
course, there is no need to place the zero of the ruler on point A. In the
figure below, the 2-inch mark is on point A. In this case, AB, measured in
inches, is |7 - 2| = |2 - 7| = 5, as before.
A
2
AB denotes the distance
between points A and B. If A
and B are the
_ endpoints of a
segment ( AB ),_then AB denotes
the length of AB .
B
3
4
5
7
6
8
9
The number obtained as a measure of distance depends on the unit of
measure. On many rulers one edge is marked in centimeters. Using
the centimeter scale, the distance between the points A and B above is
about 12.7 cm.
1. Determine the length of each segment in centimeters.
MATH TERMS
© 2010 College Board. All rights reserved.
D
a. DE =
E
F
b. EF =
The Ruler Postulate
a. To every pair of points
there corresponds a unique
positive number called the
distance between the points.
c. DF =
2. Determine the length of each segment in centimeters.
G
H
a. KH =
b. HG =
K
c. GK =
b. The points on a line can
be matched with the real
numbers so that the distance
between any two points is
the absolute value of the
difference of their associated
numbers.
Unit 1 • Proof, Parallel and Perpendicular Lines
39
ACTIVITY 1.5
Segment and Angle Measurement
continued
It All Adds Up
My Notes
SUGGESTED LEARNING STRATEGIES: Look for a Pattern,
Think/Pair/Share, Vocabulary Organizer, Interactive Word
Wall, Create Representations
3. Using your results from Items 1 and 2, describe any patterns that
you notice.
Item 4 and your answer together
form a statement of the Segment
Addition Postulate.
For each part of Item 5, make a
sketch so that you can identify
the parts of the segment.
CONNECT TO AP
You will frequently be asked to
find the lengths of horizontal,
vertical, and diagonal segments
in the coordinate plane in
AP Calculus.
40
4. Given that N is a point between endpoints__
M and P of line segment MP,
describe how to determine the__
length of__
MP, without measuring,
if you are given the lengths of MN and NP.
5. Use the Segment Addition Postulate and the given information to
complete each statement.
a. If B is between C and D, BC = 10 in., and BD = 3 in., then
CD = _______.
b. If Q is between R and T, RT = 24 cm, and QR = 6 cm, then
QT = _______.
c. If P is between L and A, PL = x + 4, PA = 2x - 1, and
LA = 5x - 3, then x = _______ and LA = _______.
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
MATH TERMS
Segment and Angle Measurement
ACTIVITY 1.5
continued
It All Adds Up
SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer,
Interactive Word Wall
My Notes
The midpoint of a segment is the point on the segment that divides__
it
into two
segments. For example, if B is the midpoint of AC ,
__
__ congruent
then AB BC.
A
B
3
3
ACADEMIC VOCABULARY
C
The midpoint of a segment is
a point on the segment that
divides it into two congruent
segments.
__
6. Given: M is the midpoint of RS. Complete each statement.
a. If RS = 10, then SM = _______.
Congruent segments are
segments that have equal
length. To indicate
__
__ congruence,
you can write AB BC.
b. If RM = 12, then MS = _______, and RS = _______.
You can use the definition of midpoint and properties of algebra to
determine the length of a segment.
EXAMPLE 1
__
If Q is the midpoint of PR
, PQ = 4x - 5, and QR = 11 + 2x,
__
determine the length of PQ.
© 2010 College Board. All rights reserved.
__
__
1. PQ QR
1. Definition of midpoint
2. PQ = QR
2. Definition of congruent segments
__
Use AB when you talk about
segment AB.
3. 4x - 5 = 11 + 2x
3. Substitution Property
4. 2x - 5 = 11
4. Subtraction Property of Equality
5. 2x = 16
5. Addition Property of Equality
6. x = 8
6. Division Property of Equality
7. PQ = 4(8) - 5 = 27
7. Substitution Property
Use AB when you talk about
__ the
measure, or length, of AB .
Unit 1 • Proof, Parallel and Perpendicular Lines
41
ACTIVITY 1.5
Segment and Angle Measurement
continued
It All Adds Up
SUGGESTED LEARNING STRATEGIES: Create Representations,
Think/Pair/Share
My Notes
EXAMPLE 2
__
If Y is the midpoint
__ of WZ, YZ = x + 3 and WZ = 3x - 4, determine
the length of WZ.
1.
2.
3.
4.
5.
6.
7.
8.
WZ = WY + YZ
__
__
1.
2.
WY YZ
3.
WY = YZ
4.
WZ = YZ + YZ
3x - 4 = (x + 3) + (x + 3) 5.
6.
x-4=6
7.
x = 10
8.
WZ = 3(10) - 4 = 26
TRY THESE A
Segment Addition Postulate
Definition of midpoint
Definition of congruent segments
Substitution Property
Substitution Property
Subtraction Property of Equality
Addition Property of Equality
Substitution Property
__
Given: M is the midpoint of RS. Use the given information to find the
missing values.
a. RM = x + 3 and MS = 2x - 1
x = _____ and RM = _____
b. RM = x + 6 and RS = 5x + 3
x = _____ and SM = _____
© 2010 College Board. All rights reserved.
You measure angles with a protractor. The number of degrees in an angle
is called its measure.
C
0
80
100
90
100
80
110
70
12
0
60
14
30
0
15
30
20
160
160
20
O
180
0
0
180
10
170
170
10
42
E
0
15
The astronomer Claudius Ptolemy
(about 85–165 CE) based his
observations of the solar system
on a unit that resulted from
dividing the distance around a
circle into 360 parts. This later
became known as a degree.
50
0
40
CONNECT TO ASTRONOMY
B
13
0
0
12
50
0
13
70
110
14
D
40
60
A
7. Determine the measure of each angle.
a. m∠AOB = 50°
b. m∠BOC = ____
c. m∠AOC = ____
d. m∠EOD = ____
e. m∠BOD = ____
f. m∠BOE = ____
SpringBoard® Mathematics with Meaning™ Geometry
Segment and Angle Measurement
ACTIVITY 1.5
continued
It All Adds Up
SUGGESTED LEARNING STRATEGIES: Look for a Pattern,
Quickwrite, Vocabulary Organizer, Interactive Word Wall,
Think/Pair/Share
My Notes
8. Use a protractor to determine the measure of each angle.
P
T
MATH TERMS
The Protractor Postulate
Q
a. m∠TQP =
b. m∠TQR =
R
c. m∠RQP =
9. Using your results from Items 7 and 8, describe any patterns that
you notice.
a. To each angle there
corresponds a unique real
number between 0 and 180
called the measure of the
angle.
b. The measure of an angle
formed by a pair of rays is
the absolute value of the
difference of their associated
numbers.
10. Given that point D is in the interior of ∠ABC, describe how to
determine the measure of ∠ABC, without measuring, if you are given
the measures of ∠ABD and ∠DBC.
MATH TERMS
© 2010 College Board. All rights reserved.
11. Use the angle addition postulate and the given information to
complete each statement.
Item 10 and your answer
together form a statement of
the Angle Addition Postulate.
a. If P is in the interior of ∠XYZ, m∠XYP = 25°, and
m∠PYZ = 50°, then m∠XYZ =
.
b. If M is in the interior of ∠RTD, m∠RTM = 40°, and
m∠RTD = 65°, then m∠MTD =
.
c. If H is in the interior of ∠EFG, m∠EFH = 75°, and
m∠HFG = (10x)°, and m∠EFG = (20x - 5)°, then
x=
and m∠HFG =
.
Unit 1 • Proof, Parallel and Perpendicular Lines
43
ACTIVITY 1.5
Segment and Angle Measurement
continued
It All Adds Up
SUGGESTED LEARNING STRATEGIES: Vocabulary
Organizer, Interactive Word Wall, Create Representations
My Notes
11d. Lines DB and EC intersect at
point F. If m∠BFC = 44° and
m∠AFB = 61°, then
ACADEMIC VOCABULARY
m∠AFC =
angle bisector
m∠AFE =
A
E
D
B
F
C
m∠EFD =
The bisector of an angle is a ray that divides the angle into two congruent
adjacent angles. For example, in the figure to the left, if BD
bisects
∠ABC, then ∠ABD ∠DBC.
A
30°
30°
D
B
12. Given: AH
bisects ∠MAT. Determine the missing measure.
C
a. m∠MAT = 70°, m∠MAH =
b. m∠HAT = 80°, m∠MAT =
Adjacent angles are coplanar
angles that share a vertex and
side, but no interior points. In
the figure above, ∠ABD is
adjacent to ∠DBC.
Congruent angles are angles
that have the same measure. To
indicate congruence, you can
either write m∠ABD = m∠DBC.
or ∠ABD ∠DBC.
44
You can use definitions, postulates, theorems, and properties of algebra to
determine the measures of angles.
EXAMPLE 3
If QP
bisects ∠DQL, m∠DQP = 5x - 7 and m∠PQL = 11 + 2x,
determine the measure of ∠DQL.
∠DQP ∠PQL
m∠DQP = m∠PQL
5x - 7 = 11 + 2x
3x - 7 = 11
3x = 18
x=6
m∠DQP = 5(6) - 7 = 23
m∠PQL = 11 + 2(6) = 23
m∠DQL = m∠DQP +
m∠PQL
10. m∠DQL = 23 + 23 = 46°
1.
2.
3.
4.
5.
6.
7.
8.
9.
SpringBoard® Mathematics with Meaning™ Geometry
1.
2.
3.
4.
5.
6.
7.
8.
9.
Definition of Angle Bisector
Definition of congruent angles
Substitution Property
Subtraction Property of Equality
Addition Property of Equality
Division Property of Equality
Substitution Property
Substitution Property
Angle Addition Postulate
10. Substitution Property
© 2010 College Board. All rights reserved.
MATH TERMS
Segment and Angle Measurement
ACTIVITY 1.5
continued
It All Adds Up
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Think/Pair/Share, Group Presentation
My Notes
GUIDED EXAMPLE
Complete the missing statements and reasons.
∠A and ∠B are complementary, m∠A = 3x + 7, and m∠B = 6x + 11.
Determine the measure of each angle.
Statements
1.
2.
3.
4.
5.
6.
7.
Reasons
m∠A + m∠B = 90°
+
= 90°
9x + 18 = 90
9x = 72
x=8
m∠A = 3(8) + 7 = 31
m∠B = 6(
) +11 =
1.
2.
3.
4.
5.
6.
7.
Definition of
Substitution Property
Substitution Property
Property of Equality
Property of Equality
Substitution Property
Substitution Property
TRY THESE B
Given: FL
bisects ∠AFM. Determine each missing value.
a. m∠LFM = 11x + 4 and m∠AFL = 12x - 2
x=
, m∠LFM =
, and m∠AFM =
b. m∠AFM = 6x - 2 and m∠AFL = 4x - 10
© 2010 College Board. All rights reserved.
x=
and m∠LFM =
In this diagram, lines AC and DB intersect as shown. Determine the
missing values.
A
c. x =
m∠AEB =
m∠CEB =
.
D (5x + 68)° (9x)°
E
B
C
(Diagram is not drawn to scale.)
d. ∠P and ∠Q are complementary. m∠P = 2x + 25 and
m∠Q = 4x + 11. Determine the measure of each angle.
Unit 1 • Proof, Parallel and Perpendicular Lines
45
ACTIVITY 1.5
Segment and Angle Measurement
continued
It All Adds Up
CHECK YOUR UNDERSTANDING
Writeyour
youranswers
answersononnotebook
notebook
paper.
Show your work.
Write
paper.
Show
7. Given: EA
⊥ ED
,
your work.
EB
bisects ∠AEC,
m∠AEB = 4x + 1,
1. Given: Point K is between points H and J,
and m∠CED = 3x.
HK = x - 5, KJ = 5x - 12 and HJ = 25.
a. x =
Find the value of x.
b. 2
c. 6
2. If K is the midpoint of HJ, HK = x + 6, and
HJ = 5x - 6, then KJ = ? .
a. 6
b. 3
c. 12
b. 23°
c. 17°
b. 29
c. 0
b. 20°
c. 6°
d. 2°
a. x =
46
4
b. m∠4 =
d. 14
6. Given: m∠1 = 4x + 30 and m∠3 = 2x + 48
c. m∠2 =
3
a. x =
Use the given diagram and information to
determine the missing measures. Show all work.
b. m∠3 =
2
1
d. 77°
5. Ray QS bisects ∠PQR. If m∠PQS = 5x and
m∠RQS = 2x + 6, then m∠PQR = ? .
a. 10°
D
d. 9
4. If ∠P and ∠Q are complementary,
m∠P = 5x + 3, and m∠Q = x + 3, then
x= ?
a. 30
E
8. Given: m∠1 = 2x + 8, m∠2 = x + 4, and
m∠3 = 3x + 18
3. Point D is in the interior of ∠ABC,
m∠ABC = 10x - 7, m∠ABD = 6x + 5, and
m∠DBC = 36°. What is m∠ABD?
a. 3°
C
b. m∠BEC =
d. 9
___
B
1
2
3
SpringBoard® Mathematics with Meaning™ Geometry
c. Is ∠4 complementary to ∠2? Explain.
9. MATHEMATICAL How could you model two
R E F L E C T I O N angles with a common
vertex and a common side that are not
adjacent angles?
© 2010 College Board. All rights reserved.
a. 7
A
Parallel and Perpendicular Lines
Patios by Madeline
ACTIVITY
1.6
SUGGESTED LEARNING STRATEGIES: Summarize/
Paraphrase/Retell
My Notes
ACADEMIC VOCABULARY
Parallel lines are coplanar lines
that do not intersect.
Gas Line
Matt works for Patios by Madeline. A new customer has asked him to
design a brick patio and walkway. The customer requires a blueprint
showing both the patio and the walkway. The customer would like the
rows of bricks in the completed patio to be parallel to the walkway. To
help create his blueprint, Matt will attach a string between two stakes on
either side of the patio as shown in the diagram below, and continue the
pattern with parallel strings. In his blueprint, Matt must include a line
to indicate the path of the underground gas line, which is located below
the patio. (Matt drew in the location of the gas line from information
supplied by the local gas company in order to avoid accidents during
construction of the patio.)
Stake
© 2010 College Board. All rights reserved.
ing
Str
ay
lkw
Wa
e
Lin
Stake
House
Unit 1 • Proof, Parallel and Perpendicular Lines
47
ACTIVITY 1.6
Parallel and Perpendicular Lines
continued
Patios by Madeline
SUGGESTED LEARNING STRATEGIES: Visualization,
Create Representations
My Notes
CONNECT TO OTHER
GEOMETRIES
___
1. Follow these steps to draw a line parallel to XY.
Step A
Determine the measure of ∠HAX.
Step B
Locate the third
___› stake on the left edge of the patio at point P
by drawing BP so that m∠HBP = m∠HAX.
Step C
Locate the fourth stake
___› at point J on the right edge of the
patio by extending BP in the opposite direction.
© 2010 College Board. All rights reserved.
Euclidean geometry is based
on five postulates proposed by
Euclid (about 300 BCE). His fifth
postulate, “Through a point
outside a line, there is exactly one
line parallel to the given line” was
a source of contention for many
mathematicians. Their thought
was that this fifth postulate could
be proven from the first four
postulates and should therefore
be called a theorem. Since Euclid
could not prove it, he kept it as a
postulate. Some mathematicians
thought that since the postulate
could not be proven they should
be able to replace it with a
contrary postulate. This created
other “non-Euclidean” geometries.
Continue designing the patio using the blueprint Matt has already begun.
Matt’s beginning blueprint is on the next page. Add information to the
blueprint as you work through this Activity.
Matt has placed the gas line on the blueprint from the information
he received from the gas company. To assure that the bricks will be
parallel to the walkway, Matt extended the string line along the edge of
the walkway and labeled points X, A, and Y. He also labeled additional
points B, C, and D where the string lines will cross the gas line.
48
SpringBoard® Mathematics with Meaning™ Geometry
Parallel and Perpendicular Lines
ACTIVITY 1.6
continued
Patios by Madeline
SUGGESTED LEARNING STRATEGIES: Visualization;
Create Representations
My Notes
Matt’s Blueprint
D
C
Y
B
A
X
© 2010 College Board. All rights reserved.
H
Unit 1 • Proof, Parallel and Perpendicular Lines
49
ACTIVITY 1.6
Parallel and Perpendicular Lines
continued
Patios by Madeline
SUGGESTED LEARNING STRATEGIES: Interactive Word
Wall, Quickwrite, Create Representations
My Notes
MATH TERMS
A transversal is a line that
intersects two or more coplanar
lines in different points.
∠HAX and ∠HBP are called corresponding angles because they are in
corresponding positions on the same side of the transversal (gas line).
According to the Corresponding Angles Postulate, when parallel lines
are cut by a transversal, the corresponding angles will always have the
same measure.
2. Postulates and theorems are often stated in if-then form.
a. State the Corresponding Angles Postulate in if-then form.
b. State the converse of the Corresponding Angles Postulate.
d. State the inverse of the Corresponding Angles Postulate.
e. Determine the validity of the inverse of the Corresponding Angles
Postulate. Support your answer with illustrations.
50
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
c. How does the statement in part b ensure that Matt’s string lines
are parallel?
Parallel and Perpendicular Lines
ACTIVITY 1.6
continued
Patios by Madeline
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Quickwrite, Self/Peer Revision
My Notes
3. Use the diagram shown to complete the table.
a. Fill in the corresponding angles column in the table below.
l
2
1
4 3
6 5
8
7
9 10
11 12
Angle
m
n
Corresponding Angles
∠1
∠2
∠3
∠4
© 2010 College Board. All rights reserved.
b. What conditions are necessary in order to apply the Corresponding
Angles Postulate?
c. What conditions are necessary in order to apply the inverse of the
Corresponding Angles Postulate?
Unit 1 • Proof, Parallel and Perpendicular Lines
51
ACTIVITY 1.6
Parallel and Perpendicular Lines
continued
Patios by Madeline
SUGGESTED LEARNING STRATEGIES: Quickwrite,
Group Presentation
My Notes
4. Other types of special angle pairs formed when lines are intersected
by a transversal are alternate interior angles. In the diagram below,
∠3 and ∠6 are alternate interior angles.
l
2
1
4 3
6 5
8
7
m
a. Explain why you think ∠3 and ∠6 are alternate interior angles.
c. Name the remaining pair of alternate interior angles in the
diagram. What appears to be true about them?
d. Using what you know about corresponding and vertical angles,
write a convincing argument to prove your response to part b.
52
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
b. Lines l and m in the diagram are parallel. What appears to be true
about the measures of ∠3 and ∠6?
Parallel and Perpendicular Lines
ACTIVITY 1.6
continued
Patios by Madeline
SUGGESTED LEARNING STRATEGIES: Interactive Word
Wall, Quickwrite, Create Representations
My Notes
5. Use your results from Item 4 to state a theorem in if-then form about
alternate interior angles.
6. State the inverse of the Alternate Interior Angles Theorem you wrote
in Item 5.
7. Determine the validity of the inverse of the Alternate Interior Angles
Theorem. Support your answer with illustrations.
© 2010 College Board. All rights reserved.
8. State the converse of the Alternate Interior Angles Theorem.
9. Matt realizes that he is not limited to using corresponding angles to
draw parallel lines. Use a protractor to draw another parallel string
line through point C on the blueprint, using alternate interior angles.
Mark the angles that were used to draw this new parallel line.
10. How does the converse of the Alternate Interior Angles Theorem
ensure that Matt’s string lines are parallel?
Unit 1 • Proof, Parallel and Perpendicular Lines
53
ACTIVITY 1.6
Parallel and Perpendicular Lines
continued
Patios by Madeline
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share
My Notes
11. Another angle pair relationship is same-side interior angles. In the
diagram below, ∠3 and ∠5 are same-side interior angles.
l
1 2
4 3
m
6 5
7 8
a. Explain why you think ∠3 and ∠5 are same-side interior angles.
c. Name the remaining pair of same-side interior angles in the
diagram. What appears to be true?
d. Provide a convincing argument to prove your response to part b.
54
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
b. Lines l and m in the diagram are parallel. What appears to be true
about the measures of ∠3 and ∠5?
Parallel and Perpendicular Lines
ACTIVITY 1.6
continued
Patios by Madeline
SUGGESTED LEARNING STRATEGIES: Interactive Word
Wall, Quickwrite, Create Representations
My Notes
12. Use your results from Item 11 to state a theorem in if-then form,
about same-side interior angles.
13. State the inverse of the Same-Side Interior Angles Theorem you wrote
in Item 12.
14. Determine the validity of the inverse of the Same-Side Interior Angles
Theorem. Support your answer with illustrations.
© 2010 College Board. All rights reserved.
15. State the converse of the Same-Side Interior Angles Theorem.
16. Use a protractor to draw another parallel string line through point D
on the blueprint using same-side interior angles. Mark the angles you
used to draw this line.
Unit 1 • Proof, Parallel and Perpendicular Lines
55
ACTIVITY 1.6
Parallel and Perpendicular Lines
continued
Patios by Madeline
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge
My Notes
ACADEMIC VOCABULARY
Perpendicular figures
intersect to form right angles.
A second customer of Madeline’s company has commissioned them to
build a patio with bricks that are perpendicular to the walkway. To help
plan this design, Matt will attach a string between two stakes on either
side of the patio as he did for the previous diagram. Continue drawing the
blueprint for the patio using the diagram below. To assure that the bricks
will be perpendicular to the walkway, Matt extended the string line along
the edge of the walkway and labeled points X and Y. He also attached a
string to another stake labeled point W as shown below.
W
Y
This theorem guarantees that
the line you draw in Item 17 is
perpendicular to line XY:
In a given plane containing a line,
there is exactly one line
perpendicular to the line through
a given point not on the line.
X
House
‹___›
18. Use your protractor and your knowledge
of parallel lines to draw a
‹___›
line, across the patio, parallel to WZ. Explain how you know the lines
are parallel.
56
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
17. Use a protractor to draw the line perpendicular to XY that
passes
‹___›
through W. Label the point at which the line intersects XY point Z.
Parallel and Perpendicular Lines
ACTIVITY 1.6
continued
Patios by Madeline
SUGGESTED LEARNING STRATEGIES: RAFT
‹___›
My Notes
19. Describe the relationship between the newly drawn line and XY.
20. Complete the statement to form a true conditional statement.
“If a transversal is perpendicular to one of two parallel lines, then it is
to the other line also.’’
© 2010 College Board. All rights reserved.
21. Matt’s boss Madeline is so impressed with Matt’s work that she asks
him to write an instruction guide for creating rows of bricks parallel
to patio walkways. Madeline asks that the guide provide enough
direction so that a bricklayer can use any pair of angles to determine
the location of each pair of stakes since not all patios will allow
for measuring the same pair of angles for each string line. She also
requests that Matt include directions for creating rows of bricks that
are perpendicular to the walkway. Use the space below to write Matt’s
instruction guide, following Madeline’s directions.
Unit 1 • Proof, Parallel and Perpendicular Lines
57
ACTIVITY 1.6
Parallel and Perpendicular Lines
continued
Patios by Madeline
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper. Show
your work.
Use this figure for Items 1-14.
l
m
6 5
8
7
9 10
11 12
c. ∠1 and ∠8
b. ∠3 and ∠5
d. ∠4 and ∠5
7. If l m, m∠4 = 15x − 7, and m∠6 = 20x +
12, then m∠8 = ?
a. 112°
c. 68°
b. 5°
d. 3°
In the diagram, l m and m ∦ n.
n
Determine whether each statement is true or
false. Justify your response with the appropriate
postulate or theorem.
1. List all pairs of alternate interior angles.
8. ∠5 is supplementary to m∠3
2. List all pairs of same-side interior angles.
9. m∠9 = m∠6
3. If l m and m∠5 = 80°, then what is m∠2?
10. ∠3 ∠8
a. 100°
c. 80°
11. ∠4 ∠10
b. 50°
d. Not enough information
12. ∠4 ∠6
4. If l m and m∠6 = 110°, then what is m∠3?
a. 70°
c. 80°
b. 110°
d. Not enough information
5. If l m, m∠4 = 12x + 5, and m∠5 = 8x + 17,
then what is m∠2?
58
a. ∠1 and ∠6
a. 139°
c. 3°
b. 7.9°
d. 41°
SpringBoard® Mathematics with Meaning™ Geometry
13. m∠8 + m∠10 = 180°
14. ∠4 is supplementary to ∠9.
15. MATHEMATICAL How could the types of
R E F L E C T I O N angles you studied in this
activity be useful to a city planner?
© 2010 College Board. All rights reserved.
1 2
4 3
6. If l m, which two angles are supplementary?
Two-Column Proofs
Now I'm Convinced
ACTIVITY
1.7
SUGGESTED LEARNING STRATEGIES: Close Reading, Activating
Prior Knowledge, Think/Pair/Share
My Notes
A proof is an argument, a justification, or a reason that something is true.
A proof is an answer to the question “why?” when the person asking
wants an argument that is indisputable.
There are three basic requirements for constructing a good proof.
1. Awareness and knowledge of the definitions of the terms related to
what you are trying to prove.
2. Knowledge and understanding of postulates and previous proven
theorems related to what you are trying to prove.
3. Knowledge of the basic rules of logic.
To write a proof, you must be able to justify
statements. The statements in the Examples 1–5
are based on the diagram to the right in which
lines AC, EG, and DF all intersect at point B.
Each of the statements is justified using a
property, postulate, or definition.
F
E
A
D
2
B
5
6
C
1
G
© 2010 College Board. All rights reserved.
EXAMPLE 1
Statement
If ∠ABE is a right angle,
then m∠ABE = 90°.
Justification
Definition of right angle
EXAMPLE 2
Statement
If ∠2 ∠1 and ∠1 ∠5,
then ∠2 ∠5.
Justification
Transitive Property
EXAMPLE 3
Statement
___
AC.
Given: ___
B is the
midpoint
of
___
Prove: AB BC
Justification
Definition of midpoint
Unit 1 • Proof, Parallel and Perpendicular Lines
59
ACTIVITY 1.7
Two-Column Proofs
continued
Now I’m Convinced
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share
My Notes
EXAMPLE 4
Statement
m∠2 + m∠ABE = m∠DBE
Justification
Angle Addition Postulate
EXAMPLE 5
Statement
If ∠1 is supplementary to ∠FBG,
then m∠1 + m∠FBG = 180°.
F
E
A
D
2
B
1
5
6
Justification
Definition of supplementary
angles
TRY THESE A
C
Using the diagram from the previous page, reproduced to the left, name
the property, postulate, or definition that justifies each statement.
a.
G
Statement
Justification
EB + BG = EG
b.
Statement
Justification
c.
d.
e.
Statement
If m∠1 + m∠6 = 90, then
∠1 is complementary to ∠6.
Justification
Statement
If m∠1 + m∠5 = m∠6 + m∠5,
then m∠1 = m∠6.
Justification
Statement
Justification
⊥ EG
Given: AC
Prove: ∠ABG is a right angle.
60
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
If ∠5 ∠6, then BF
bisects
∠EBC.
Two-Column Proofs
ACTIVITY 1.7
continued
Now I’m Convinced
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Vocabulary Organizer
My Notes
One way to learn how to write proofs is to start by studying completed
proofs and then practicing simple proofs before moving on to more
challenging proofs.
EXAMPLE 6
Theorem: Vertical angles are congruent.
1
Given: ∠1 and ∠2 are vertical angles.
Prove: ∠1 ∠2
1.
2.
3.
4.
5.
Statements
m∠1 + m∠3 = 180°
m∠2 + m∠3 = 180°
m∠1 + m∠3 = m∠2 + m∠3
m∠1 = m∠2
∠1 ∠2
1.
2.
3.
4.
5.
3 2
Reasons
Angle Addition Postulate
Angle Addition Postulate
Substitution Property
Subtraction Property of Equality
Definition of congruent angles
EXAMPLE 7
D
Theorem: The sum of the angles of a
triangle is 180°.
5
A
1
4
Given: ABC
Prove: m∠2 + m∠1 + m∠3 = 180°
MATH TERMS
© 2010 College Board. All rights reserved.
B
Statements
,
1. Through point A, draw AD
.
|| BC
so that AD
2. m∠5 + m∠1 + m∠4 = 180°
3. ∠5 ∠2; ∠4 ∠3
4. m∠5 = m∠2; m∠4 = m∠3
5. m∠2 + m∠1 + m∠3 = 180°
2
3
C
Reasons
1. Through any point not on a line,
there is exactly one line || to the
given line.
2. Angle Addition Postulate
3. If parallel lines are cut by a
transversal, then alternate interior
angles are congruent.
4. Definition of congruent angles
5. Substitution Property
Line AD is an example of an
auxiliary line. An auxiliary line
is a line that is added to a figure
to aid in the completion of a
proof.
Unit 1 • Proof, Parallel and Perpendicular Lines
61
ACTIVITY 1.7
Two-Column Proofs
continued
Now I’m Convinced
SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer
My Notes
GUIDED EXAMPLE
b
Supply the missing statements and reasons.
Theorem: In a plane, two lines perpendicular
a
to the same line are parallel.
∠1 and ∠2 are right angles.
m∠1 = 90°; m∠2 = 90°
m∠1 = m∠2
∠1 ∠2
a || c
Reasons
1. Given information
2. Definition of
3. Definition of
4.
5.
6.
b
TRY THESE B
a
a. Complete the proof.
Given: a || b, c || d
Prove: ∠2 ∠16
c
d
Statements
62
Reasons
1. c || d
1.
2. ∠2 ∠7
2.
3. a || b
3.
4. ∠7 ∠16
4.
5. ∠2 ∠16
5.
SpringBoard® Mathematics with Meaning™ Geometry
12
9
2 10 11
1
14
3
4
15
13
7 16
6
8
5
© 2010 College Board. All rights reserved.
1.
2.
3.
4.
5.
6.
2
c
Given: a ⊥ b, c ⊥ b
Prove: a || c
Statements
1
Two-Column Proofs
ACTIVITY 1.7
continued
Now I’m Convinced
SUGGESTED LEARNING STRATEGIES: Group Presentation
My Notes
TRY THESE B (continued)
b. Write a two-column proof for the following.
b
a
c
d
12
9 11
2 10
1
14
3
4
15 13
6 7 16
8
5
Given: a || b, c || d
Prove: ∠1 ∠15
Reasons
© 2010 College Board. All rights reserved.
Statements
Unit 1 • Proof, Parallel and Perpendicular Lines
63
ACTIVITY 1.7
Two-Column Proofs
continued
Now I’m Convinced
CHECK YOUR UNDERSTANDING
Writeyour
youranswers
answersononnotebook
notebook
paper.
Show your work.
Write
paper.
Show
9. Supply the Statements and Reasons.
your work.
Given: ∠1 is complementary to ∠2;
BE
bisects ∠DBC.
Lines CF, DH, and EA intersect at point B. Use this
Prove:
∠1
is complementary to ∠3.
figure for Items 1–8. Write the definition, postulate,
property, or theorem that justifies each statement.
A
C
D
1
E
B 6
2 3
D
A
5
4
H
B
G
1
E
2
3
F
C
1. ∠1 ∠5
Statements
2. If ∠2 is supplementary to ∠CBE,
then m∠2 + m∠CBE = 180°.
1. BE
bisects ∠DBC.
2. ∠2 ∠3
3. If ∠2 ∠3, then BF
bisects ∠GBE.
6. If m∠3 = m∠6, then m∠3 + m∠2 =
m∠6 + m∠2.
___
7. If AB BE, then B is the midpoint of AE.
8. m∠2 + m∠3 = m∠EBG
3. Definition of
congruent angles
4. ∠1 is complementary 4.
to ∠2.
5. m∠1 + m∠2 = ____ 5.
6. m∠1 + m∠3 = 90°
6.
7.
7. Definition of
complementary
angles
10. MATHEMATICAL Choose one of the theorems
R E F L E C T I O N about angles formed when
parallel lines are cut by a transversal
(Activity 1–6: Patios by Madeline). Explain
how you would set up and prove the theorem
in two-column format.
64
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
.
⊥ CF
5. If ∠DBF is a right angle, then HD
___
1.
2.
3.
4. CB + BF = CF
___
Reasons
Angles and Parallel Lines
Embedded
Assessment 2
THROUGH THE LOOKING GLASS
Use after Activity 1.7.
Light rays are bent as they pass through glass. Since a block of glass is a rectangular
prism, the opposite sides are parallel and a ray is bent the same amount entering a
piece of glass as exiting the glass.
B
G
I
R
E
F
S
A
1. Extend
and
through the block of glass. Label the points of intersection
__IR
__EF
with SB and AG; X and Y, respectively.
__
2. Name a pair of same side interior angles formed by the transversal RE.
© 2010 College Board. All rights reserved.
3. Identify the geometric term that describes the relationship between the pair of
angles ∠IRE and ∠REF.
4. Explain why a ray exits the glass in a path parallel to the path in which it
entered the glass.
Unit 1 • Proof, Parallel and Perpendicular Lines
65
Embedded
Assessment 2
Angles and Parallel Lines
Use after Activity 1.7.
THROUGH THE LOOKING GLASS
Exemplary
Proficient
Emerging
The student:
• Gives the name
of a pair of
angles that are
not same side
interior angles.
• Gives an incorrect
geometric term for
the relationship.
The student:
• Gives the correct
name for the pair
of angles. (2)
• Gives the correct
geometric
term for the
relationship
between the pair
of angles. (3)
Representations
#1
The student
correctly extends
the rays and
labels the points
of intersection
correctly. (1)
The student
correctly extends
the rays and labels
some, but not
all of the points
of intersection
correctly.
The student
extends the rays,
but does not
label the points
of intersection
correctly.
Communication
#4
The student
gives a complete
and correct
explanation. (4)
The student gives
an incomplete
explanation that
does not contain
mathematical
errors.
The student gives
an explanation
that is not correct.
© 2010 College Board. All rights reserved.
Math Knowledge
#2, 3
66
SpringBoard® Mathematics with Meaning™ Geometry
Equations of Parallel and
Perpendicular Lines
Skateboard Geometry
ACTIVITY
1.8
SUGGESTED LEARNING STRATEGIES: Look for a Pattern
My Notes
The new ramp at the local skate park is shown above. In addition to the
wooden ramp, an aluminum rail (not shown) is mounted to the edge of
the ramp. While the image of the ramp may conjure thoughts of kickflips,
nollies, and nose grinds, there are mathematical forces at work here as well.
1. Use the diagram of the ramp to complete the chart below:
© 2010 College Board. All rights reserved.
Describe two parts of the
ramp that appear to be
parallel.
Describe two parts of the
ramp that appear to be
perpendicular.
Describe two parts of
the ramp that appear to
be neither parallel nor
perpendicular.
Unit 1 • Proof, Parallel and Perpendicular Lines
67
ACTIVITY 1.8
Equations of Parallel and Perpendicular Lines
continued
Skateboard Geometry
SUGGESTED LEARNING STRATEGIES: Look for a Pattern,
Work Backward
My Notes
y Y
E
A
F
–12
CONNECT TO ALGEBRA
y y2 − y1
Slope = ____ = ______
x x2 − x1
V
X W
G
–8
S
T
U
R
8
O
J
I
H
4
B
P
K
L
–4
C
N
4
M
D
8
x
2. The diagram above shows a cross-section of the skate ramp transposed
onto a coordinate grid. Find the slopes of the following segments.
__
a. AB
__
b. XT
__
c. WG
_
d. TJ
__
3. Compare and contrast the slopes of the segments. Make conjectures
about how the slopes of these segments relate to the positions of the
segments on the graph.
68
SpringBoard® Mathematics with Meaning™ Geometry
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e. AD
Equations of Parallel and Perpendicular Lines
ACTIVITY 1.8
continued
Skateboard Geometry
SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge,
Predict and Confirm, Look for a Pattern, Identify a Subtask
My Notes
__
2 x + 4.
4. The equation of the line containing BD is y = −__
3
Identify the slope and the y-intercept of this line.
CONNECT TO ALGEBRA
5. Based on your conjectures
in Item 3, what would be the slope of the
__
line containing JM? Use the formula for slope to verify your answer
or provide an explanation of the method you used to determine the
slope.
The equation of a line in
slope-intercept form is
y = mx + b, where m represents
the slope and b represents the
y-intercept of the line.
__
6. Identify the y-intercept of the line containing JM. Use the slope and
the y-intercept to write the equation of this line.
__
_
7. If JM ⊥ RJ, identify
_ the slope, y-intercept, and the equation for the
line containing RJ.
__ _
© 2010 College Board. All rights reserved.
8. Is BD⊥RJ? Explain your reasoning.
Unit 1 • Proof, Parallel and Perpendicular Lines
69
ACTIVITY 1.8
Equations of Parallel and Perpendicular Lines
continued
Skateboard Geometry
CHECK YOUR UNDERSTANDING
Writeyour
youranswers
answersononnotebook
notebook
paper.
Show your work.
Write
paper.
Show
5. Consider the lines described below:
your work.
3x - 2
line a: a line with the equation y = __
5
1. What is the slope of a line parallel to the line
line
b:
a
line
containing
the
points
(-6,
1)
4 x - 7?
with equation y = __
and (3, -14)
5
2. What is the slope of a line perpendicular to
line c: a line containing the points (-1, 8)
the line with equation y = -2x + 1?
and (4, 11)
3. Find the equation of the line that has a
y-intercept of -4 and is parallel to the line
with equation y = 9x + 11.
Use the vocabulary from the lesson to write
a paragraph that describes the relationships
among lines a, b, c, and d.
6. MATHEMATICAL What is the slope of a line
R E F L E C T I O N parallel to the line with
equation y = -3? What is the slope of a line
perpendicular to the line with equation y = -3?
© 2010 College Board. All rights reserved.
4. Triangle ABC has vertices at the points
A(-2, 3), B (7, 4), and C(5, 22). What kind of
triangle is ABC? Explain your answer.
line d: a line with the equation -5x + 3y = 6
70
SpringBoard® Mathematics with Meaning™ Geometry
Distance and Midpoint
We Descartes
ACTIVITY
1.9
SUGGESTED LEARNING STRATEGIES: Predict and Confirm,
Visualization, Simplify the Problem
My Notes
If mathematics concept popularity contest were ever held, the Pythagorean
Theorem might be the runaway victor. With well over 300 known proofs,
people from Plato to U. S. President James A. Garfield have spent at least
part of their lives studying various proofs and applications of the theorem.
While some of these proofs involve a level of complexity comparable
to figuring out how to land a remote control car on the surface of Mars,
the following is a very simple example of how the Pythagorean Theorem
actually works.
c
MATH TERMS
The Pythagorean Theorem:
The square of the hypotenuse
of a right triangle is equal to the
sum of the squares of the legs
of the right triangle.
a
b
© 2010 College Board. All rights reserved.
1. Suppose you painted the large square extending from the hypotenuse
of the triangle above. If it required exactly one can of paint to cover the
area of the large square, make a conjecture as to how many cans you
would need to paint the other two squares. Explain your reasoning.
Unit 1 • Proof, Parallel and Perpendicular Lines
71
ACTIVITY 1.9
Distance and Midpoint
continued
We Descartes
SUGGESTED LEARNING STRATEGIES: Simplify the Problem,
Identify a Subtask, Create Representations
My Notes
You can use a common Pythagorean triple to explore the idea of the
painted squares further.
Pythagorean triples are sets of
whole numbers that represent the
side lengths of a right triangle.
Some common Pythagorean
triples are (3, 4, 5), (5, 12, 13),
(7, 24, 25), and (8, 15, 17).
5 cm
3 cm
4 cm
a=3
b=4
c=5
Area = a2 = 32 = 9
Area = b2 = 42 = 16
Area = c2 = 52 = 25
25 = 9 + 16
c2 = a2 + b2
Therefore,
52 = 32 + 42
72
SpringBoard® Mathematics with Meaning™ Geometry
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From the information in the figure above, you can derive that the area
of the large square is (5 cm)2 = 25 cm2, and that the areas of the smaller
squares are (3 cm)2 = 9 cm2 and (4 cm)2 = 16 cm2, respectively.
Distance and Midpoint
ACTIVITY 1.9
continued
We Descartes
SUGGESTED LEARNING STRATEGIES: Create Representations,
Identify a Subtask, Simplify the Problem
My Notes
You can use the Pythagorean theorem to determine the distance between
two points on a coordinate grid by drawing auxiliary lines to create
right triangles.
EXAMPLE 1
What is the distance between the points with coordinates (1, 4) and (13, 9)?
Step 1:
Start by plotting the points and connecting them.
MATH TERMS
The Cartesian Coordinate
System was developed by
the French philosopher and
mathematician René Descartes
in 1637 as a way to specify the
position of a point or object on
a plane.
y (13, 9)
(1, 4)
x
Step 2:
Add some auxiliary line segments to create a right triangle.
y © 2010 College Board. All rights reserved.
(13, 9)
(1, 4)
x
Unit 1 • Proof, Parallel and Perpendicular Lines
73
ACTIVITY 1.9
Distance and Midpoint
continued
We Descartes
SUGGESTED LEARNING STRATEGIES: Create
Representations, Identify a Subtask, Simplify the Problem
My Notes
EXAMPLE 1 (continued)
Step 3:
Count units along the horizontal and vertical legs to determine the
coordinates of the vertex of the right angle.
The coordinates of the vertex of the right angle are (13, 4).
y (13, 9)
5 units
(1, 4)
12 units
x
Step 4:
Use the Pythagorean theorem to find the distance between the two
points (the endpoints of the hypotenuse of the right triangle).
hypotenuse 2 = leg 2 + leg 2
c2 = 52 + 122
__
________
__
____
√c2 = √25 + 144
√c2 = √169
Solution: The distance between the points with coordinates (1, 4) and
(13, 9) is 13 units.
TRY THESE A
Plot each pair of points on grid paper, and demonstrate how you can use
the Pythagorean Theorem to find the distance between the points.
a. (-3, 6) and (3, 14)
74
SpringBoard® Mathematics with Meaning™ Geometry
b. (-8, 5) and (7, -3)
© 2010 College Board. All rights reserved.
c = 13
Distance and Midpoint
ACTIVITY 1.9
continued
We Descartes
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Simplify the Problem, Identify a Subtask
My Notes
While this method for finding the distance between two points will
always work, it may not be practical to plot points on a coordinate
grid and draw auxiliary lines each time you want to find the distance
between them.
In Example 1, you determined that the lengths of the legs of the right
triangle were 5 and 12 units, respectively. You could also have determined
5.
that the slope of the segment between the points (1, 4) and (13, 9) is ___
12
To understand why getting the numbers 5 and 12 is more than a
coincidence, recall some ways in which slope can be defined.
Δy y2 - y1
Slope = rate of change = ___ = ______
Δx x2 - x1
CONNECT TO SCIENCE
To find the length of the horizontal leg, you simply counted to get
12 units. However, if you think of the x-coordinates of the points as
x-values on a horizontal number line, the distance between x-values =
(Δx) = 13 − 1 = 12.
1
13
The Greek letter Δ (delta) is used
in mathematical formulas to
denote a change in quantities such
as value, magnitude, or position.
The expression Δ x would
represent a change in x. Similarly,
in Physics, the expression Δv
(where v = velocity) would
represent a change in velocity.
In other words, the distance between the x-values is the same as the
length of the horizontal leg of the right triangle.
Likewise, the distance between the y-values is the same as the length
of the vertical leg of the right triangle: (Δy) = 9 − 4 = 5
9
© 2010 College Board. All rights reserved.
4
Remember that distances are
always measured in positive
units.
Unit 1 • Proof, Parallel and Perpendicular Lines
75
ACTIVITY 1.9
Distance and Midpoint
continued
We Descartes
SUGGESTED LEARNING STRATEGIES: Identify a Subtask,
Simplify the Problem, Create Representations
My Notes
You can use algebraic methods to derive a formula for determining the
distance between two points.
From Example 1, the Pythagorean equation looked like this:
hypotenuse 2 = leg 2 + leg 2
c2 = 52 + 122
c2 = 25 + 144
__
________
__
____
√c2 = √25 + 144
√c2 = √169
13 = c
Now, redefine the variables based on exploration of slope on the
previous page.
a = horizontal leg = Δx = x2 - x1
b = vertical leg = Δy = y2 - y1
c = hypotenuse = distance between two points = d
Use substitution:
hypotenuse 2 = leg 2 + leg 2
c2 = a2 + b2
distance2 = (Δx)2 + (Δy)2
d 2 = (x2 - x1)2 + (y2 - y1)2
CONNECT TO AP
The coordinate geometry formulas
introduced in this activity are used
frequently in AP Calculus in a wide
variety of applications.
76
3. In the space to the right, create a Venn diagram that compares and
contrasts the mathematical operations performed in the Pythagorean
theorem and the distance formula.
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
2. Finish simplifying the equation to find the formula for determining
the distance d between two points.
Distance and Midpoint
ACTIVITY 1.9
continued
We Descartes
SUGGESTED LEARNING STRATEGIES: Guess and Check, Create
Representations, Work Backward, Visualization
My Notes
TRY THESE B
Use the distance formula to find the distance between the points with the
coordinates shown.
a. (-4, 6) and (15, 6)
b. (12, -5) and (-3, 7)
c. (8, 1) and (14, -2)
In Activity 1–5, you defined midpoint as the point on the segment that
divides it into two congruent segments.
4. Explore various methods, and explain how you could determine the
midpoint of the following segments:
a.
© 2010 College Board. All rights reserved.
b.
c.
–3
5
17
0
12
y (9, 12)
(0, 0)
x
Unit 1 • Proof, Parallel and Perpendicular Lines
77
ACTIVITY 1.9
Distance and Midpoint
continued
We Descartes
SUGGESTED LEARNING STRATEGIES: Identify a Subtask,
Simplify the Problem, Create Representations
My Notes
5. Consider the right triangle below:
y x
a. Using the definition of midpoint, identify and label the midpoints
of the legs of the triangle
b. Draw an auxiliary line vertically from the midpoint of the
horizontal leg through the hypotenuse
d. Identify the coordinates of the point of intersection of the two
auxiliary lines.
This point represents the midpoint of the hypotenuse!
TRY THESE C
Find the midpoint of the segment whose endpoints have the given
coordinates.
a. (2, 3) and (-5, 11)
b. (-8, -1) and (-7, -5)
c. (6, -11) and (22, 16)
78
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
c. Draw an auxiliary line horizontally from the midpoint of the
vertical leg through the hypotenuse.
Distance and Midpoint
ACTIVITY 1.9
continued
We Descartes
SUGGESTED LEARNING STRATEGIES: Identify a Subtask,
Simplify the Problem
My Notes
As was the case with the distance formula, it is not always practical to plot
the points on a coordinate grid and use auxiliary lines to find the midpoint
of a segment. Mathematically, the coordinates of the midpoint of a segment
are simply the averages of the x- and y-coordinates of the endpoints.
EXAMPLE 2
__
What are the coordinates of the midpoint of AC with endpoints A(1,4) and
C(11,10)?
Step 1:
Find the average of the x-coordinates:
x-coordinate of point A = 1
Step 2:
x-coordinate of C = 11
1 + 11 = 6
Average = ______
2
Find the average of the y-coordinates.
y-coordinate of point A = 4
y-coordinate of C = 10
4 + 10 = 7
Average = ______
2
___
Solution: The coordinates of the midpoint of AC are (6, 7).
TRY THESE D
__
a. Find the midpoint of QR with endpoints Q(-3, 14) and R(7, 5).
© 2010 College Board. All rights reserved.
b. Find the midpoint of S(13, 7) and T(-2, -7)
c. Find the midpoint of each side of the right triangle with vertices
D(-1, 2), E(8, 2), and F(8, 14).
6. Let (x1, y1) and (x2, y2) represent the coordinates of two points. Write
a formula for finding the coordinates of midpoint M of the segment
connecting those two points.
Unit 1 • Proof, Parallel and Perpendicular Lines
79
ACTIVITY 1.9
Distance and Midpoint
continued
We Descartes
CHECK YOUR UNDERSTANDING
Writeyour
youranswers
answersononnotebook
notebook
paper.
Show your work.
Write
paper
or grid
5. Find and explain the errors that were made
paper. Show your work.
in the following calculation of midpoint.
Then fix the errors, and determine the
1. Find the distance between the points
correct answer.
A(-8, -6) and B(4, 10).
Find the midpoint of the segment with
2. Find the distance between the points C(5, 14)
endpoints R(-2, 3) and S(13, -7).
and D(-3, -9)
-2 + 13 , _____
3-7
________
3. Use your Venn diagram from Item 3 to write
2
2
a RAFT .
-15 , ___
-4
= ____
2
2
Role: Teacher
= (-7.5, -2)
Audience: A classmate who was absent for the
6. The vertices of XYZ are X(-3, -6),
lesson on distance between two points
Y(21, -6) and Z(21, 4).
Format: Personal note
a. Find the length of each side of the triangle.
Topic: Explain the mathematical similarities
b. Find the coordinates of the midpoint of
and differences between the Pythagorean
each side of the triangle.
theorem and the distance formula.
7. MATHEMATICAL Segment RS is graphed on a
4. Find the midpoint of the segment with
R E F L E C T I O N coordinate grid. Explain
endpoints E(4, 11) and F(-11, -5).
how you would determine the coordinates of
the point on the segment that is one fourth of
the distance from R to S.
(
)
)
© 2010 College Board. All rights reserved.
(
80
SpringBoard® Mathematics with Meaning™ Geometry
Slope, Distance, and Midpoint
Embedded
Assessment 3
GRAPH OF STEEL
Use after Activity 1.9.
The first hill of the Steel Dragon 2000 roller coaster in Nagashima, Japan, drops
riders from a height of 318 ft. A portion of this first hill has been transposed onto a
coordinate plane and is shown to the right.
Write your answers on notebook paper or grid paper. Show your work.
y 1–6: See below.
1. The structure of the supports for the hill consists of steel beams that
run parallel and perpendicular to one another. The endpoints of the
longer of the two support beams highlighted in Quadrant I are (0, 150)
and (120, 0). If the endpoints of the other highlighted support beam
are (0, 125) and (100, 0), verify and explain why the two beams are
parallel.
2. Determine the equations of the lines containing the beams from Item
1, and explain how the equations of the lines can help you determine
that the beams are parallel.
3. The equation of a line containing another support beam is
4 x + 150. Determine whether this beam is parallel or
y = __
5
perpendicular to the other two beams, and explain your reasoning.
x
4. A linear portion of the first drop is also highlighted in the photo
and has endpoints of (62, 258) and (110, 132). To the nearest foot,
determine the distance between the endpoints of the linear section of the track.
Justify your result by showing your work.
© 2010 College Board. All rights reserved.
5. A camera is being installed at the midpoint of the linear portion of the track
described in Item 4. Determine the coordinates where this camera should be
placed.
6. Explain how you could use the distance formula to verify that the coordinates
you determined for the midpoint are correct.
Unit 1 • Proof, Parallel and Perpendicular Lines
81
Slope, Distance, and Midpoint
Use after Activity 1.9.
GRAPH OF STEEL
82
Exemplary
Proficient
Emerging
Math Knowledge
#2, 3, 4, 5
The student:
• Gives the correct
equations for the
parallel lines. (2)
• Determines the
correct relationship
among the beams. (3)
• Determines the correct
distance between the
points. (4)
• Correctly determines
the coordinates of the
midpoint of the linear
section. (5)
The student:
• Gives the correct
equation for one,
but not both, of
the parallel lines.
• Uses a correct
method to
determine the
distance but makes
a computational
error.
• Uses a correct
method to
determine the
coordinates of the
midpoint but makes
a computational
error.
The student:
• Gives incorrect
linear equations.
• Does not determine
the correct
relationship among
the beams.
• Uses an incorrect
method to
determine the
distance.
• Uses an incorrect
method to
determine the
coordinates of the
midpoint.
Representations
#2
The student gives the
correct equations for
the parallel lines. (2)
The student gives
the correct equation
for only one of the
lines.
The student gives
the correct equation
for neither of the
lines.
Communication
#1, 2, 3, 4, 6
The student:
• Verifies and writes a
correct explanation
for the reason the two
beams are parallel. (1)
• Gives a correct
explanation for the
connection between
the equations and the
parallel relationship
of the lines. (2)
• Gives a complete
explanation for the
parallelism and
perpendicularity of
the beams. (3)
• Shows work that is
complete. (4)
• Gives a complete
explanation of how
the distance formula
could be used to
verify the coordinates
of the midpoint. (6)
The student:
• Verifies and writes
an incomplete
explanation for
the reason the two
beams are parallel.
• Gives an
incomplete
explanation for
the connection
between the
equations and the
parallel relationship
of the lines.
• Gives a complete
explanation for the
parallelism or the
perpendicularity of
the beams but not
both.
• Shows incomplete
work that contains
no mathematical
errors.
• Gives an incomplete
explanation of how
the distance formula
could be used for
verification.
The student:
• Says the beams
are parallel, but
does not give an
explanation.
• Gives an incorrect
explanation for the
connection.
• Gives an incorrect
explanation for
the relationship
among the beams.
• Shows incorrect
work
• Gives an incorrect
explanation for the
use of the distance
formula.
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
Embedded
Assessment 3
Practice
UNIT 1
Use this circle Q for Items 4–7.
ACTIVITY 1.1
1. Which is the correct name for this line?
U
G
E
W
R
M
____
a. G
c. MG
b. GM
d. ME
Q
2. Use the diagram to name each of the following.
T
L
M
J
N
4. Name the radii of circle Q.
5. Name the diameter(s) in circle Q.
P
Q
6. Name the chord(s) in circle Q.
R
7. Which statement below must be true about
circle Q?
a. parallel lines
b. perpendicular lines
3. In this diagram, m∠SUT = 25°.
R
a. The distance from U to W is the same as the
distance from R to T.
b. The distance from U to W is the same as the
distance from Q to J.
c. The distance from R to T is half the distance
from Q to R.
d. The distance from R to T is twice the
distance from Q to J.
© 2010 College Board. All rights reserved.
Q
ACTIVITY 1.2
P
U
S
T
a. Name another angle that has measure 25°.
b. Name a pair of complementary angles.
c. Name a pair of supplementary angles.
8. Use inductive reasoning to determine the next
two terms in the sequence.
a. 1, 3, 7, 15, 31, …
b. 3, -6, 12, -24, 48, …
9. Write the first five terms of two different
sequences for which 24 is the third term.
10. Generate a sequence using the description:
the first term in the sequence is 2 and the
terms increased by consecutive odd numbers
beginning with 3.
Unit 1 • Proof, Parallel and Perpendicular Lines
83
UNIT 1
Practice
11. Use this picture pattern.
ACTIVITY 1.4
Construct a truth table for each of the following
compound statements.
12. Use expressions for odd integers to confirm the
conjecture that the product of two odd integers
is an odd integer.
ACTIVITY 1.3
13. Identify the property that justifies the statement:
5(x – 3) = 5x – 15
a. multiplication
c. subtraction
b. transitive
d. distributive
23. p ∧ ˜q
24. ˜(˜p ∨ q)
25. p ∨ (q → p)
ACTIVITY 1.5
26. If Q is between A and M and MQ = 7.3 and
AM = 8.5, then QA =
a. 5.8
Given: 5(x – 2) = 2x – 4
Prove:
15. Write the statement in if-then form.
The sum of two even numbers is even.
Use this statement for Items 16–19.
If today is Thursday, then tomorrow is Friday.
16. State the conclusion of the statement.
17. Write the converse of the statement.
18. Write the inverse of the statement.
19. Write the contrapositive of the statement.
20. Given the false conditional statement, “If a
vehicle is built to fly, then it is an airplane”,
write a counter example.
21. Give an example of a statement that is false and
logically equivalent to its inverse.
84
SpringBoard® Mathematics with Meaning™ Geometry
c. 7.3
d. 14.6
27. Given: K is between H and J, HK = 2x - 5,
KJ = 3x + 4, and HJ = 24. What is the value
of x?
a. 9
b. 5
c. 19
d. 3
__
28. If K is the midpoint of HJ , HK = x + 6, and
HJ = 4x - 6, then KJ = ? .
a. 15
14. Complete the prove statement and write a two
column proof for the equation:
b. 1.2
b. 9
c. 4
d. 10
29. P lies in the interior of ∠RST. m∠RSP = 40° and
m∠TSP = 10°. m∠RST = ?
a. 100°
b. 50°
c. 30°
d. 10°
30. QS
bisects ∠PQR. If m∠PQS = 3x and
m∠RQS = 2x + 6, then m∠PQR = ? .
a. 18°
b. 36°
c. 30°
d. 6°
31. ∠P and ∠Q are supplementary. m∠P = 5x + 3
and m∠Q = x + 3. x = ?
a. 14
b. 0
c. 29
d. 30
© 2010 College Board. All rights reserved.
a. Draw the next shape in the pattern,
b. Write a sequence of numbers that could be
used to express the pattern,
c. Verbally describe the pattern of the
sequence.
22. p → ˜q
Practice
, m∠BHC = 20°, and
⊥ CG
In this figure, AE
m∠DHE = 60°. Use this figure determine the
angle measures for Items 32–35.
UNIT 1
37. In this figure, m∠1 = 4x + 50 and
m∠3 = 2x + 66. Show all work for parts a, b,
and c.
A
F
B
1
G
H
2
C
3
D
E
32. m∠CHD
33. m∠AHF
34. m∠GHF
35. m∠DHA
36. In this figure, m∠3 = x + 18, m∠4 = x + 15,
and m∠5 = 4x + 3. Show all work for parts a, b,
and c.
a. What is the value of x?
b. What is the measure of m∠3?
c. What is the measure of m∠2?
38. Given: AE
⊥ AB
, AD
bisects ∠EAC,
m∠CAB = 2x - 4 and m∠CAE = 3x + 14.
Show all work for parts a and b.
E
© 2010 College Board. All rights reserved.
1
D
C
5
3
4
A
B
a. What is the value of x?
b. What is the measure of m∠DAC?
a. What is the value of x?
b. What is the measure of ∠1
c. Is ∠3 complementary to ∠1? Explain.
Unit 1 • Proof, Parallel and Perpendicular Lines
85
Practice
UNIT 1
ACTIVITY 1.6
49. If l n and m ⊥ n, then m∠3 =
?
Use this diagram for Items 39–50.
50. If l n and m ⊥ l, then m∠13 =
? .
ACTIVITY 1.7
n
Use this diagram to identify the property,
postulate, or theorem that justifies each
statement in Items 51–55.
l
12
11
10
16
9
1
13
2
14
3
5
P
D
4
2
15
8
A
4 Q
E
6
7
m
t
R
39. List all angles in the diagram that form a
corresponding angle pair with ∠5.
51. PQ + QR = PR
41. List all angles in the diagram that form a same
side interior angle pair with ∠14.
42. Given l n. If m∠6 = 120°, then m∠16 = ? .
43. Given l n. If m∠12 = 5x +10 and
m∠14 = 6x - 4, then x = ? and
m∠4 = ? .
44. Given l n. If m∠14 = 75°, then m∠2 =
? .
? .
47. Given l n. If m∠16 = 5x + 18 and
m∠15 = 3x+ 2, then x = ? and
m∠8 = ? .
48. If m∠10 = 135°, m∠12 = 75°, and l n determine
the measure of each angle. Justify each answer.
a. m∠16
c. m∠8
e. m∠3
Angle addition postulate
Addition property
Definition of congruent segments
Segment addition postulate
___
___
b. m∠15
d. m∠14
SpringBoard® Mathematics with Meaning™ Geometry
___
52. If Q is the midpoint of PR, then PQ QR.
a.
b.
c.
d.
Definition of midpoint
Definition of congruent segments
Definition of segment bisector
Segment addition postulate
53. ∠3 ∠4
45. Given l n. If m∠9 = 3x + 12 and
m∠15 = 2x + 27, then x = ? and
m∠10 = ? .
46. Given l n. If m∠3 = 100°, then m∠2 =
a.
b.
c.
d.
a.
b.
c.
d.
Definition of linear pair
Definition of congruent angles
Vertical angles are congruent.
Definition of angle bisector
54. If ∠1 is complementary to ∠2, then
m∠1 + m∠2 = 90
a.
b.
c.
d.
Angle Addition Postulate
Addition property
Definition of perpendicular
Definition of complementary
© 2010 College Board. All rights reserved.
40. List all angles in the diagram that form an
alternate interior angle pair with ∠14.
86
3
1
Practice
55. If m∠3 = 90, then ∠3 is a right angle.
ACTIVITY 1.9
a. Definition of perpendicular
b. Definition of complementary angles
c. Definition of right angle
d. Vertical angles are congruent.
56. Write a two-column proof.
61. Calculate the distance between the points
A(-4, 2) and B(15, 6).
Prove: ∠5 ∠12
b
a
9
1
2
4
62. Calculate the distance between the points
R(1.5, 7) and S(-2.3, -8).
63. Kevin and Clarice both live on a street that runs
through the center of town. If the police station
marks the midpoint between their houses, at
what point is the police station on the number
line?
Given: a b, c d
12
10
-11
11
15
3
6
7
5
14
16
13
8
c
d
ACTIVITY 1.8
57. Determine the slope of the line that contains the
points with coordinates (1, 5) and (-2, 7).
© 2010 College Board. All rights reserved.
UNIT 1
58. Which of the following is NOT an equation for a
1 x - 6?
line parallel to y = __
2
2
__
a. y = x + 6
b. y = 0.5x - 3
4
1x + 1
d. y = 2x - 4
c. y = __
2
59. Determine the slope of a line perpendicular to
the line with equation 6x - 4y = 30.
Kevin’s
House
1
Clarice’s
House
64. Determine the coordinates of the midpoint of
the segment with endpoints R(3, 16) and
S(7, -6).
65. Determine the coordinates of the midpoint of
the segment with endpoints W(-5, 10.2) and
X(12, 4.5).
66. Two explorers on an expedition to the Arctic
Circle have radioed their coordinates to base
camp. Explorer A is at coordinates (-26, -15).
Explorer B is at coordinates (13, 21). The base
camp is located at the origin.
a. Determine the linear distance between the
two explorers.
b. Determine the midpoint between the two
explorers.
c. Determine the distance between the
midpoint of the explorers and the base camp.
60. Line m contains the points with coordinates
(-4, 1) and (5, 8), and line n contains the points
with coordinates (6, -2) and (10, 7). Are the
lines parallel, perpendicular, or neither? Justify
your answer.
Unit 1 • Proof, Parallel and Perpendicular Lines
87
UNIT 1
Reflection
An important aspect of growing as a learner is to take the time to reflect on
your learning. It is important to think about where you started, what you have
accomplished, what helped you learn, and how you will apply your new knowledge in
the future. Use notebook paper to record your thinking on the following topics and to
identify evidence of your learning.
Essential Questions
1. Review the mathematical concepts and your work in this unit before you write
thoughtful responses to the questions below. Support your responses with
specific examples from concepts and activities in the unit.
Why are properties, postulates, and theorems important in mathematics?
How are angles and parallel and perpendicular lines used in real-world settings?
Academic Vocabulary
2. Look at the following academic vocabulary words:
angle bisector
counterexample
perpendicular
complementary angles
deductive reasoning
postulate
conditional statement
inductive reasoning
proof
congruent
midpoint of a segment
supplementary angles
conjecture
parallel
theorem
Choose three words and explain your understanding of each word and why each is
important in your study of math.
Self-Evaluation
3. Look through the activities and Embedded Assessments in this unit. Use a table
similar to the one below to list three major concepts in this unit and to rate your
understanding of each.
Is Your Understanding
Strong (S) or Weak (W)?
Concept 1
Concept 2
Concept 3
a. What will you do to address each weakness?
b. What strategies or class activities were particularly helpful in learning the
concepts you identified as strengths? Give examples to explain.
4. How do the concepts you learned in this unit relate to other math concepts and to
the use of mathematics in the real world?
88
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
Unit
Concepts
Math Standards
Review
Unit 1
1. Keisha walked from her house to Doris’ house. This can best be
represented by which of the following:
A. a ray
C. a line
B. a segment
D. a point
2. Sam wanted to go fishing. He took his fishing pole and walked
6 blocks due south to pick up his friend. Together, they walked
8 blocks due east before they reached the lake. There is a path
that goes directly from Sam’s house to the lake. If he had taken
that path, how many blocks would he have walked?
1. Ⓐ Ⓑ Ⓒ Ⓓ
2.
‒
○
⊘⊘⊘⊘
• ○
• ○
• ○
• ○
• ○
•
○
⓪⓪⓪ ⓪ ⓪ ⓪
①①① ① ① ①
②②② ② ② ②
③③③ ③ ③ ③
④④④ ④ ④ ④
⑤⑤⑤ ⑤ ⑤ ⑤
⑥⑥⑥ ⑥ ⑥ ⑥
⑦⑦⑦ ⑦ ⑦ ⑦
⑧⑧⑧ ⑧ ⑧ ⑧
⑨⑨⑨ ⑨ ⑨ ⑨
3. To amuse her brother, Shirka was making triangular formations 3.
out of rocks. She made the following pattern:
‒
○
⊘⊘⊘⊘
© 2010 College Board. All rights reserved.
• ○
• ○
• ○
• ○
• ○
•
○
How many rocks will be in the next triangular formation in the
pattern?
⓪⓪⓪ ⓪ ⓪ ⓪
①①① ① ① ①
②②② ② ② ②
③③③ ③ ③ ③
④④④ ④ ④ ④
⑤⑤⑤ ⑤ ⑤ ⑤
⑥⑥⑥ ⑥ ⑥ ⑥
⑦⑦⑦ ⑦ ⑦ ⑦
⑧⑧⑧ ⑧ ⑧ ⑧
⑨⑨⑨ ⑨ ⑨ ⑨
Unit 1 • Proof, Parallel and Perpendicular Lines
89
Math Standards
Review
Unit 1 (continued)
4. When camping, Sudi and Raul did not want to get lost. They
made a coordinate grid for their camp site. Camp was at the
Solve
point (0, 0). On their grid, Sudi went to the point (-3, 10) and
Explain
Raul went to the point (0, 6).
Read
Part A: On the grid below, draw a diagram of the camp site and
label the campsite location, Sudi’s location, and Raul’s location.
y
12
10
8
6
4
2
–12 –10 –8
–6
–4
–2
2
4
6
8
10
12
x
–2
–4
–6
–8
–10
Part B: Find the distance between Sudi and Raul.
Part C: Find the midpoint between Sudi and Raul.
90
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