Proof, Parallel and 1 Perpendicular Lines · midpoint of a segment parallel perpendicular postulate ... Write the equation of a line that has slope __1 3 ... Unit 1 • Proof, Parallel
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Why are properties, postulates, and theorems important in mathematics?
How are angles and parallel and perpendicular lines used in real-world settings?
Unit OverviewIn 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.
Academic VocabularyAdd these words and others you encounter in this unit to your Math Notebook.
midpoint of a segment parallel perpendicular postulate proof supplementary angles theorem
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 fi gures.
Unit 1 • Proof, Parallel and Perpendicular Lines 3
ACTIVITY
1.1Geometric FiguresWhat’s My Name?SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Activating Prior Knowledge, Group Presentation
Below are some types of fi gures you have seen in earlier mathematics courses. Describe each fi gure. Using geometric terms and symbols, list as many names as possible for each fi gure.
1. Q 2.
3. X Y Z 4. D E
5. � m 6.
N
χ
P
KT
J
7.
D B
A
C
TRY THESE A
Identify each geometric fi gure. Th en use symbols to write two diff erent names for each.
Unit 1 • Proof, Parallel and Perpendicular Lines 7
My Notes 13. In the space below, draw a circle with center P and radius PQ = 1 in. Locate a point A so that PA = 1 1 __ 2 in. Locate a point B so that PB = 3 __ 4 in.
14. Use your diagram to complete these statements.
a. A lies _________________ the circle because ___________________________________.
b. B lies _________________ the circle because ___________________________________.
15. In your diagram above, draw circles with radii PA and PB.Th ese three circles are called _______________________.
Unit 1 • Proof, Parallel and Perpendicular Lines 9
My Notes
ACTIVITY
1.2Logical ReasoningRiddle Me ThisSUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Vocabulary Organizer, Think/Pair/Share, Look for a Pattern
Th e 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. Th is 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 fi ne. Judy didn’t eat anything and also felt fi ne. 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.
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.
a. A, 4, C, 8, E, 12, G, 16, ,
b. 3, 9, 27, 81, 243, ,
c. 3, 8, 15, 24, 35, 48, ,
d. 1, 1, 2, 3, 5, 8, 13, ,
ACADEMIC VOCABULARY
conjectureinductive reasoning
CONNECT TO HISTORYHISTORY
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.
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My Notes
Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This
3. Use the four circles on the next page. From each of the given points, draw all possible chords. Th ese chords will form a number of non-overlapping regions in the interior of each circle. For each circle, count the number of these regions. Th en enter this number in the appropriate place in the table below.
Number of Points on the Circle
Number of Non-Overlapping Regions Formed
2345
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.
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.
SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Group Presentation
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My NotesMy Notes
Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This
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?
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.
d. Try to fi nd a new pattern for predicting the number of regions formed by chords joining points on a circle when two, three, four, fi ve, and six points are placed on a circle. Describe the pattern algebraically or in words.
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My Notes
Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This
ACTIVITY 1.2continued
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.
b. Compare your description above with the descriptions that others in your group or class have written. Below, record any descriptions that were diff erent from yours.
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 fi ndings below.
Unit 1 • Proof, Parallel and Perpendicular Lines 15
My Notes
ACTIVITY 1.2continued
Logical ReasoningRiddle Me ThisRiddle Me This
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
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 Squares Shaded in this Round
Total Number of Shaded Squares
Round 1: A 1 1Round 2: B 3 4Round 3: CRound 4: DRound 5: ERound 6: F
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Marking the Text, Predict and Confirm, Look for a Pattern, Group Presentation, Discussion Group
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My Notes
Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle 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
9. Th ink about the process you used to complete the table for Item 8c. Use your fi ndings in Item 8 to answer the following questions.
a. One method of describing the pattern in Item 6 would be to say, “Th e 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.
b. Another method of describing the pattern in Item 6 would be to say, “Th e 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.
Unit 1 • Proof, Parallel and Perpendicular Lines 17
My Notes
ACTIVITY 1.2continued
Logical ReasoningRiddle Me ThisRiddle Me This
SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Vocabulary Organizer, Marking the Text
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 confi rming 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. Th ese 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 defi nitions 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.
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 inte-ger and one odd integer. Show the examples you use and make a conjecture about the sum of an even integer and an odd integer.
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My Notes
Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Look for a Pattern, Use Manipulatives
Th e 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 fi gure represents an integer determined by counting the square pieces in the fi gure. Use these fi gures to answer the following questions.
a. Which of the fi gures can be used to model an even integer?
b. Which of the fi gures can be used to model an odd integer?
c. Compare and contrast the models of even and odd integers.
Unit 1 • Proof, Parallel and Perpendicular Lines 19
My Notes
ACTIVITY 1.2continued
Logical ReasoningRiddle Me ThisRiddle Me This
SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Discussion Group
14. Which pairs of puzzle pieces can fi t together to form rectangles? Make sketches to show how they fi t.
15. Explain how the fi gures (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.
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My Notes
Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This
SUGGESTED LEARNING STRATEGIES: Discussion Group
Th is is an algebraic defi nition 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.)
16. Use the expressions 2p and 2m, where p and m are integers, to confi rm the conjecture that the sum of two even integers is an even integer.
Th is is an algebraic defi nition 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.)
17. Use expressions for odd integers to confi rm the conjecture that the sum of two odd integers is an even integer.
18. Use expressions for even and odd integers to confi rm the conjecture that the sum of an even integer and an odd integer is an odd integer.
CONNECT TO PROPERTIESPROPERTIES
• A set has closure under an operation if the result of the operation on members of the set is also in the same set.
• In symbols, distributive property of multiplication over addition can be written as a(b + c) = a(b) + a(c).
Unit 1 • Proof, Parallel and Perpendicular Lines 21
My Notes
ACTIVITY 1.2continued
Logical ReasoningRiddle Me ThisRiddle Me This
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Summarize/Paraphrase/Retell, Create Representations, Group Presentation
In Items 14 and 15, you developed a geometric puzzle-piece argument to confi rm conjectures about the sums of even and odd integers. In Items 16–18, you developed an algebraic argument to confi rm these same conjectures. Even though one method is considered geometric and the other algebraic, they are oft en seen as the same basic argument.
19. Explain the link between the geometric and the algebraic methods of the proof.
20. State three theorems that you have proved about the sums of even and odd integers.
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Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This
Write your answers on notebook paper. Show your work.
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper. Show your work.
1. Use inductive reasoning to determine the next two terms in the sequence.
a. 1 __ 2 , 1 __ 4 , 1 __ 8 , 1 ___ 16 , …
b. A, B, D, G, K, …
2. Write the fi rst fi ve terms of two diff erent sequences that have 10 as the second term.
3. Generate a sequence using the description: the fi rst term in the sequence is 4 and each term is three more than twice the previous term.
Use this picture pattern for Item 4 at the top of the next column.
4. 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.
5. Use expressions for even and odd integers to confi rm the conjecture that the product of an even integer and an odd integer is an even integer.
6. MATHEMATICAL R E F L E C T I O N
Suppose you are given the fi rst fi ve terms of a
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.
Unit 1 • Proof, Parallel and Perpendicular Lines 23
My Notes
ACTIVITY
1.3The Axiomatic System of GeometryBack to the BeginningSUGGESTED LEARNING STRATEGIES: Close Reading, Quickwrite, Think/Pair/Share, Vocabulary Organizer, Interactive Word Wall
Geometry is an axiomatic system. Th at 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 defi nitions, for words used in a technical manner. For a defi nition to be helpful, it must be expressed in words whose meanings are already known and understood.
Compare the following defi nitions.
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.
1. Which of the two defi nitions above is easier to understand? Why?
For a new vocabulary term to be helpful, it should be defi ned using words that have already been defi ned. Th e fi rst defi nitions used in building a system, however, cannot be defi ned in terms of other vocabulary words because no other vocabulary words have been defi ned yet. In geometry, it is traditional to start with the simplest and most fundamental terms—without trying to defi ne them—and use these terms to defi ne other terms and develop the system of geometry. Th ese fundamental undefi ned terms are point, line, and plane.
2. Defi ne each term using the undefi ned terms.
a. Ray
b. Collinear points
c. Coplanar points
The term, line segment, can be defi ned in terms of undefi ned terms: A line segment is part of a line bounded by two points on the line called endpoints.
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My Notes
The Axiomatic System of Geometry ACTIVITY 1.3continued Back to the BeginningBack to the Beginning
Aft er a term has been defi ned, it can be used to defi ne other terms. For example, an angle is defi ned as a fi gure formed by two rays with a common endpoint.
3. Defi ne each term using the already defi ned terms.
a. Complementary angles
b. Supplementary angles
Th e 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. Th ese assertions are called axioms, or postulates.
When you solve algebraic equations, you are using deduction. Th e 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 - 1 _ 2 x. Name each operation or property used to justify each step.
Unit 1 • Proof, Parallel and Perpendicular Lines 25
My Notes
ACTIVITY 1.3continued
The Axiomatic System of Geometry Back to the BeginningBack to the Beginning
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. Th is format is called a two-column proof.
EXAMPLE 1
Given: 3(x + 2) - 1 = 5x + 11 Prove: x = -3
Statements Reasons1. 3(x + 2) - 1 = 5x + 11 1. Given equation2. 3(x + 2) = 5x + 12 2. Addition Property of Equality3. 3x + 6 = 5x + 12 3. Distributive Property4. 6 = 2x + 12 4. Subtraction Property of Equality5. -6 = 2x 5. Subtraction Property of Equality6. -3 = x 6. Division Property of Equality7. x = -3 7. Symmetric Property of Equality
TRY THESE A
a. Supply the reasons to justify each statement in the proof below.
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My NotesMy Notes
The Axiomatic System of Geometry ACTIVITY 1.3continued Back to the BeginningBack to the Beginning
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.
Given: 4x + 9 = 18 - 1 __ 2 x Prove:
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 oft en 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.
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
Use the conditional statement: If x + 7 = 10, then x = 3.
a. What is the hypothesis?
b. What is the conclusion?
c. State the property of equality that justifi es the conclusion of the statement.
ACADEMIC VOCABULARY
conditional statement
WRITING MATH
The letters p and q are often used to represent the hypo thesis and conclusion, respectively, in a conditional statement. The basic form of an if-then statement would then be, “If p, then q.”
SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Think/Pair/Share, Interactive Word Wall, Marking the Text, Group Presentation
Unit 1 • Proof, Parallel and Perpendicular Lines 27
My Notes
ACTIVITY 1.3continued
The Axiomatic System of Geometry Back to the BeginningBack to the Beginning
SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall
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.
b. Th ere is smoke only if there is fi re.
c. x = 4 implies x2 = 16.
An if-then statement is false if an example can be found for which the hypothesis is true and the conclusion is false. Th is type of example is a counterexample.
6. Th is is a false conditional statement.
If two numbers are odd, then their sum is odd.
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.
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My Notes
The Axiomatic System of Geometry ACTIVITY 1.3continued Back to the BeginningBack to the Beginning
SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall, Think/Pair/Share
Every conditional statement has three related conditionals. Th ese are the converse, the inverse, and the contrapositive of the conditional statement. Th e converse of a conditional is formed by interchanging the hypothesis and conclusion of the statement. Th e 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.
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 fi gure is a triangle, then it is a polygon.
Complete the table.
Form of the statement
Write the statement
True or False?
If the statement is false, give a
counterexample.
Conditional statement
If a fi gure is a triangle, then it is a polygon.
Converse of the conditional statementInverse of the conditional statementContrapositive of the conditional statement
converseinversecontrapositive
MATH TERMS
CONNECT TO APAP
When both a statement and its converse are true, you can connect the hypothesis and conclusion with the words “if and only if.”
Unit 1 • Proof, Parallel and Perpendicular Lines 29
My Notes
ACTIVITY 1.3continued
The Axiomatic System of Geometry Back to the BeginningBack to the Beginning
SUGGESTED LEARNING STRATEGIES: Group Presentation
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.
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 defi nitions you have learned can be written as “if and only if ” statements.
10. Write the defi nition 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.
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.
The truth value of a statement is the truth or falsity of that statement
Unit 1 • Proof, Parallel and Perpendicular Lines 31
My Notes
ACTIVITY
1.4Truth Tables The Truth of the MatterSUGGESTED LEARNING STRATEGIES: Close Reading, Activating Prior Knowledge, Group Discussion, Interactive Word Wall, Vocabulary Organizer, Think/Pair/Share
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. Th e 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.
Truth tables are used to determine the conditions under which a statement is true or false. Th is truth table displays the truth values for p → q, which are dependent on the truth values for p and q.
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.
p q p → qRow 1 T T TRow 2 T F FRow 3 F T TRow 4 F F T
p q p → qRow 1 T T TRow 2 T F FRow 3 F T TRow 4 F F T
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My Notes
Truth Tables ACTIVITY 1.4continued The Truth of the MatterThe Truth of the Matter
Th e symbol ~p is the negation of p and can be read as “not p.” Th is 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.
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.
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.
EXAMPLE
Make a truth table for ~p → q.
Step 1 Step 2Write down all Add a columnpossible T and F for ~p andcombinations for negate p.p and q.
Step 3Add a column for ~p → q and evaluate ~p → q.
SUGGESTED LEARNING STRATEGIES: Close Reading, Create Representations, Think/Pair/Share
Unit 1 • Proof, Parallel and Perpendicular Lines 33
My Notes
ACTIVITY 1.4continued
Truth TablesThe Truth of the MatterThe 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
8. Follow the steps and complete the truth table for ~(p → q).
Step 1Find all possible T and F combinations for p and q.
Step 2Evaluate p → q.
Step 3Negate p → q.
9. Write the steps and complete the truth table for ~q → ~p.
10. Th e 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 “Th e chord is a diameter.” Write the statement that is represented by p ↔ q.
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My NotesMy Notes
Truth Tables ACTIVITY 1.4continued The Truth of the MatterThe Truth of the Matter
Th e truth table for p ↔ q isshown to the right. Notice thatp ↔ q is true only whenp and q are both true or both false.
11. Construct a truth table for (p → q) ↔ ~q.
12. Th e symbol ∧ represents a conjunction. You read it as “and.” Let p represent “the fi gure is a rectangle,” and let q represent “the fi gure is a rhombus.”
a. Write the statement that is represented by p ∧ q.
b. Write the statement that is represented by p ∧ ~q.
Th e truth table for p ∧ q is shown to the right. Notice that p ∧ q is only true when both p and q are true.
SUGGESTED LEARNING STRATEGIES: Close Reading, Think/Pair/Share, Create Representations, Identify a Subtask, Discussion Group, Group Presentation, Interactive Word Wall, Vocabulary Organizer
p q p ↔ qT T TT F FF T FF F T
p q p ∧ qT T TT F FF T FF F F
A rectangle is a parallelogram with a right angle.
A rhombus is a parallelogram with consecutive congruent sides.
MATH TERMS
Because biconditionals are true only when both parts have the same truth value, many defi ni-tions are written in the form of a biconditional.
Unit 1 • Proof, Parallel and Perpendicular Lines 35
My Notes
ACTIVITY 1.4continued
Truth TablesThe Truth of the MatterThe Truth of the Matter
SUGGESTED LEARNING STRATEGIES: Close Reading, Create Representations, Identify a Subtask
13. Th e symbol ∨ represents a disjunction. You read it as “or.” Let p represent “the fi gure is not a rectangle,” and let q represent “the fi gure is a rhombus.”
a. Write the statement that is represented by p ∨ q.
b. Write the statement that is represented by ~p ∨ q.
Th e truth table for p ∨ q is shown to the right. Notice that p ∨ q is only false when both p and q are false.
Unit 1 • Proof, Parallel and Perpendicular Lines 37
Embedded Assessment 1 Conditional Statements and Logic
HEALTHY HABITS
Statements given in advertisements oft en 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 fi rst 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 infl uence 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 infl uence people to buy the vitamin.
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. Th ey have all heard cholesterol-lowering strategies discussed on talk shows and in commercials. Th ose strategies included: taking two red yeast rice tablets daily; eating oatmeal daily; practicing meditation for 1 __ 2 hour each day; and getting at least twenty minutes of aerobic exercise daily.
3. For one month, they each used strategies daily. Th ey 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 eff ective in lowering cholesterol levels? Explain your reasoning.
Unit 1 • Proof, Parallel and Perpendicular Lines 39
My Notes
ACTIVITYSegment and Angle MeasurementIt All Adds UpSUGGESTED LEARNING STRATEGIES: Close Reading
If two points are no more than one foot apart, you can fi nd the distance between them by using an ordinary ruler. (Th e inch rulers below have been reduced to fi t on the page.)
1inches
2 3 4 5 6 7
BA
In the fi gure, 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 fi gure below, the 2-inch mark is on point A. In this case, AB, measured in inches, is |7 - 2| = |2 - 7| = 5, as before.
2 3 4 5 6 7 8 9
BA
Th e 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.
D FE
a. DE = b. EF = c. DF =
2. Determine the length of each segment in centimeters.
G KH
a. KH = b. HG = c. GK =
The Ruler Postulate
a. To every pair of points there corresponds a unique positive number called the distance between the points.
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.
MATH TERMS
1.5
READING MATH
AB denotes the distance between points A and B. If A and B are the endpoints of a segment (
WZ , YZ = x + 3 and WZ = 3x - 4, determine the length of
__ WZ .
1. WZ = WY + YZ 1. Segment Addition Postulate2.
__ WY �
__ YZ 2. Defi nition of midpoint
3. WY = YZ 3. Defi nition of congruent segments4. WZ = YZ + YZ 4. Substitution Property5. 3x - 4 = (x + 3) + (x + 3) 5. Substitution Property6. x - 4 = 6 6. Subtraction Property of Equality7. x = 10 7. Addition Property of Equality8. WZ = 3(10) - 4 = 26 8. Substitution Property
TRY THESE A
Given: M is the midpoint of __
RS . Use the given information to fi nd the missing values.
a. RM = x + 3 and MS = 2x - 1 b. RM = x + 6 and RS = 5x + 3x = _____ and RM = _____ x = _____ and SM = _____
You measure angles with a protractor. Th e number of degrees in an angle is called its measure.
90
16020
180017010
1503014040
13050
12060
11070
10080
8010070
11060
12050130
40140
30150
20160
10 170
0 180
O AE
B
C
D
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 = ____
CONNECT TO ASTRONOMYASTRONOMY
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.
Unit 1 • Proof, Parallel and Perpendicular Lines 43
My Notes
ACTIVITY 1.5continued
Segment and Angle MeasurementIt All Adds UpIt All Adds Up
SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Vocabulary Organizer, Interactive Word Wall, Think/Pair/Share
8. Use a protractor to determine the measure of each angle.
R
TP
Q
a. m∠TQP = b. m∠TQR = c. m∠RQP =
9. Using your results from Items 7 and 8, describe any patterns that you notice.
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.
11. Use the angle addition postulate and the given information to complete each statement.
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 = .
The Protractor Postulate
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.
MATH TERMS
Item 10 and your answer together form a statement of the Angle Addition Postulate.
44 SpringBoard® Mathematics with Meaning™ Geometry
My Notes
Segment and Angle MeasurementACTIVITY 1.5continued It All Adds UpIt All Adds Up
SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall, Create Representations
11d. Lines DB and EC intersect at point F. If m∠BFC = 44° and m∠AFB = 61°, then
m∠AFC =
m∠AFE =
m∠EFD =
Th e bisector of an angle is a ray that divides the angle into two congruent adjacent angles. For example, in the fi gure to the left , if �� � BD bisects ∠ABC, then ∠ABD � ∠DBC.
Unit 1 • Proof, Parallel and Perpendicular Lines 47
My Notes
ACTIVITY
1.6Parallel and Perpendicular LinesPatios by Madeline
Matt works for Patios by Madeline. A new customer has asked him to design a brick patio and walkway. Th e customer requires a blueprint showing both the patio and the walkway. Th e 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.)
Euclidean geometry is based on fi ve postulates proposed by Euclid (about 300 BCE). His fi fth 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 fi fth postulate could be proven from the fi rst 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.
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
50 SpringBoard® Mathematics with Meaning™ Geometry
My Notes
Parallel and Perpendicular LinesACTIVITY 1.6continued Patios by MadelinePatios by Madeline
SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Quickwrite, Create Representations
A transversal is a line that intersects two or more coplanar lines in different points.
MATH TERMS
∠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 oft en stated in if-then form.
a. State the Corresponding Angles Postulate in if-then form.
b. State the converse of the Corresponding Angles Postulate.
c. How does the statement in part b ensure that Matt’s string lines are parallel?
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.
52 SpringBoard® Mathematics with Meaning™ Geometry
My Notes
Parallel and Perpendicular LinesACTIVITY 1.6continued Patios by MadelinePatios by Madeline
SUGGESTED LEARNING STRATEGIES: Quickwrite, Group Presentation
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.
1 2
3456
7 8
l
m
a. Explain why you think ∠3 and ∠6 are alternate interior angles.
b. Lines l and m in the diagram are parallel. What appears to be true about the measures of ∠3 and ∠6?
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.
Unit 1 • Proof, Parallel and Perpendicular Lines 53
My Notes
ACTIVITY 1.6continued
Parallel and Perpendicular LinesPatios by MadelinePatios by Madeline
SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Quickwrite, Create Representations
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 Th eorem you wrote in Item 5.
7. Determine the validity of the inverse of the Alternate Interior Angles Th eorem. Support your answer with illustrations.
8. State the converse of the Alternate Interior Angles Th eorem.
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 Th eorem ensure that Matt’s string lines are parallel?
Unit 1 • Proof, Parallel and Perpendicular Lines 55
My Notes
ACTIVITY 1.6continued
Parallel and Perpendicular LinesPatios by MadelinePatios by Madeline
SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Quickwrite, Create Representations
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 Th eorem you wrote in Item 12.
14. Determine the validity of the inverse of the Same-Side Interior Angles Th eorem. Support your answer with illustrations.
15. State the converse of the Same-Side Interior Angles Th eorem.
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.
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.
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.
X
W
Y
House
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.
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.
ACADEMIC VOCABULARY
Perpendicular fi gures intersect to form right angles.
Unit 1 • Proof, Parallel and Perpendicular Lines 57
My Notes
ACTIVITY 1.6continued
Parallel and Perpendicular LinesPatios by MadelinePatios by Madeline
SUGGESTED LEARNING STRATEGIES: RAFT
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.’’
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.
A proof is an argument, a justifi cation, 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.
Th ere are three basic requirements for constructing a good proof.
1. Awareness and knowledge of the defi nitions 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. Th e 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 justifi ed using a property, postulate, or defi nition.
EXAMPLE 1
Statement Justifi cationIf ∠ABE is a right angle,then m∠ABE = 90°. Defi nition of right angle
Unit 1 • Proof, Parallel and Perpendicular Lines 61
My Notes
ACTIVITY 1.7continued
Two-Column Proofs Now I’m ConvincedNow I’m Convinced
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
Th eorem: Vertical angles are congruent.
Given: ∠1 and ∠2 are vertical angles.Prove: ∠1 � ∠2
64 SpringBoard® Mathematics with Meaning™ Geometry
Two-Column Proofs Now I’m ConvincedNow I’m Convinced
ACTIVITY 1.7continued
Write your answers on notebook paper. Show your work.
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper. Show your work.
Lines CF, DH, and EA intersect at point B. Use this fi gure for Items 1–8. Write the defi nition, postulate, property, or theorem that justifi es each statement.
H
A
D
C
56
3 41 2
F
GE
B
1. ∠1 � ∠5
2. If ∠2 is supplementary to ∠CBE, then m∠2 + m∠CBE = 180°.
3. If ∠2 � ∠3, then � � BF bisects ∠GBE.
4. CB + BF = CF
5. If ∠DBF is a right angle, then � � � HD ⊥ � � � CF .
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
9. Supply the Statements and Reasons.
Given: ∠1 is complementary to ∠2; � � BE bisects ∠DBC.
Prove: ∠1 is complementary to ∠3.
B
D
E1 23
A
C
Statements Reasons
1. � � BE bisects ∠DBC. 1. 2. ∠2 � ∠3 2. 3. 3. Defi nition of
Unit 1 • Proof, Parallel and Perpendicular Lines 65
Embedded Assessment 2 Use after Activity 1.7.
Angles and Parallel LinesTHROUGH THE LOOKING GLASS
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.
I
B G
R E
F
AS
1. Extend � � IR and � � EF through the block of glass. Label the points of intersection with
__ SB and
__ AG ; X and Y, respectively.
2. Name a pair of same side interior angles formed by the transversal __
RE .
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 67
My Notes
ACTIVITY
1.8Equations of Parallel andPerpendicular LinesSkateboard GeometrySUGGESTED LEARNING STRATEGIES: Look for a Pattern
Th e 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 kickfl ips, 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:
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.
68 SpringBoard® Mathematics with Meaning™ Geometry
My Notes
Equations of Parallel and Perpendicular Lines ACTIVITY 1.8continued Skateboard GeometrySkateboard Geometry
SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Work Backward
x
y
CAFE
Y X WV
UT S
J
IH
G
B
D
ML
NP
O
K
R
–12 –8 –4 4
4
8
8
2. Th e 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
e. __
AD
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.
Unit 1 • Proof, Parallel and Perpendicular Lines 71
My Notes
ACTIVITY
1.9Distance and MidpointWe � DescartesSUGGESTED LEARNING STRATEGIES: Predict and Confirm, Visualization, Simplify the Problem
If mathematics concept popularity contest were ever held, the Pythagorean Th eorem might be the runaway victor. With well over 300 known proofs, people from Plato to U. S. President James A. Garfi eld 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 fi guring out how to land a remote control car on the surface of Mars, the following is a very simple example of how the Pythagorean Th eorem actually works.
b
c a
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.
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.
72 SpringBoard® Mathematics with Meaning™ Geometry
My Notes
Distance and Midpoint ACTIVITY 1.9continued We We �� Descartes Descartes
SUGGESTED LEARNING STRATEGIES: Simplify the Problem, Identify a Subtask, Create Representations
You can use a common Pythagorean triple to explore the idea of the painted squares further.
5 cm3 cm
4 cm
From the information in the fi gure 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.
a = 3 b = 4 c = 5
Area = a2 = 32 = 9 Area = b2 = 42 = 16 Area = c2 = 52 = 25
Th erefore,
52 = 32 + 42 25 = 9 + 16 c2 = a2 + b2
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).
Unit 1 • Proof, Parallel and Perpendicular Lines 73
My Notes
ACTIVITY 1.9continued
Distance and Midpoint We We �� Descartes Descartes
SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask, Simplify the Problem
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.
x
y
(13, 9)
(1, 4)
Step 2: Add some auxiliary line segments to create a right triangle.
x
y
(13, 9)
(1, 4)
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.
Unit 1 • Proof, Parallel and Perpendicular Lines 75
My Notes
ACTIVITY 1.9continued
Distance and Midpoint We We �� Descartes Descartes
SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Simplify the Problem, Identify a Subtask
While this method for fi nding 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 fi nd 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 that the slope of the segment between the points (1, 4) and (13, 9) is 5 ___ 12 .
To understand why getting the numbers 5 and 12 is more than a coincidence, recall some ways in which slope can be defi ned.
Slope = rate of change = Δy
___ Δx =
y2 - y1 ______ x2 - x1
To fi nd 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
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
4 9
CONNECT TO SCIENCESCIENCE
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 wouldrepresent a change in x. Similarly, in Physics, the expression Δv (where v = velocity) wouldrepresent a change in velocity.
Remember that distances are always measured in positive units.
76 SpringBoard® Mathematics with Meaning™ Geometry
My Notes
Distance and Midpoint ACTIVITY 1.9continued We We �� Descartes Descartes
SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Simplify the Problem, Create Representations
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, redefi ne 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
2. Finish simplifying the equation to fi nd the formula for determining the distance d between two points.
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.
CONNECT TO APAP
The coordinate geometry formulas introduced in this activity are used frequently in AP Calculus in a wide variety of applications.
Unit 1 • Proof, Parallel and Perpendicular Lines 79
My Notes
ACTIVITY 1.9continued
Distance and Midpoint We We �� Descartes Descartes
SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Simplify the Problem
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 fi nd 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 = 1x-coordinate of C = 11Average = 1 + 11 ______ 2 = 6
Step 2: Find the average of the y-coordinates.y-coordinate of point A = 4y-coordinate of C = 10Average = 4 + 10 ______ 2 = 7
Solution: Th e 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).
b. Find the midpoint of S(13, 7) and T(-2, -7)
c. Find the midpoint of each side of the right triangle with verticesD(-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 fi nding the coordinates of midpoint M of the segment connecting those two points.
Unit 1 • Proof, Parallel and Perpendicular Lines 81
Embedded Assessment 3 Use after Activity 1.9.
Slope, Distance, and MidpointGRAPH OF STEEL
Th e fi rst hill of the Steel Dragon 2000 roller coaster in Nagashima, Japan, drops riders from a height of 318 ft . A portion of this fi rst 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.
1. Th e structure of the supports for the hill consists of steel beams that run parallel and perpendicular to one another. Th e 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. Th e equation of a line containing another support beam is y = 4 __ 5 x + 150. Determine whether this beam is parallel or
perpendicular to the other two beams, and explain your reasoning.
4. A linear portion of the fi rst 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.
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 83
ACTIVITY 1.1
1. Which is the correct name for this line?
G E M
a. �� � G b. ____
GM c. � ��
MG d. �� � ME
2. Use the diagram to name each of the following.
L NM
P RQ
a. parallel lines b. perpendicular lines
3. In this diagram, m∠SUT = 25°.
P S
R
U
T
Q
a. Name another angle that has measure 25°. b. Name a pair of complementary angles. c. Name a pair of supplementary angles.
Use this circle Q for Items 4–7.
WR
J
Q
T
U
4. Name the radii of circle Q. 5. Name the diameter(s) in circle Q. 6. Name the chord(s) in circle Q. 7. Which statement below must be true about
circle Q? 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.
ACTIVITY 1.2
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 fi rst fi ve terms of two diff erent sequences for which 24 is the third term.
10. Generate a sequence using the description: the fi rst term in the sequence is 2 and the terms increased by consecutive odd numbers beginning with 3.
84 SpringBoard® Mathematics with Meaning™ Geometry
11. Use this picture pattern.
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.
12. Use expressions for odd integers to confi rm the conjecture that the product of two odd integers is an odd integer.
ACTIVITY 1.3
13. Identify the property that justifi es the statement: 5(x – 3) = 5x – 15
a. multiplication b. transitive c. subtraction d. distributive
14. Complete the prove statement and write a two column proof for the equation:
Given: 5(x – 2) = 2x – 4 Prove: 15. Write the statement in if-then form. Th e sum of two even numbers is even.
Use this statement for Items 16–19.
If today is Th ursday, 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 fl y, 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.
ACTIVITY 1.4
Construct a truth table for each of the following compound statements.
22. p → ˜q 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 b. 1.2 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 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 = ?
Unit 1 • Proof, Parallel and Perpendicular Lines 87
55. If m∠3 = 90, then ∠3 is a right angle. 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. Given: a ‖ b, c ‖ d Prove: ∠5 � ∠12
a
c
d
b
12
34
912
1110
1514
1316
67
85
ACTIVITY 1.8
57. Determine the slope of the line that contains the points with coordinates (1, 5) and (-2, 7).
58. Which of the following is NOT an equation for a line parallel to y = 1 __ 2 x - 6?
a. y = 2 __ 4 x + 6 b. y = 0.5x - 3
c. y = 1 __ 2 x + 1 d. y = 2x - 4
59. Determine the slope of a line perpendicular to the line with equation 6x - 4y = 30.
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.
ACTIVITY 1.9
61. Calculate the distance between the points A(-4, 2) and B(15, 6).
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?
Kevin’sHouse
-11 1
Clarice’sHouse
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). Th e 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.
88 SpringBoard® Mathematics with Meaning™ Geometry
An important aspect of growing as a learner is to take the time to refl ect 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 specifi c 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.
Unit Concepts
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 identifi ed 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?
088_SB_Geom_1-SR_SE.indd 88088_SB_Geom_1-SR_SE.indd 88 2/2/10 11:15:56 AM2/2/10 11:15:56 AM
Unit 1 • Proof, Parallel and Perpendicular Lines 89
Unit 1
Math Standards Review
1. Keisha walked from her house to Doris’ house. Th is 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 fi shing. He took his fi shing pole and walked 6 blocks due south to pick up his friend. Together, they walked 8 blocks due east before they reached the lake. Th ere 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?
3. To amuse her brother, Shirka was making triangular formations out of rocks. She made the following pattern:
How many rocks will be in the next triangular formation in the pattern?
90 SpringBoard® Mathematics with Meaning™ Geometry
Unit 1 (continued)
Math Standards Review
4. When camping, Sudi and Raul did not want to get lost. Th ey made a coordinate grid for their camp site. Camp was at the point (0, 0). On their grid, Sudi went to the point (-3, 10) and Raul went to the point (0, 6).
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.