Flow Chart and Paragraph Proofs ACTIVITY Go with …tristanbates.wikispaces.com/file/view/Geometry - Unit 2...Unit 2 • Congruence, Triangles, and Quadrilaterals 139 My Notes ACTIVITY

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Unit 2 • Congruence, Triangles, and Quadrilaterals 139

My Notes

ACTIVITY

2.6Flow Chart and Paragraph Proofs Go with the FlowSUGGESTED LEARNING STRATEGIES: Marking the Text, Prewriting, Self/Peer Revision,

You know how to write two-column and paragraph proofs. In this activity you will use your knowledge of congruent triangles to explore ! ow chart proofs.

As lawyers must gather physical evidence and agree on facts before they proceed to trial and attempt to prove their claim, you must organize the information you need before beginning a formal geometry proof. Usually, this consists of a diagram, some given information, and a statement to prove.

You will use the diagram and statements below to prove that the opposite sides in the diagram are parallel.

A B

M

CD

Given: M is the midpoint of __

BD ; M is the midpoint of __

AC Prove:

__ AB !

__ DC

1. Use what you know about congruent triangles to write a paragraph proof to justify that the opposite sides in the diagram are parallel.

CONNECT TO APAP

The free response portions of the AP Calculus and Statistics examinations require you to justify your work. Learning to use logical reasoning in geometry through fl ow chart and paragraph proofs will help develop your ability to document and explain your thinking in future courses.

139-150_SB_Geom_2-6_SE.indd 139139-150_SB_Geom_2-6_SE.indd 139 1/21/10 4:21:20 PM1/21/10 4:21:20 PM

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140 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Flow Chart and Paragraph Proofs ACTIVITY 2.6continued Go with the FlowGo with the Flow

One way to organize your thoughts into a logical sequence is a ! ow chart. A signi" cant di# erence between paragraph and ! ow chart proofs is justi" cation. Many times, in paragraph proofs, justi" cations that are expected to be clear are omitted. However, in a ! ow chart proof each statement must be justi" ed. $ e categories of valid reasons in a proof are:

• Algebraic properties• De" nitions• Given• Postulates• $ eorems

A ! ow chart proof can begin with statements based on given information or information that can be assumed from the diagram. Flow chart proofs will end with the statement you are trying to prove is true.

2. Use the diagram on the previous page and the prove statements to prove that the opposite sides in the diagram are parallel. Begin with the statement you are trying to prove. Put that statement in the box below and in box #8 on page 142 for the proof.

3. Start your ! ow chart for this proof with the given information.

a. Put your statements in the boxes below and in boxes #1 and #2 on page 142 for the proof.

SUGGESTED LEARNING STRATEGIES: Marking the Text, Close Reading, Create Representations, Identify a Subtask

CONNECT TO CAREERSCAREERS

Computer programmers and game designers use fl ow charts to describe an algorithm, or plan, for the logic in a program.

A fl ow chart is a concept map showing a procedure. The boxes represent specifi c actions and the arrows connect actions to show the fl ow of the logic.

MATH TERMS


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My Notes

ACTIVITY 2.6continued

Flow Chart and Paragraph Proofs Go with the FlowGo with the Flow

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Identify a Subtask

b. Remember that in a ! ow chart proof, each statement must be justi" ed. Write given on the line under the boxes in #1 and #2 on page 142.

4. Once a statement has been placed on the ! ow chart proof and has been justi" ed, that statement can be used to support other statements.

a. Use mathematical notation to write a statement about ___

BM and ____ DM that can be supported with the given statement “M is the

midpoint of ___

BD .” Put your statement in box #3 on the ! ow chart on page 142.

b. Mark the information on your diagram using appropriate symbols.

5. # e statement you made in Item 4 can be justi" ed with a de" nition. In a proof, when a de" nition is used as a reason, it is written as de! nition of . Write the reason that justi" es the statement in box #3 on the line below the box on page 142.

6. An argument similar to the one in Items 4 and 5 can be made concerning point M and

___ AC . Use this argument to " ll in box #4 on

page 142 and supply the reason that justi" es the statement in the box on the line below the box. Mark the information on your diagram using appropriate symbols.

7. Is there more information that can be assumed from the diagram? If so, what is it?

a. Mark the information on your diagram using appropriate symbols.


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My Notes


SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask

A B

M

CD

7.

3.1.

4. 6.2.

5.

8.


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SUGGESTED LEARNING STRATEGIES: Think/Pair/Share

b. Use appropriate mathematical notation to add the information pertaining to !AMB and !DMC to your ! ow chart in box #5 on page 142.

c. Include a valid reason for your statement on the line below the box.

8. " ere is no arrow from any prior statements to box #5. Why is it unnecessary to draw an arrow from any prior statements to the statement in box #5?

9. " ere are two triangles in the diagram that appear to be congruent.

a. Is there enough information in your diagram and ! ow chart to prove conclusively that the two triangles are indeed congruent? Explain your answer.

b. Justify your answer in part (a) by naming the appropriate congruent triangle method.


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My Notes


c. Record the congruence statement in box #6 on your ! ow chart. Include the reason on the line below the box.

d. " ere are 3 arrows connecting box #6 to previous boxes. Explain why the three arrows are necessary.

10. " ere is some information that can be concluded based on the fact that !CMD " !AMB.

a. Write this information below, using mathematical statements.

b. What valid mathematical reason supports all of the statements listed in part (a)?

c. Because the triangles are congruent, the corresponding parts are congruent. " is is o# en referred to as CPCTC (corresponding parts of congruent triangles are congruent) in proofs. Since these statements have a valid reason, mark them onto your diagram.

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations


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My Notes



d. Which of the congruence statements from part (a) can be used to prove the fact that

___ AB !

___ DC ? Explain your answer.

e. Record the relevant statement from part (a) in box #7 on your ! ow chart. Include the reason given in part (b) on the line below the box.

11. Based on your knowledge of parallel line postulates and theorems, add a reason on the line below the prove statement written in box #8 on your ! ow chart. Explain your reasoning in the space below this question.

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations


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My Notes


SUGGESTED LEARNING STRATEGIES: Create Representations

Remember that a two column proof lists the statements in the le! column and the corresponding reasons that justify each statement in the right column.

12. Refer to the " ow chart proof that you wrote on page 142.

a. Record the diagram you drew with the relevant information indicated, the given information, and the statement to be proven.

b. List the statements and the reasons for each box on your ! ow chart proof to create a two column proof.

Statements Reasons1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.


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My Notes



Writing your ! ow chart proof involved several steps. First the diagram, given information, and prove statement were added to the ! ow chart. Next, information that could be justi" ed using a valid reason was added to the proof in a logical sequence. Finally, the statement to be proved true was validated.

13. Write a paragraph proof for the following.

Given: ____

MA ‖ ___

HT ; ∠MAT # ∠THMProve:

____ MH #

___ AT

SUGGESTED LEARNING STRATEGIES: Self/Peer Revision, Quickwrite

AM

TH


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My Notes


14. Write a ! owchart proof for the proof in Item 13.

15. Write a two column proof for the proof in Item 13.

Statements Reasons

SUGGESTED LEARNING STRATEGIES: Self/Peer Revision, Think/Pair/Share

2.

3. 5.

6.

1.

4.


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CHECK YOUR UNDERSTANDING

1. If the ! rst statement in a " ow chart proof is ! " BX is the bisector of ∠ABC, then which of the following should be the second statement in the proof?

a. ∠ABC $ ∠XBC

b. m∠ABX + m∠XBC = m∠ABC

c. ∠ABX $ ∠CBX

d. ∠ABC $ ∠CBX

2. Which theorem or de! nition justi! es the statement: If E is the midpoint of

__ AF ,

then __

AE $ __

EF ?

3. Supply the missing statements and/or reasons below for the following proof.

Given: __

AD ‖ __

BC ; __

AD $ __

BC

Prove: &ABD $ &CDB

2. ∠2" ∠3

3. 5.

1.

Given

4. BD" BD

AD ‖ BC

12

34

D C

A B


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CHECK YOUR UNDERSTANDING (continued)

4. Complete a ! ow chart proof for the following. Write your proof on notebook paper.

Given: ___

CD bisects ___

AB ; ∠A " ∠BProve: #AED " #BEC

D B

A

E

C

5. A cell phone tower on level ground is supported at Q by 3 wires of equal length. " e wires are attached to the ground at points D, E and F, which are equidistant from a point T at the base of the tower. Explain in a paragraph how you can prove that the angles the wires make with the ground are all congruent.

Q

D

F

E

T

6. MATHEMATICAL R E F L E C T I O N

You have studied proofs with two columns, ! ow

chart proofs, and paragraph proofs. What are some of the advantages and disadvantages of each?



Flow Chart and Paragraph Proofs ACTIVITY Go with …tristanbates.wikispaces.com/file/view/Geometry - Unit 2...Unit 2 • Congruence, Triangles, and Quadrilaterals 139 My Notes ACTIVITY

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