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

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1 Flow Chart and Paragraph Proofs SUGGESTED 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 flow 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. D A Given: M is the midpoint of BD ; M is the midpoint of AC Prove: AB DC M C B CONNECT TO AP ACTIVITY 2.6 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 flow chart and paragraph proofs will help develop your ability to document and explain your thinking in future courses. 1. Use what you know about congruent triangles to write a paragraph proof to justify that the opposite sides in the diagram are parallel. Unit 2 Congruence, Triangles, and Quadrilaterals 139

2 ACTIVITY 2.6 Flow Chart and Paragraph Proofs MATH TERMS A flow chart is a concept map showing a procedure. The boxes represent specific actions and the arrows connect actions to show the flow of the logic. CONNECT TO CAREERS Computer programmers and game designers use flow charts to describe an algorithm, or plan, for the logic in a program. SUGGESTED LEARNING STRATEGIES: Marking the Text, Close Reading, Create Representations, Identify a Subtask One way to organize your thoughts into a logical sequence is a flow chart. A significant difference between paragraph and flow chart proofs is justification. Many times, in paragraph proofs, justifications that are expected to be clear are omitted. However, in a flow chart proof each statement must be justified. The categories of valid reasons in a proof are: Algebraic properties Definitions Given Postulates Theorems A flow 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 flow 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. 140 SpringBoard Mathematics with Meaning Geometry

3 Flow Chart and Paragraph Proofs ACTIVITY 2.6 SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Identify a Subtask b. Remember that in a flow chart proof, each statement must be justified. Write given on the line under the boxes in #1 and #2 on page Once a statement has been placed on the flow chart proof and has been justified, 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 flow chart on page 142. b. Mark the information on your diagram using appropriate symbols. 5. The statement you made in Item 4 can be justified with a definition. In a proof, when a definition is used as a reason, it is written as definition of. Write the reason that justifies the statement in box #3 on the line below the box on page An argument similar to the one in Items 4 and 5 can be made concerning point M and AC. Use this argument to fill in box #4 on page 142 and supply the reason that justifies 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. Unit 2 Congruence, Triangles, and Quadrilaterals 141

4 ACTIVITY 2.6 Flow Chart and Paragraph Proofs SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask A B M D C SpringBoard Mathematics with Meaning Geometry

5 Flow Chart and Paragraph Proofs ACTIVITY 2.6 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share b. Use appropriate mathematical notation to add the information pertaining to AMB and DMC to your flow chart in box #5 on page 142. c. Include a valid reason for your statement on the line below the box. 8. There 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. There are two triangles in the diagram that appear to be congruent. a. Is there enough information in your diagram and flow 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. Unit 2 Congruence, Triangles, and Quadrilaterals 143

6 ACTIVITY 2.6 Flow Chart and Paragraph Proofs SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations c. Record the congruence statement in box #6 on your flow chart. Include the reason on the line below the box. d. There are 3 arrows connecting box #6 to previous boxes. Explain why the three arrows are necessary. 10. There 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. This is often 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. 144 SpringBoard Mathematics with Meaning Geometry

7 Flow Chart and Paragraph Proofs ACTIVITY 2.6 SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations 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 flow 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 flow chart. Explain your reasoning in the space below this question. Unit 2 Congruence, Triangles, and Quadrilaterals 145

8 ACTIVITY 2.6 Flow Chart and Paragraph Proofs SUGGESTED LEARNING STRATEGIES: Create Representations Remember that a two column proof lists the statements in the left column and the corresponding reasons that justify each statement in the right column. 12. Refer to the flow 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 flow chart proof to create a two column proof. Statements Reasons SpringBoard Mathematics with Meaning Geometry

9 Flow Chart and Paragraph Proofs ACTIVITY 2.6 SUGGESTED LEARNING STRATEGIES: Self/Peer Revision, Quickwrite Writing your flow chart proof involved several steps. First the diagram, given information, and prove statement were added to the flow chart. Next, information that could be justified 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. M A H T Given: MA HT ; MAT THM Prove: MH AT Unit 2 Congruence, Triangles, and Quadrilaterals 147

10 ACTIVITY 2.6 Flow Chart and Paragraph Proofs SUGGESTED LEARNING STRATEGIES: Self/Peer Revision, Think/Pair/Share 14. Write a flowchart proof for the proof in Item Write a two column proof for the proof in Item 13. Statements Reasons 148 SpringBoard Mathematics with Meaning Geometry

11 Flow Chart and Paragraph Proofs ACTIVITY 2.6 CHECK YOUR UNDERSTANDING 1. If the first statement in a flow 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 definition justifies 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 A 4 3 B 2 D 1 C 1. AD BC Given 4. BD BD Unit 2 Congruence, Triangles, and Quadrilaterals 149

12 ACTIVITY 2.6 Flow Chart and Paragraph Proofs CHECK YOUR UNDERSTANDING () 4. Complete a flow chart proof for the following. Write your proof on notebook paper. Given: CD bisects AB ; A B Prove: AED BEC A E C 5. A cell phone tower on level ground is supported at Q by 3 wires of equal length. The 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. D B Q F D T E 6. MATHEMATICAL REFLECTION You have studied proofs with two columns, flow chart proofs, and paragraph proofs. What are some of the advantages and disadvantages of each? 150 SpringBoard Mathematics with Meaning Geometry