ACTIVITY 15 Continued Lesson 15-2

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1 Continued PLAN Pacing: 1 class period Chunking the Lesson Examples A, B Try These A B #1 2 Example C Lesson Practice TEACH Bell-Ringer Activity Read the introduction with students and remind them of the definition of vertical. Encourage them to add this term and the Vertical Angles Theorem to their Vocabulary Organizers and to their list of theorems. Example A, Guided Example B Vocabulary Organizer, Think-Pair- Share, Close Reading, Discussion Groups, Construct an Argument, Critique Reasoning This is the first opportunity for students to see formal geometric proofs. The relationship between statements and reasons in a two-column proof should be discussed. Asking students to explain how the reason justifies the statement in several steps of the proof is one method of having students analyze the proofs in this Example. Students will prove that vertical have the same measure. They will then use that conclusion to prove other theorems. Because this is the first theorem that students have encountered, note that the only justifications that can be used are definitions, properties, and postulates. Then they will be able to use the Vertical Angles Theorem to prove other theorems. Observe the structure of the proof above the two columns. There is a statement of the theorem that will be proved, along with the statement. In Example A, the information and the Prove statement is stated in terms of the diagram. After the proof of the Vertical Angles Theorem in Example A, students are asked to complete the proof of another theorem in Example B, namely that all right are congruent. Students should add both of these theorems as well as the term theorem to their Vocabulary Organizers. MATH TERMS The Vertical Angles Theorem states that vertical are congruent. DISCUSSION GROUP TIPS As you listen to the group discussion, take notes to aid comprehension and to help you describe your own ideas to others in your group. Ask questions to clarify ideas and to gain further understanding of key concepts. Differentiating Instruction Support students who struggle to distinguish given information from what they are to prove by having them rewrite the theorem as an if-then statement: If two are vertical, then they are congruent. Have students identify the hypothesis as the information, and the conclusion is what they need to write for the Prove statement. Extend the lesson and challenge students to rewrite the theorem All right are congruent as an if-then statement. Ask them to draw a diagram to accompany the and Prove statements. Learning Targets: Complete two-column proofs to prove theorems about segments. Complete two-column proofs to prove theorems about. SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Think-Pair-Share, Close Reading, Discussion Groups, Self Revision/Peer Revision, Group Presentation When writing two-column proofs to solve algebraic equations, you justify each statement in these proofs by using an algebraic property. Now you will use two-column proofs to prove geometric theorems. You must justify each statement by using a definition, a postulate, a property, or a previously proven theorem. Recall that vertical are opposite formed by a pair of intersecting lines. In the figure below, 1 and 2 are vertical. The following example illustrates how to prove that vertical are congruent. Example A Theorem: Vertical are congruent. : 1 and 2 are vertical. Prove: m 1 + m 3 = Definition of supplementary 2. m 2 + m 3 = Definition of supplementary 3. m 1 + m 3 = m 2 + m 3 3. Substitution Property 4. m 1 = m 2 4. Subtraction Property of Equality Definition of congruent Guided Example B Supply the missing statements and reasons. Theorem: All right are congruent. : A and B are right. Prove: A B A and B are right m A = 90 ; m B = Definition of right 3. m A = m B 3. Transitive Property 4. A B 4. Definition of congruent 260 SpringBoard Integrated Mathematics I, Unit 3 Lines, Segments, and Angles

2 Try These A B a. Complete the proof. : Q is the midpoint of PR. QR RS Prove: PQ RS 1. Q is the midpoint of PR PQ 2. QR b. Complete the proof. : 1 and 2 are supplementary. m 1 = 68 Prove: m 2 = and 2 are supplementary. 1. P Q R Definition of midpoint 3. QR 3. RS 4. PQ 4. RS Transitive Property of Congruence m 1 + m 2 = Definition of supplementary 3. m 1 = m 2 = Substitution Property 5. m 2 = Subtraction Property of Equality in a proof must be arranged in a logical order. 1. If you are given the information that A and B are straight, what is the logical order for the two statements below? Explain your reasoning. m A = m B m A = 180 and m B = 180 You need to state the measures of the before stating that the measures are equal, so the logical order is the statement m A = m B followed by the statement m A = 180 and m B = 180. S More than one statement in a two-column proof can be given information. Continued Try These A B Vocabulary Organizer, Think-Pair-Share, Close Reading, Discussion Groups, Construct an Argument, Critique Reasoning In each proof, note that there are two statements that are given information and that both are necessary for proving the conclusion. Have students discuss these proofs and monitor discussions to ensure that all students are participating. Encourage students to challenge others in their group to justify or explain the statements they make in the discussion. Then have groups together present their proofs to the rest of the class. Monitor students group discussions to ensure that complex mathematical concepts are being verbalized precisely and that all group members are actively participating in discussions through the sharing of ideas and through asking and answering questions appropriately. Developing Math Language Evaluate students use of all types of vocabulary in their written responses to ensure that they are using everyday words as well as academic vocabulary and math terms correctly. 1 2 Activating Prior Knowledge, Close Reading In these items, students build on their knowledge of the nature of a proof. Without writing full proofs, they put the essential steps of an argument in the correct order. Confirm that students understand that each step in a proof follows from an earlier step or steps. Activity 15 Proofs About Line Segments and Angles 261

3 Continued Example C Vocabulary Organizer, Think-Pair-Share, Close Reading, Discussion Groups, Construct an Argument, Critique Reasoning Ask students to cover the bottom of the page with a sheet of paper, and challenge them to work together to place the statements and reasons in logical order. When they have finished, they can compare their work with the proof at the bottom of the page. Students have not used the Transitive Property with congruence before this time. Remind students that two are congruent when their measures are equal and that the Transitive Property of Equality can be applied to angle measures. Point out that there are several algebra properties that can be used with congruence: Reflexive Property (a = a) A figure is always congruent to itself. Symmetric Property (If a = b, then b = a.) For example, if AB CD, then CD AB. Transitive Property (If a = b and b = c, then a = c.) For example, if A B and B C, then A C. Encourage students to add these properties to their Vocabulary Organizers. Have students add the theorem from Example C to their Vocabulary Organizers, and tell them that they will be able to use this theorem to justify other statements in future proofs. The theorem stated in Example C is called the Congruent Complements Theorem. 2. You are given that X and Y are complementary, and that m X = 45. What is the logical order for the statements below? Explain m Y = 90 m X + m Y = 90 You need to use the definition of complementary to state that the sum of the measures of X and Y is 90. Then you can substitute 45 for m X. So, the logical order is m X + m Y = 90, followed by the statement 45 + m Y = 90. Example C Arrange the statements and reasons below in a logical order to complete the proof. Theorem: If two are complementary to the same angle, then the two are congruent. A : A and B are each Prove: A B B C m A + m C = m B + m C A and B are each A B m A = m B m A + m C = 90 ; m B + m C = 90 Transitive Property Definition of congruent segments Subtraction Property of Equality Definition of complementary Start the proof with the given information. Then decide which statement and reason follow logically from the first statement. Continue until you have proved that A B. 1. A and B are each m A + m C = 90 m B + m C = Definition of complementary 3. m A + m C =m B + m C 3. Transitive Property 4. m A = m B 4. Subtraction Property of Equality 5. A B 5. Definition of congruent 262 SpringBoard Integrated Mathematics I, Unit 3 Lines, Segments, and Angles

4 a. Attend to precision. Arrange the statements and reasons below in a logical order to complete the proof. : 1 and 2 are vertical ; 1 3. Prove: and 2 are vertical. Vertical are congruent. Transitive Property b. Write a two-column proof of the following theorem. Theorem: If two are supplementary to the same angle, then the two are congruent. : R and S are each supplementary to T. Prove: R S Continued Vocabulary Organizer, Think-Pair-Share, Close Reading, Discussion Groups, Construct an Argument, Critique Reasoning Students may notice the parallelism between part b and Example C. Encourage them to add each new theorem to their Vocabulary Organizers. Monitor students written work carefully and note that the structure is important to help them organize their proofs in a logical manner. The graphic organizer Proof Builder may be a helpful tool for students who struggle with organizing their proofs. As students use the graphic organizer, encourage them to draw diagrams based on the statements they are given. Answers a. Sample proof is shown. 3. If you know that D and F are both complementary to J, what statement could you prove using the Congruent Complements Theorem? 4. What types of information can you list as reasons in a two-column geometric proof? 5. Kenneth completed this two-column proof. K What mistake did he make? How could L you correct the mistake? : uru 35 JL bisects KJM; m KJL = 35. Prove: m LJM = 35 J M uru 1. JL bisects KJM KJL LJM 2. Definition of congruent 3. m KJL = m LJM 3. Definition of angle bisector 4. m KJL = m LJM = Transitive Property Answers 3. D F 4. given information, postulates, properties, definitions, and previously proven theorems 5. The reasons for 2 and 3 do not make sense logically based on the previous statements of the proof. Based on Statement 1 (JL bisects KJM), Kenneth can conclude that KJL LJM by the definition of an angle bisector. Based on Statement 2 ( KJL LJM), he can conclude that m KJL = m LJM by the definition of congruent and 2 1. are vertical Vertical are congruent Transitive Property b. Sample proof is shown. 1. R and S are 1. each supplementary to T. 2. m R + m T = 180 ; m S + m T = m R + m T = m S + m T 2. Definition of supplementary 3. Transitive Property 4. m R = m S 4. Subtraction Property of Equality 5. R S 5. Definition of congruent Debrief students answers to these items to ensure that students understand the structure of a two-column proof. Discuss the answers as a class and be sure to address any misconceptions that students may have. Students who have difficulty may benefit from additional practice with putting statements and reasons for a proof in logical order. Activity 15 Proofs About Line Segments and Angles 263

5 Continued ASSESS Students answers to the Lesson Practice items will provide a formative assessment of their understanding of writing two-column proofs to prove theorems about segments and, and of students ability to apply their learning. Short-cycle formative assessment items for are also available in the Assessment section on SpringBoard Digital. Refer back to the graphic organizer the class created when unpacking Embedded Assessment 2. Ask students to use the graphic organizer to identify the concepts or skills they learned in this lesson. LESSON 15-2 PRACTICE BE bisects 1. DBC Def. of bisector 3. m 2 = m 3 3. Def. of 4. 1 is 4. complementary to m 1 + m 2 = m 1 + m 3 = is complementary to 3. ADAPT 5. Def. of compl. 6. Substitution 7. Def. of compl. Check students answers to the Lesson Practice to ensure that they understand the basics of two-column proofs. Call students attention to the Math Tip about drawing a diagram if one is not provided. For students who struggle, provide additional practice with placing statements and reasons in logical order and have them work with another student. See the Activity Practice on page 275 and the Additional Unit Practice in the Teacher Resources on SpringBoard Digital for additional problems for this lesson. You may wish to use the Teacher Assessment Builder on SpringBoard Digital to create custom assessments or additional practice. If the given information does not include a diagram, it may be helpful to make a sketch to represent the information. 7. Sample proof is shown. 1. M is the midpoint 1. of LN. 2. LM MN 2. Def. of mdpt. 3. LM = MN 3. Def. of segments 4. LM = MN = 8 5. Trans. Prop. 6. LN = LM + MN 6. Segment Addition Postulate 7. LN = = Substitution LESSON 15-2 PRACTICE 6. Supply the missing statements and reasons. ur uu : 1 is complementary to 2; BE bisects DBC. Prove: 1 is complementary to 3. uru 1. BE bisects DBC Definition of congruent 4. 1 is complementary to m 2 = m 1 + m 3 = Definition of complementary Construct viable arguments. Write a two-column proof for Items : M is the midpoint of LN; LM = 8. Prove: LN = : BD ur uu bisects ABC; m DBC = 90. Prove: ABC is a straight angle. 9. : PQ QR, QR = 14, PR = 14 Prove: PQ PR A B Reason abstractly. What type of triangle is shown in Item 9? Explain how you know. 8. Sample proof is shown. 1. BD bisects ABC. D C E D A B C P 1. Q 2. ABD DBC 2. Def. of bisector 3. m ABD = m DBC 3. Def. of 4. m DBC = m ABD = Trans. Prop. 6. m ABC = m ABD + m DBC 6. Add. Post. 7. m ABC = = Substitution Property 8. ABC is a straight angle. 8. Definition of straight R 264 SpringBoard Integrated Mathematics I, Unit 3 Lines, Segments, and Angles