SUGGESTED LEARNING STRATEGIES:

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1 Two-olumn Proofs Now I'm onvinced SUGGST LRNING STRTGIS: lose Reading, ctivating Prior Knowledge, Think/Pair/Share proof is an argument, a justification, or a reason that something is true. proof is an answer to the question why? when the person asking wants an argument that is indisputable. There are three basic requirements for constructing a good proof.. wareness and knowledge of the definitions of the terms related to what you are trying to prove.. 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. TIVITY.7 To write a proof, you must be able to justify statements. The statements in the xamples 5 are based on the diagram to the right in which lines, G, and F all intersect at point B. ach of the statements is justified using a property, postulate, or definition. B 5 6 G F XMPL 00 ollege Board. ll rights reserved. If B is a right angle, then m B = 90. XMPL If and 5, then 5. XMPL 3 efinition of right angle Transitive Property Given: B is the midpoint of. Prove: B B efinition of midpoint Unit Proof, Parallel and Perpendicular Lines 59

2 TIVITY.7 Two-olumn Proofs SUGGST LRNING STRTGIS: Think/Pair/Share XMPL 4 m + m B = m B ngle ddition Postulate XMPL 5 If is supplementary to FBG, then m + m FBG = 80. efinition of supplementary angles B 5 6 G F TRY THS Using the diagram from the previous page, reproduced to the left, name the property, postulate, or definition that justifies each statement. a. B + BG = G b. If 5 6, then BF bisects B. c. d. e. If m + m 6 = 90, then is complementary to 6. If m + m 5 = m 6 + m 5, then m = m ollege Board. ll rights reserved. Given: G Prove: BG is a right angle. 60 SpringBoard Mathematics with Meaning Geometry

3 Two-olumn Proofs TIVITY.7 SUGGST LRNING STRTGIS: ctivating Prior Knowledge, Vocabulary Organizer 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. XMPL 6 Theorem: Vertical angles are congruent. Given: and are vertical angles. Prove: 3 s. m + m 3 = 80. ngle ddition Postulate. m + m 3 = 80. ngle ddition Postulate 3. m + m 3 = m + m 3 3. Substitution Property 4. m = m 4. Subtraction Property of quality efinition of congruent angles 00 ollege Board. ll rights reserved. XMPL 7 Theorem: The sum of the angles of a triangle is 80. Given: B Prove: m + m + m 3 = 80 s. Through point, draw, so that B. 3. Through any point not on a line, there is exactly one line to the given line.. m 5 + m + m 4 = 80. ngle ddition Postulate 3. 5 ; If parallel lines are cut by a transversal, then alternate interior angles are congruent. 4. m 5 = m ; m 4 = m 3 4. efinition of congruent angles 5. m + m + m 3 = Substitution Property B 5 4 MTH TRMS Line is an example of an auxiliary line. n auxiliary line is a line that is added to a figure to aid in the completion of a proof. Unit Proof, Parallel and Perpendicular Lines 6

4 TIVITY.7 Two-olumn Proofs SUGGST LRNING STRTGIS: Vocabulary Organizer GUI XMPL Supply the missing statements and reasons. b Theorem: In a plane, two lines perpendicular to the same line are parallel. Given: a b, c b Prove: a c a c s.. Given information. and are right angles.. efinition of 3. m = 90 ; m = efinition of 4. m = m a c 6. TRY THS B a. omplete the proof. a b Given: a b, c d Prove: 6 s. c d.. 7. c d ollege Board. ll rights reserved. 3. a b SpringBoard Mathematics with Meaning Geometry

5 Two-olumn Proofs TIVITY.7 SUGGST LRNING STRTGIS: Group Presentation TRY THS B () b. Write a two-column proof for the following. a b c d Given: a b, c d Prove: 5 s 00 ollege Board. ll rights reserved. Unit Proof, Parallel and Perpendicular Lines 63

6 TIVITY.7 Two-olumn Proofs HK YOUR UNRSTNING Write your answers on on notebook paper. paper. Show Show your work. 9. Supply the s and. your work. Given: is complementary to ; Lines F, H, and intersect at point B. Use this B bisects B. figure for Items 8. Write the definition, postulate, Prove: is complementary to 3. property, or theorem that justifies each statement. B F. 5. If is supplementary to B, then m + m B = If 3, then BF bisects GB. 4. B + BF = F 5. If BF is a right angle, then H F. 6. If m 3 = m 6, then m 3 + m = m 6 + m. 7. If B B, then B is the midpoint of. 8. m + m 3 = m BG H G B 3 s. B bisects B efinition of congruent angles 4. is complementary 4. to. 5. m + m = m + m 3 = efinition of complementary angles 0. MTHMTIL RFLTION hoose one of the theorems about angles formed when parallel lines are cut by a transversal (ctivity 6: Patios by Madeline). xplain how you would set up and prove the theorem in two-column format. 00 ollege Board. ll rights reserved. 64 SpringBoard Mathematics with Meaning Geometry