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Name LESSON 2-6 Date Class Reteach Geometric Proof To write a geometric proof, start with the hypothesis of a conditional. Hypothesis Apply deductive reasoning. Deductive Reasoning • Definitions • Properties • Postulates • Theorems Prove that the conclusion of the conditional is true. Conclusion __› Conditional: If BD is the angle bisector of ABC, and ABD 1, then DBC 1. __› Given: BD is the angle bisector of ABC, and ABD 1. ! $ Prove: DBC 1 Proof: __› 1. BD is the angle bisector of ABC. 2. ABD DBC 3. ABD 1 4. DBC 1 " 1. 2. 3. 4. Given Def. of bisector Given Transitive Prop. of 1 # _ 1. Given: _ Prove: - N is the midpoint the _, Q is _ of MP _ midpoint of RP , and PQ NM . . _ PN QR 2 0 Write a justification for each step. Proof: _ 1. N is the midpoint of MP . 1. Given 2. Given 3. Def. of midpoint 4. Given 5. Transitive Prop. of 6. Def. of midpoint 7. Transitive Prop. of _ 2. Q is the midpoint of RP . _ _ 3. PN NM _ _ 4. PQ NM _ _ 5. PN PQ _ _ 6. PQ QR _ _ 7. PN QR Copyright © by Holt, Rinehart and Winston. All rights reserved. 46 1 Holt Geometry Name LESSON 2-6 Date Class Reteach Geometric Proof continued A theorem is any statement that you can prove. You can use two-column proofs and deductive reasoning to prove theorems. Congruent Supplements Theorem If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. Right Angle Congruence Theorem All right angles are congruent. Here is a two-column proof of one case of the Congruent Supplements Theorem. 4 and 5 are supplementary and 5 and 6 are supplementary. Given: 7 4 6 5 4 6 Prove: Proof: Statements Reasons 1. 4 and 5 are supplementary. 1. Given 2. 5 and 6 are supplementary. 2. Given 3. m4 m5 180 3. Definition of supplementary angles 4. m5 m6 180 4. Definition of supplementary angles 5. m4 m5 m5 m6 5. Substitution Property of Equality 6. m4 m6 6. Subtraction Property of Equality 7. 4 6 7. Definition of congruent angles Fill in the blanks to complete the two-column proof of the Right Angle Congruence Theorem. 1 2. Given: 1 and 2 are right angles. 2 Prove: 1 2 Proof: Statements 1. a. Reasons 1 and 2 are right angles. 1. Given 2. m1 90 3. c. 2. b. m2 90 3. Definition of right angle 4. m1 m2 5. e. 4. d. 1 2 Copyright © by Holt, Rinehart and Winston. All rights reserved. Definition of right angle Transitive Property of Equality 5. Definition of congruent angles 47 Holt Geometry Name Date LESSON 2-6 Class Practice A __› A. B. C. � � � � � 4. c. �1 � �2 Given 2. �1 and �2 are supplementary, and �3 and �4 are supplementary. 2. b. Linear Pair Thm. 3. c. m�1 � m�2 � 180°, and m�3 � m�4 � 180° Name LESSON 2-6 4. d. 43 Seg. Add. Post. 5. 2AB � AC, and 2EF � DF. Subst. 6. AB � EF Given 7. 2AB � 2EF Mult. Prop. of � 8. AC � DF Subst. Prop. of � � � � � Reasons �HKJ is a straight angle. 1. Given 2. b. 2. m�HKJ � 180� __› 3. c. KI bisects �HKJ 4. �IKJ � �IKH 4. Def. of � bisector 5. Def. of � � 5. m�IKJ � m�IKH 6. d. Def. of straight � 3. Given m�IKJ � m�IKH � m�HKJ 6. � Add. Post. 7. 2m�IKJ � 180� 7. e. Subst. (Steps 8. m�IKJ � 90� 8. Div. Prop. of � 9. �IKJ is a right angle. 9. f. 2, 5, 6 ) Def. of right � Add. Prop. of � Date Class Holt Geometry 2-6 Write a two-column proof. 4 Holt Geometry Hypothesis Deductive Reasoning • Definitions • Properties • Postulates • Theorems Apply deductive reasoning. Reasons 1. Given 2. Linear Pair Thm. 3. Def. of supp. � 4. Subst. Prop. of � 5. Subtr. Prop. of � Class Geometric Proof To write a geometric proof, start with the hypothesis of a conditional. 3 Statements 1. m�2 � m�3 � m�4 � 180� 2. �1 and �2 are supplementary. 3. m�1 � m�2 � 180� 4. m�1 � m�2 � m�2 � m�3 � m�4 5. m�1 � m�3 � m�4 Date Reteach LESSON 2 44 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name Geometric Proof 1 Def. of � segments Statements 1. a. Practice C 1. Given: The sum of the angle measures in a triangle is 180°. Prove: m�1 � m�3 � m�4 _ Proof: 3. Def. of supp. � 4. m�1 � m�2 � m�3 � m�4 � 360� Copyright © by Holt, Rinehart and Winston. All rights reserved. Def. of � segments 4. AB � BC � AC, and DE � EF � DF. 10. Given: �HKJ is a straight angle. __› KI bisects �HKJ. Prove: �IKJ is a right angle. Reasons 1. a. Def. of mdpt. Fill in the blanks to complete the two-column proof. Follow the plan to fill in the blanks in the two-column proof. 7. Given: �1 and �2 form a linear pair, and � � � �3 and �4 form a linear pair. � Prove: m�1 � m�2 � m�3 � m�4 � 360� Plan: The Linear Pair Theorem shows that �1 and �2 are supplementary and �3 and �4 are supplementary. The definition of supplementary says that m�1 � m�2 � 180� and m�3 � m�4 � 180�. Use the Addition Property of Equality to make the conclusion. 1. �1 and �2 form a linear pair, and �3 and �4 form a linear pair. Given _ 3. AB � BC, and DE � EF. _ 4. Def. of � � Statements _ � 9. AC � DF Def. of straight � 3. Subst. Prop. of � � 2. AB � BC, and DE � EF. 1. Given 3. m�1 � m�2 _ _ Reasons 2. b. � _ Fill in the blanks with the justifications and steps listed to complete the two-column proof. Use this list to complete the proof. �1 � �2 Def. of straight � 1 �1 and �2 are straight angles. 6. Given: �1 and �2 are straight angles. Prove: �1 � �2 2 Proof: 2. m�1 � 180�, m�2 � 180� � � � 1. B is the midpoint of AC, _ and E is the midpoint of DF. two-column 5. In a proof, each step in the proof is on the left and the reason for the step is on the right. �1 and �2 are straight angles. _ Given: AB = EF, B is the midpoint _of AC, and E is the midpoint of DF. � Definition of � bisector Given Transitive Prop. of � Statements Geometric Proof Write a justification for each step. � Class Practice B 2-6 B HJ is the bisector of �IHK. A �2 � �1 B �1 � �3 C �2 � �3 1. a. Date LESSON Geometric Proof Write the letter of the correct justification next to each step. (Use one __›justification twice.) Given: HJ is the bisector of �IHK and �1 � �3. 1. 2. 3. 4. Name Prove that the conclusion of the conditional is true. Conclusion __› Conditional: If BD is the angle bisector of �ABC, and �ABD � �1, then �DBC � �1. __› Given: BD is the angle bisector of �ABC, and �ABD � �1. 2. Peter drives on a straight road and stops at an intersection. The intersecting road is also straight. Peter notices that one of the angles formed by the intersection is a right angle. He concludes that the other three angles must also be right angles. Draw a diagram and write a two-column proof to show that Peter is correct. Possible answer: 4 1 3 Prove: �DBC � �1 Proof: __ › Statements 1. �1 is a right angle. 2. �1 and �2, �1 and �4, �2 and �3 are supplementary. 3. �1 � �3 4. �3 is a right angle. 5. m�1 � m�2 � 180�, m�1 � m�4 � 180� 6. m�1 � 90� 7. 90� � m�2 � 180�, 90� � m�4 � 180� 8. m�2 � 90�, m�4 � 90� 9. �2 and �4 are right angles. Reasons 1. Given 2. Linear Pair Thm. 1. Given 2. Def. of � bisector 3. �ABD � �1 3. Given 4. �DBC � �1 4. Transitive Prop. of � 1. Given: Prove: _ � � _ _ 2. Q is the midpoint of RP . _ _ _ _ _ _ _ _ 3. PN � NM 4. PQ � NM 5. PN � PQ 6. PQ � QR _ _ 7. PN � QR Copyright © by Holt, Rinehart and Winston. All rights reserved. 45 Copyright © by Holt, Rinehart and Winston. All rights reserved. Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 69 � PN � QR 1. N is the midpoint of MP . 8. Subtr. Prop. of � 9. Def. of rt. � � _ Proof: 6. Def. of rt. � 7. Subst. � _ N is the midpoint the _, Q is _ of MP _ midpoint of RP , and PQ � NM. Write a justification for each step. 3. Congruent Supps. Thm. 4. Rt. � � Thm. 5. Def. of supp. � 1 � � 1. BD is the angle bisector of �ABC. 2. �ABD � �DBC 2 � 46 � 1. Given 2. Given 3. Def. of midpoint 4. Given 5. Transitive Prop. of � 6. Def. of midpoint 7. Transitive Prop. of � Holt Geometry Holt Geometry Name LESSON 2-6 Date Class Name Reteach Date Challenge LESSON Geometric Proof 2-6 continued A theorem is any statement that you can prove. You can use two-column proofs and deductive reasoning to prove theorems. Prove It! In a proof, you can often determine the Given information from the figure. Congruent Supplements Theorem If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. Right Angle Congruence Theorem All right angles are congruent. Write the information that is given in each figure. Then make a conjecture about what you could prove using the given information. 2. 1. �4 and �5 are supplementary and �5 and �6 are supplementary. Prove: �4 � �6 7 4 6 � 1 5 2 3 � Proof: Statements Reasons 1. Given 2. Given 3. m�4 � m�5 � 180� 3. Definition of supplementary angles 4. m�5 � m�6 � 180� 4. Definition of supplementary angles � � 5. m�4 � m�5 � m�5 � m�6 5. Substitution Property of Equality Given: EF � EJ and FG � JH. right �. Possible answer: Prove Possible answer: Prove EH � EG. �KLM and �NML are right angles. �2 � �3 �1 � �4 6. Subtraction Property of Equality Prove: Possible answer: 2 Prove: �1 � �2 Proof: Statements Reasons 1. Given 3. c. 2. b. m�2 � 90� 5. e. 4. d. �1 � �2 47 Copyright © by Holt, Rinehart and Winston. All rights reserved. 2-6 Transitive Property of Equality 5. Definition of congruent angles Name LESSON Definition of right angle 3. Definition of right angle 4. m�1 � m�2 Date Holt Geometry Class Name 3 2-6 1. Refer to the diagram of the stained-glass window and use the given plan to write a two-column proof. 4 � Reasons 1. Given 2. Rt. � � Thm. 3. Def. of � � 4. � Add. Post. 5. Subst. Prop. of � 6. Given 7. Def. of � � 8. Subst. Prop. of � 9. Reflex. Prop. of � 10. Subtr. Prop. of � 11. Def. of � � Date Class Follow a Procedure The following five steps are used to give geometric proofs: The Proof Process Given: �1 and �3 are supplementary. �2 and �4 are supplementary. �3 � �4 1. Write the conjecture to be proven. 3 2. Draw a diagram if one is not provided. 1 Prove: �1 � �2 3. State the given information and mark it on the diagram. 4 2 Use the definition of supplementary angles to write the given information in terms of angle measures. Then use the Substitution Property of Equality and the Subtraction Property of Equality to conclude that �1 � �2. 4. State the conclusion of the conjecture in terms of the diagram. 5. Plan your argument and prove your conjecture. Mark the diagram and answer the questions about the following proof. __› Statements Reasons 1. �1 and �3 are supplementary. 1. Given �2 and �4 are supplementary. 2. m�1 � m�3 � 180� 2. Def. of supp. � m�2 � m�4 � 180� 3. m�1 � m�3 � m�2 � m�4 3. Subst. Prop. of � 4. �3 � �4 4. Given 5. m�3 � m�4 Given: FD bisects �EFC. � __› � FC bisects �DFB. Prove: �EFD � �CFB � � � Proof: __› __› 1. FD bisects �EFC, FC bisects �DFB. 1. Given 2. �EFD � �DFC, �DFC � �CFB 2. Definition of � bisector 5. Def. of � � 3. m�EFD � m�DFC, m�DFC � m�CFB 3. Definition of � � 6. m�1 � m�4 � m�2 � m�4 6. Subst. Prop. of � 4. m�EFD � m�CFB 4. Transitive Property of Equality 7. m�4 � m�4 7. Reflex. Prop. of � 5. �EFD � �CFB 5. Definition of � � 8. m�1 � m�2 8. Subtr. Prop. of � 9. �1 � �2 9. Def. of � � 1. What __ __ was the given information? › › FD bisects �EFC, FC bisects �DFB 2. What should be marked in the diagram? The position of a sprinter at the starting blocks is shown in the diagram. Which statement can be proved using the given information? Choose the best answer. All three angles should be marked congruent to each other. 3. What was the conjecture to be proved? � 2. Given: �1 and �4 are right angles. A �3 � �5 C m�1 � m�4 � 90� B �1 � �4 D m�3 � m�5 � 180� �EFD � �CFB 4. What titles should be put above the two columns? � Statements and Reasons 3. Given: �2 and �3 are supplementary. �2 and �5 are supplementary. Copyright © by Holt, Rinehart and Winston. All rights reserved. Holt Geometry Reading Strategies LESSON Geometric Proof F �3 � �5 G �2 � �5 _ � 2 48 Copyright © by Holt, Rinehart and Winston. All rights reserved. Problem Solving Plan: 1 � Statements 1. �KLM and �NML are right angles. 2. �KLM � �NML 3. m�KLM � m�NML 4. m�KLM � m�1 � m�2, m�NML � m�3 � m�4 5. m�1 � m�2 � m�3 � m�4 6. �2 � �3 7. m�2 � m�3 8. m�1 � m�2 � m�2 � m�4 9. m�2 � m�2 10. m�1 � m�4 11. �1 � �4 1 2. Given: �1 and �2 are right angles. � � Given: 7. Definition of congruent angles �1 and �2 are right angles. _ 3. Write a two-column proof. 7. �4 � �6 2. m�1 � 90� � Given: �1 � �4 and �ABC is a 6. m�4 � m�6 1. a. � 4 �2 � �3. 1. �4 and �5 are supplementary. 2. �5 and �6 are supplementary. Fill in the blanks to complete the two-column proof of the Right Angle Congruence Theorem. � � Here is a two-column proof of one case of the Congruent Supplements Theorem. Given: Class � H �3 and �5 are complementary. J �1 and �2 are supplementary. 49 Copyright © by Holt, Rinehart and Winston. All rights reserved. � 5. What property of equality was used to prove the angles congruent? � Transitive Property of Equality Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 70 50 Holt Geometry Holt Geometry