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Understanding Algebraic Proofs and Postulates in Geometry

This lesson focuses on algebraic proofs in geometry, emphasizing how to effectively use postulates to validate geometric statements. Students will explore the relationships between points, lines, and planes, and learn how to construct two-column proofs employing properties of equality and logical reasoning. The lesson includes examples and practice problems, addressing statements like collinearity and coplanarity. Ultimately, students will enhance their problem-solving skills through algebraic reasoning and proof writing, preparing for standardized tests.

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Understanding Algebraic Proofs and Postulates in Geometry

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  1. Lesson 2-6 Algebraic Proof

  2. Transparency 2-6 5-Minute Check on Lesson 2-5 In the figure shown, A, C, and DH lie in plane R, and B is on AC. State the postulate that can be used to show each statement is true. 1. A,B, and C are collinear. 2. AC lies in plane R . 3. A, H, and D are coplanar. 4. E and F are collinear. 5. DH intersects EF at point B. 6. Which statement is not supported by a postulate? Standardized Test Practice: A R and S are collinear. M lies on LM. B C P, X and Y must be collinear. D J, K and L are coplanar.

  3. Transparency 2-6 5-Minute Check on Lesson 2-5 In the figure shown, A, C, and DH lie in plane R , and B is on AC. State the postulate that can be used to show each statement is true. 1. A,B, and C are collinear. A line contains at least two points. 2. AC lies in plane R . If two points lie in a plane, then the entire line containing those points lies in that plane. 3. A, H, and D are coplanar. Through any 3 points not on the same line, there is exactly one plane. 4. E and F are collinear. Through any 2 points, there is exactly one line. 5. DH intersects EF at point B. If two lines intersect, then their intersection is exactly one point. 6. Which statement is not supported by a postulate? Standardized Test Practice: A R and S are collinear. M lies on LM. B C P, X and Y must be collinear. D J, K and L are coplanar.

  4. Objectives • Use algebra to write two-column proofs • Use properties of equality in geometry proofs

  5. Vocabulary • Deductive argument– a group of logical steps used to solve problems • Two-column proof – also known as a formal proof

  6. Algebraic Properties

  7. Solve 2(5 – 3a) – 4(a + 7) = 92 Algebraic StepsProperties 2(5 – 3a) – 4(a + 7) = 92 Original equation 10 – 6a – 4a – 28 = 92 Distributive Property –18 – 10a = 92 Substitution Property –18 + 18 – 10a = 92 + 18 Addition Property –10a = 110 Substitution Property Division Property a = – 11 Substitution Property Answer: a = – 11

  8. If Write a two-column proof. then Proof: Statements Reasons 1. Given 1. 2. 2. Multiplication Property 3. 3. Substitution 4. 4. Subtraction Property 5. 5. Substitution 6. 6. Division Property 7. 7. Substitution

  9. Proof: Statements Reasons a. 1. Given 1. 2. Multiplication Property 2. 3. Substitution 3. 4. Subtraction Property 4. 5. Substitution 5. 6. Division Property 6. 7. Substitution 7. Write a two-column proof.

  10. Proof: Statements Reasons 1. Given 1. 2. Multiplication Property 2. 3. Distributive Property 3. 4. Subtraction Property 4. 5. Substitution 5. 6. Subtraction Property 6. 7. Substitution 7. Write a two-column proof. b.Given: Prove: a = –5

  11. MULTIPLE-CHOICE TEST ITEM If and then which of the following is a valid conclusion? I II III A I only B I and II C I and III D I, II, and III Read the Test Item Determine whether the statements are true based on the given information. Answer: B

  12. MULTIPLE- CHOICE TEST ITEM If and then which of the following is a valid conclusion? I. II. III. A I only B I and II C I and III D II and III Answer: C

  13. Proof: Statements Reasons Given: 1. 1. Given m leg 1 22 cm 2. 2. Transitive Property 3. m leg 1 m leg 3 3. Definition of congruence Prove: m leg 3 22 cm 4. m leg 1 22 cm 4. Given 5. m leg 3 22 cm 5. Transitive Property SEA LIFEA starfish has five legs. If the length of leg 1 is 22 centimeters, and leg 1 is congruent to leg 2, and leg 2 is congruent to leg 3, prove that leg 3 has length 22 centimeters.

  14. Summary & Homework • Summary: • Algebraic properties of equality can be applied to the measures of segments and angles to prove statements • Homework: • pg 97-8: 4-9, 15-18, 24, 25

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