1. Lesson 2-6 Algebraic Proof

2. 5-Minute Check on Lesson 2-5 Transparency 2-6 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 C B D R and S are collinear. M lies on LM. P, X and Y must be collinear. J, K and L are coplanar.

3. 5-Minute Check on Lesson 2-5 Transparency 2-6 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 C B D R and S are collinear. M lies on LM. P, X and Y must be collinear. 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 Properties of Equality for Real Numbers Reflexive For every a, a = a Symmetric For all numbers a and b, if a = b, then b = a Transitive For all numbers a, b, and c, if a = b and b = c, then a = c Addition & Subtraction For all numbers a, b, and c, if a = b, then a + c = b + c and a – c = b - c Multiplication & Division For all numbers a, b, and c, if a = b, then ac = bc and if c ≠ 0, a/c = b/c Substitution For all numbers a and b, if a = b, then a may be replaced by b in any equation or expression Distributive For all numbers a, b, and c, a(b + c) = ab + ac

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

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

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

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

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

13. SEA LIFE A 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. Given: m leg 1 22 cm Prove: m leg 3 22 cm Proof: Statements Reasons 1. Given1. 2. Transitive Property2. 3. Definition of congruencem leg 1 m leg 33. 4. Givenm leg 1 22 cm4. 5. Transitive Propertym leg 3 22 cm5.

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