1. 2.6 Algebraic Proof

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

3. ALGEBRAIC PROPERTIES OF EQUALITY Reflexive Property a = a. Symmetric Property If a = b, then b = a. Symmetric Property If a = b, then b = a. Addition Property of Equality If a = b, then a + c = b + c. Addition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a – c = b – c. Subtraction Property of Equality If a = b, then a – c = b – c. Multiplication Property of Equality If a = b, then ac = bc. Multiplication Property of Equality If a = b, then ac = bc. Division Property of Equality If a = b, then a/c = b/c. Division Property of Equality If a = b, then a/c = b/c. Transitive Property of Equality If a = b and b = c, then a = c. Transitive Property of Equality If a = b and b = c, then a = c. Distributive Property a(b + c) = ab + ac. Distributive Property a(b + c) = ab + ac. Substitution Property of Equality If a = b, then you may replace b with a in any expression. Substitution Property of Equality If a = b, then you may replace b with a in any expression. WE USE THE PROPERTIES TO JUSTIFY ALGEBRAIC STEPS AND SOLVE PROBLEMS. THIS IS DEDUCTIVE REASONING.

4. Original equation Algebraic StepsProperties Solve Distributive Property Substitution Property Addition Property Example 1:

5. Substitution Property Division Property Substitution Property Answer: Example 1:

6. Original equation Algebraic StepsProperties Distributive Property Substitution Property Subtraction Property Solve Your Turn:

7. Substitution Property Division Property Substitution Property Answer: Your Turn:

8. Two-Column Proof Two-Column Proof – A proof format used in geometry in which an argument is presented with two columns, statements and reasons, to prove conjectures and theorems are true. Also referred to as a formal proof. Two-Column Proof – A proof format used in geometry in which an argument is presented with two columns, statements and reasons, to prove conjectures and theorems are true. Also referred to as a formal proof.

9. StatementsReasons Proof: Two-Column Proof

10. 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 Example 2a:

11. Write a two-column proof. If then 1. Given 1. 2. Multiplication Property 2. 3. Distributive Property 3. 4. Subtraction Property 4. 5. Substitution 5. 6. Addition Property 6. Proof: Statements Reasons Example 2b:

12. Proof: Statements Reasons 8. Division Property8. 9. Substitution9. 7. Substitution 7. Write a two-column proof. If then Example 2b:

13. Write a two-column proof for the following. a. Your Turn:

14. 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. Your Turn:

15. Prove: b. Given: Write a two-column proof for the following. Your Turn:

16. 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. Your Turn:

17. Geometric Proof Since geometry also uses variables, numbers, and operations, many of the algebraic properties of equality are true in geometry. For example: Since geometry also uses variables, numbers, and operations, many of the algebraic properties of equality are true in geometry. For example: PropertySegmentsAngles Reflexive AB = AB m 1 = m 1 Symmetric If AB = CD, then CD = AB. If m 1 = m 2, then m 2 = m 1. Transitive If AB = CD and CD = EF, then AB = EF. If m 1 = m 2 and m 2 = m 3, then m 1 = m 3.           

18. 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 Example 3:

19. Solve the Test Item Statement II: Since the order you name the endpoints of a segment is not important, and TS = PR. Thus, Statement II is true. Statement I: Examine the given information, GH JK ST and. From the definition of congruence of segments, if, then ST RP. You can substitute RP for ST in GH JK ST to get GH JK RP. Thus, Statement I is true. Example 3:

20. Because Statements I and II only are true, choice B is correct. Answer: B Statement III If GH JK ST, then. Statement III is not true. Example 3:

21. 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 Your Turn:

22. 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 Example 4:

23. 1. Given1. 2. Transitive Property2. Proof: Statements Reasons 3. Definition of congruencem leg 1 m leg 33. 4. Givenm leg 1 22 cm4. 5. Transitive Propertym leg 3 22 cm5. Example 4:

24. DRIVING A stop sign as shown below is a regular octagon. If the measure of angle A is 135 and angle A is congruent to angle G, prove that the measure of angle G is 135. Your Turn:

25. Proof: StatementsReasons 1. Given 2. Given 3. Definition of congruent angles 4. Transitive Property 1. 2. 3. 4. Your Turn:

26. Assignment Geometry: Pg. 97 – 98 Geometry: Pg. 97 – 98 #4 – 9, 14 – 25 Pre-AP Geometry: Pre-AP Geometry: Pg. 97 – 98 #14 – 31