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2.6 Algebraic Proof

2.6 Algebraic Proof. Objectives. Use algebra to write two-column proofs Use properties of equality in geometry proofs. ALGEBRAIC PROPERTIES OF EQUALITY. Reflexive Property a = a. Symmetric Property If a = b , then b = a .

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2.6 Algebraic Proof

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  1. 2.6 Algebraic Proof

  2. Objectives • Use algebra to write two-column 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. Addition 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. 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. Distributive Property a(b + c) = ab + ac. 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. Solve Example 1: Algebraic Steps Properties Original equation Distributive Property Substitution Property Addition Property

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

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

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

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

  9. Proof: Statements Reasons Two-Column Proof

  10. 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 Example 2a:

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

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

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

  14. Proof: Statements Reasons 1. Given 1. 2. Multiplication Property 2. 3. Substitution 3. 4. Subtraction Property 4. 5. Substitution 5. 6. Division Property 6. 7. Substitution 7. Your Turn:

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

  16. 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. 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:            

  18. 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 Example 3: Read the Test Item Determine whether the statements are true based on the given information.

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

  20. Statement IIIIf GHJKST, then . Statement III is not true. Example 3: Because Statements I and II only are true, choice B is correct. Answer: B

  21. 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 Your Turn: Answer: C

  22. Given: m leg 1 22 cm Prove: m leg 3 22 cm Example 4: 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.

  23. Proof: Statements Reasons 1. 1. Given 2. 2. Transitive Property 3. m leg 1 m leg 3 3. Definition of congruence 4. m leg 1 22 cm 4. Given 5. m leg 3 22 cm 5. Transitive Property 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: Statements Reasons 1. Given 1. 2. Given 2. 3. Definition of congruent angles 3. 4. Transitive Property 4. Your Turn:

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

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