Transcript Lesson
Lesson 2-6 Algebraic Proof 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. Standardized Test Practice: Which statement is not supported by a postulate? A C R and S are collinear. P, X and Y must be collinear. B D M lies on LM. J, K and L are coplanar. 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. Standardized Test Practice: Which statement is not supported by a postulate? A C R and S are collinear. P, X and Y must be collinear. B D M lies on LM. J, K and L are coplanar. Objectives • Use algebra to write two-column proofs • Use properties of equality in geometry proofs Vocabulary • Deductive argument – a group of logical steps used to solve problems • Two-column proof – also known as a formal proof 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 For all numbers a, b, and c, if a = b, & Subtraction 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 Solve 2(5 – 3a) – 4(a + 7) = 92 Algebraic Steps Properties 2(5 – 3a) – 4(a + 7) = 92 10 – 6a – 4a – 28 = 92 –18 – 10a = 92 –18 + 18 – 10a = 92 + 18 –10a = 110 Original equation Distributive Property Substitution Property Addition Property Substitution Property Division Property a = – 11 Answer: a = – 11 Substitution Property Write a two-column proof. If then Proof: Statements Reasons 1. 1. Given 2. 2. Multiplication Property 3. 3. Substitution 4. 4. Subtraction Property 5. 5. Substitution 6. 6. Division Property 7. 7. Substitution Write a two-column proof. a. Proof: Statements Reasons 1. 1. Given 2. 2. Multiplication Property 3. 4. 5. 3. Substitution 4. Subtraction Property 5. Substitution 6. 6. Division Property 7. 7. Substitution Write a two-column proof. b. Given: Prove: a = –5 Proof: Statements Reasons 1. 1. Given 2. 2. Multiplication Property 3. 4. 3. Distributive Property 4. Subtraction Property 5. 5. Substitution 6. 6. Subtraction Property 7. 7. Substitution 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 MULTIPLE- CHOICE TEST ITEM If and then which of the following is a valid conclusion? I. II. III. A I only Answer: C B I and II C I and III D II and III 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. Prove: m leg 3 Given: m leg 1 22 cm 22 cm 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 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