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Mastering Algebraic Proofs: Techniques for Two-Column and Paragraph Formats

Explore the fundamentals of algebraic proofs, focusing on the use of postulates about points, lines, and planes. This guide will walk you through writing two-column proofs using properties of equality, as well as paragraph and flowchart proofs. Understand that an algebraic proof consists of a series of logical statements, each supported by accepted truths. Learn to justify each algebraic step, solve equations, and structure your proofs effectively. Enhance your reasoning skills and solidify your understanding of geometric concepts.

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Mastering Algebraic Proofs: Techniques for Two-Column and Paragraph Formats

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  1. 2-6 Algebraic Proofp. 136 You used postulates about points, lines, and planes to write paragraph proofs. • Use algebra to write two-column proofs. • Use properties of equality to write geometric proofs.

  2. Proofs A proof is a logical argument in which each statement you make is supported by a statement that is accepted as true. It can be written as: • A paragraph • Two column or formal • Flow chart.

  3. Page 136

  4. Algebraic Proofs You just saw a table summarizing the properties of real numbers you studied in Algebra. Now you will use these properties in Algebraic Proofs. An algebraic proof is a proof that is made up of a series of algebraic statements.

  5. Justify Each Step When Solving an Equation Solve 2(5 – 3a) – 4(a + 7) = 92. Algebraic Steps Properties 2(5 – 3a) – 4(a + 7) = 92 Original equation 10 – 6a – 4a – 28 = 92 Distributive Property –18 – 10a = 92 Substitution Property –18 – 10a + 18 = 92 + 18 Addition Property –10a = 110 Substitution Property Division Property a = –11 Substitution Property Answer:a = –11

  6. Solve –3(a + 3) + 5(3 – a) = –50. A.a = 12 B.a = –37 C.a = –7 D.a = 7

  7. Begin by stating what is given and what you are to prove.

  8. Proof: d = 20t + 5 1.Given 1. Statements Reasons 2. d – 5 = 20t 2. Addition Property of Equality = t 3. 3. Division Property of Equality 4. 4. Symmetric Property of Equality *Hint* always start with GIVEN Always end with PROVE

  9. If the formula for the area of a trapezoid is , then the height h of the trapezoid is given by . Which of the following statements would complete the proof of this conjecture?

  10. Proof: Statements Reasons 1. Given 1. ? 2. _____________ 2. Multiplication Property of Equality 3. 3. Division Property of Equality 4. 4. Symmetric Property of Equality 2A = (b1 + b2)h

  11. Proof: If A B, mB = 2mC, and mC = 45, then mA = 90. Write a two-column proof to verify this conjecture. Statements Reasons mA = 2(45) 4. Substitution 4. 2.mA = mB 2. Definition of angles A B; mB = 2mC; mC = 45 1. Given 1. 5. mA = 90 5. Substitution 3. Transitive Property of Equality 3. mA = 2mC

  12. Proof: 1. Given 1. Statements Reasons ? 2. 2. _______________ 3.AB = RS 3. Definition of congruent segments 4. AB = 12 5. RS = 12 4. Given 5. Substitution Transitive Property of Equality

  13. 2-6 Assignment Page 139, 2-16 even, 17, 18 Write out all of the information in the book for 17 & 18 That includes: Given Prove Statements Reasons

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