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Mastering Theorems: Formal Proof Strategies

Learn how to formally prove theorems using statements, drawings, proofs, and given information with logical flow and justifications. Example theorems on angles and segment congruence included.

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Mastering Theorems: Formal Proof Strategies

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  1. Section 1.7The Formal Proof of a Theorem • Statement: States the theorem to be proved. • Drawing: Represents hypothesis of the theorem. • Given: Describes the drawing according to the information found in the hypothesis. • Prove: Describes the drawing according to the claim made in the conclusion of the theorem. • Proof: Orders a list of claims (Statements) and justifications (Reasons), beginning with the Given and ending with the Prove; there must be a logical flow in this Proof. H: Hypothesis ← statement of proof P: Principle ← reason of proof ∴C: Conclusion← next statement in proof Ex 2. p 55

  2. Converse of a Statement Ex. Consider the statement: If I study, then I will do well on my test. A new statement can be formed: If I did well on my test, then I studied The new statement formed is called the converse of the original statement. • The converse of a true statement may not be assumed to be true. • Another ex. If I am 12 years old, then I am not eligible to vote. Converse: If I am not eligible to vote, then I am 12 years old.

  3. Proving a Theorem • Theorem 1.7.1: If two lines meet to form a right angle, then these lines are perpendicular • Fig 1.66 • Strategy: • Look at the diagram • What are we given? • What does the given tell us to use? • How will using these postulates, theorems, definitions help us to prove our theorem • Example 3 p. 56

  4. Additional Theorems on Angles • Theorem 1.7.2: If two angles are complementary to the same angle (or to congruent angles), then these angles are congruent. (Ex. 21) • Theorem 1.7.3: If two angles are supplementary to the same angle (or to congruent angles), then these angles are congruent .(Ex. 22) • Theorem 1.7.4: Any two right angles are congruent. • Theorem 1.7.5: If the exterior sides of two adjacent acute angles form perpendicular rays, then these angles are complementary. Picture proof p. 57

  5. Other Theorems to use in Proofs • Theorem 1.7.6: If the exterior sides of two adjacent angles form a straight line, then these angles are supplementary. Proof: Ex 4 p. 57. • Theorem 1.7.7: If two line segments are congruent, then their midpoints separate these areas into four congruent segments. • Theorem 1.7.8: If two angles are congruent, then their bisectors separate these angles into four congruent angles. • Example in class #32: Prove: The bisectors of two adjacent supplementary angles form a right angle .

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