Planning a proof
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The document outlines the typical structure for proving a theorem, which consists of 5 parts: 1) stating the theorem, 2) illustrating it with a diagram, 3) listing what is given, 4) listing what is to be proved, and 5) a series of statements and reasons leading from the givens to the conclusion. It then provides an example proof of the theorem: "If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent." The proof uses the standard structure and logical reasoning to show that if two pairs of angles are supplementary, then the corresponding angles are congruent.
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Planning a proof
- 1. Planning a Proof of a Theorem (p. 60) A proof of a theorem consists of five parts: 1. A statement of the theorem 2. A diagram that illustrates the given information 3. A list, in terms of the figure, of what is given 4. A list, in terms of the figure, of what you are to prove 5. A series of statements and reasons that lead from the given information to the statement that is to be proved
- 2. Theorem 2-7 (p. 61) If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent. 1 2 3 4
- 3. 1 2 3 4 Given: 1 and 2 are supplementary; 3 and 4 are supplementary 2 4 Prove: 1 3 Statements Reasons 1. 1 and 2 are supplementary; 3 and 4 are supplementary 1. Given 2. m 1 + m 2 = 180; m 3 + m 4 = 180 2. Definition of supplementary angles 3. m 1 + m 2 = m 3 + m 4 3. Substitution 4. 2 4, or m 2 = m 4 4. Given, Definition of 5. m 1 = m 3, or 1 3 5. Subtraction Prop of =, Definition of
- 4. Theorem 2-8 If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent. TRY ON YOUR OWN: prove the theorem above in your notes… 1 2 3 4 Given: 1 and 2 are complementary; 3 and 4 are complementary 2 4 Prove: 1 3
- 5. Try for fun!
Editor's Notes
- #5: Proof is in practice assignment – Exercise #18