The Segment-Addition Postulate can be generalized as

Recall from Section 1.3 that statements that follow logically from known undefined terms, definitions, and postulates are called theorems. The formal proof of a theorem has several parts; to understand how these parts are related, consider carefully the terms hypothesis and conclusion. The hypothesis of a statement describes the given situation Given, whereas the conclusion describes what you need to establish Prove. When a statement has the form “If H, then C,” the hypothesis is H and the conclusion is C. Some theorems must be reworded to fit into “If . . . , then . . .” form so that the hypothesis and conclusion are easy to recognize. Formal Proof of a Theorem Converse of a Theorem Picture Proof Informal of a Theorem The Formal Proof of a Theorem 1.7 KEY CONCEPTS EXAMPLE 1 Give the hypothesis H and conclusion C for each of these statements. a If two lines intersect, then the vertical angles formed are congruent. b All right angles are congruent. c Parallel lines do not intersect. d Lines are perpendicular when they meet to form congruent adjacent angles. Solution a As is H: Two lines intersect. C: The vertical angles formed are congruent.

28. In the proof to the right, provide the missing reasons.

Given: ⬔1 and ⬔2 are complementary ⬔1 is acute Prove: ⬔2 is also acute PROOF Statements Reasons 1. ⬔1 and ⬔2 are 1. ? complementary 1 2. m ⬔1 + m ⬔2 = 90 2. ? 3. ⬔1 is acute 3. ? 3 4. Where m ⬔1 = x , 4. ? 0 x 90 2 5. x + m ⬔2 = 90 5. ? 5 6. m ⬔2 = 90 - x 6. ? 4 7. -x 0 90 - x 7. ? 7 8. 90 - x 90 180 - x 8. ? 7, 8 9. 0 90 - x 90 9. ? 6, 9 10. 0 m⬔2 90 10. ? 10 11. ⬔2 is acute 11. ? The most difficult aspect of a formal proof is the thinking process that must take place between parts 4 and 5. This game plan or analysis involves deducing and order- ing conclusions based on the given situation. One must be somewhat like a lawyer, se- lecting the claims that help prove the case while discarding those that are superfluous. In the process of ordering the statements, it may be beneficial to think in reverse order, like so: The Prove statement would be true if what else were true? The final proof must be arranged in an order that allows one to reason from an ear- lier statement to a later claim by using deduction perhaps several times. H: hypothesis —¬¬ statement of proof P: principle —¬¬ reason of proof ‹ C: conclusion —¬¬ next statement in proof Consider the following theorem, which was proved in Example 1 of Section 1.6. ESSENTIAL PARTS OF THE FORMAL PROOF OF A THEOREM 1. Statement: States the theorem to be proved. 2. Drawing: Represents the hypothesis of the theorem. 3. Given: Describes the Drawing according to the information found in the hypothesis of the theorem. 4. Prove: Describes the Drawing according to the claim made in the conclusion of the theorem. 5. 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. If two lines are perpendicular, then they meet to form right angles. THEOREM 1.6.1 Why do we need to distinguish between the hypothesis and the conclusion? For a theorem, the hypothesis determines the Drawing and the Given, providing a description of the Drawing’s known characteristics. The conclusion determines the relationship the Prove that you wish to establish in the Drawing. THE WRITTEN PARTS OF A FORMAL PROOF The five necessary parts of a formal proof are listed in the accompanying box in the or- der in which they should be developed. Exs. 1–3 b Reworded If two angles are right angles, then these angles are congruent. H: Two angles are right angles. C: The angles are congruent. c Reworded If two lines are parallel, then these lines do not intersect. H: Two lines are parallel. C: The lines do not intersect. d Reordered When if two lines meet to form congruent adjacent angles, these lines are perpendicular. H: Two lines meet to form congruent adjacent angles. C: The lines are perpendicular. 쮿