The Formal Proof of a Theorem

Transcription

1 .7 The Formal Proof of a Theorem 5.7 The Formal Proof of a Theorem KY ONPTS Formal Proof of a Theorem onverse of a Theorem Picture Proof (Informal) of a Theorem Recall from Section. that statements that can be proved are called theorems. To understand the formal proof of a theorem, we begin by considering 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, the hypothesis is H and the conclusion is. Some theorems must be reworded to fit into If..., then... form so that the hypothesis and conclusion are easy to recognize. XMPL Give the hypothesis H and conclusion for each of these statements. a) If two lines intersect, then the vertical angles formed are congruent. b) ll right angles are congruent. c) Parallel lines do not intersect. d) Lines are perpendicular when they meet to form congruent adjacent angles. SOLUTION a) s is H: Two lines intersect. : The vertical angles formed are congruent. b) Reworded If two angles are right angles, then H: Two angles are right angles. : The angles are congruent. c) Reworded If two lines are parallel, then these lines do not intersect. H: Two lines are parallel. : 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. : The lines are perpendicular. XS. Why do we need to distinguish between the hypothesis and the conclusion? For a theorem, the hypothesis determines the Given and the rawing. The Given provides a description of the rawing s known characteristics. The conclusion (Prove) determines the relationship that you wish to establish in the rawing. TH WRITTN PRTS OF FORML The five necessary parts of a formal proof are listed in the following box in the order in which they should be developed. opyright 04 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

2 5 HPTR LIN N NGL RLTIONSHIPS SSNTIL PRTS OF TH FORML OF THORM. Statement: States the theorem to be proved.. rawing: Represents the hypothesis of the theorem.. Given: escribes the rawing according to the information found in the hypothesis of the theorem. 4. Prove: escribes the rawing according to the claim made in the conclusion of the theorem. 5. Proof: Orders a list of claims () and justifications (), beginning with the Given and ending with the Prove; there must be a logical flow in this Proof. 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 ordering conclusions based on the given situation. One must be somewhat like a lawyer, selecting 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 earlier statement to a later claim by using deduction (perhaps several times). Where principle P has the form If H, then, the logical order follows. H: hypothesis statement of proof P: principle reason of proof : conclusion next statement in proof onsider the following theorem, which was proved in xample of Section.6. THORM.6. If two lines are perpendicular, then they meet to form right angles. XMPL Write the parts of the formal proof of Theorem.6.. XS. 4, 5 SOLUTION. State the theorem. If two lines are perpendicular, then they meet to form right angles.. The hypothesis is H: Two lines are perpendicular. Make a rawing to fit this description. (See Figure.65.). Write the Given statement, using the rawing and based on the hypothesis H: Two lines are. Given: intersecting at 4. Write the Prove statement, using the rawing and based on the conclusion : They meet to form right angles. Prove: is a right angle. Figure onstruct the Proof. This formal proof is found in xample, Section.6. Unless otherwise noted, all content on this page is engage Learning. opyright 04 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

3 .7 The Formal Proof of a Theorem 5 Warning You should not make a drawing that embeds qualities beyond those described in the hypothesis; nor should your drawing indicate fewer qualities than the hypothesis prescribes! ONVRS OF STTMNT The converse of the statement If P, then Q is If Q, then P. That is, the converse of a given statement interchanges its hypothesis and conclusion. onsider the following: Statement: onverse: If a person lives in London, then that person lives in ngland. If a person lives in ngland, then that person lives in London. s shown above, the given statement is true, whereas its converse is false. Sometimes the converse of a true statement is also true. In fact, xample presents the formal proof of Theorem.7., which is the converse of Theorem.6.. Once a theorem has been proved, it may be cited thereafter as a reason in future proofs. Thus, any theorem found in this section can be used for justification in proof problems found in later sections. The proof that follows is nearly complete! It is difficult to provide a complete formal proof that explains the how to and simultaneously presents the final polished form. xample illustrates the polished proof. You do not see the thought process and the scratch paper needed to piece this puzzle together. The proof of a theorem is not unique! For instance, students rawings need not match, even though the same relationships should be indicated. ertainly, different letters are likely to be chosen for the rawing that illustrates the hypothesis. THORM.7. If two lines meet to form a right angle, then these lines are perpendicular. XMPL Give a formal proof for Theorem.7.. If two lines meet to form a right angle, then these lines are perpendicular. GIVN: and intersect at so that is a right angle (Figure.66) PROV: Figure.66. and intersect so that. Given is a right angle. m 90. If an is a right, its measure is 90. is a straight,. If an is a straight, so m 80 its measure is m m 4. ngle-ddition Postulate m (), (), (4) m Substitution (5) 6. m Subtraction Property of quality (), (6) 7. m m 7. Substitution If two s have measures, 9.! the s are 9. If two lines form adjacent s, these lines are ecause perpendicular lines lead to right angles, and conversely, a square (see Figure.66) may be used to indicate perpendicular lines or a right angle. Unless otherwise noted, all content on this page is engage Learning. opyright 04 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

4 54 HPTR LIN N NGL RLTIONSHIPS XS. 6 8 Several additional theorems are now stated, the proofs of which are left as exercises. This list contains theorems that are quite useful when cited as reasons in later proofs. formal proof is provided only for Theorem.7.6. THORM.7. If two angles are complementary to the same angle (or to congruent angles), then these angles are congruent. See xercise 7 for a drawing describing Theorem.7.. THORM.7. If two angles are supplementary to the same angle (or to congruent angles), then these angles are congruent. See xercise 8 for a drawing describing Theorem.7.. THORM.7.4 ny two right angles are congruent. THORM.7.5 If the exterior sides of two adjacent acute angles form perpendicular rays, then these angles are complementary. For Theorem.7.5, we create an informal proof called a picture proof. lthough such a proof is less detailed, the impact of the explanation is the same! This is the first of several picture proofs found in this textbook. In Figure.67, the square is used to indicate that!! PITUR OF THORM.7.5 Given:!! Prove: and are complementary Proof: With!!, we see that and are parts of a right angle. Then m m 90, so and are complementary. Figure.67 STRTGY FOR The Final Reason in the Proof General Rule: The last reason explains why the last statement must be true. Never write the word Prove for any reason in a proof. Illustration: The final reason in the proof of Theorem.7.6 is the definition of supplementary angles: If the sum of measures of two angles is 80, the angles are supplementary. Unless otherwise noted, all content on this page is engage Learning. opyright 04 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

5 .7 The Formal Proof of a Theorem 55 Technology xploration Use computer software if available.. raw G containing point F. lso draw FH! as in Figure.68.. Measure and 4.. Show that m m (nswer may not be perfect.) XMPL 4 Study the formal proof of Theorem.7.6. THORM.7.6 If the exterior sides of two adjacent angles form a straight line, then these angles are supplementary. GIVN: and 4 and G (Figure.68) PROV: and 4 are supplementary Figure.68 4 F H G XS. 9. and 4 and G. m m 4 m FG. FG is a straight angle 4. m FG m m and 4 are supplementary. Given. ngle-ddition Postulate. If the sides of an are opposite rays, it is a straight 4. The measure of a straight is Substitution 6. If the sum of the measures of two s is 80, the s are supplementary The final two theorems in this section are stated for convenience. We suggest that the student make drawings to illustrate Theorem.7.7 and Theorem.7.8. THORM.7.7 If two line segments are congruent, then their midpoints separate these segments into four congruent segments. THORM.7.8 XS., 4 If two angles are congruent, then their bisectors separate these angles into four congruent angles. xercises.7 In xercises to 6, state the hypothesis H and the conclusion for each statement.. If a line segment is bisected, then each of the equal segments has half the length of the original segment.. If two sides of a triangle are congruent, then the triangle is isosceles.. ll squares are quadrilaterals. 4. very regular polygon has congruent interior angles. 5. Two angles are congruent if each is a right angle. 6. The lengths of corresponding sides of similar polygons are proportional. 7. Name, in order, the five parts of the formal proof of a theorem. 8. Which part (hypothesis or conclusion) of a theorem determines the a) rawing? b) Given? c) Prove? Unless otherwise noted, all content on this page is engage Learning. opyright 04 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

6 56 HPTR LIN N NGL RLTIONSHIPS 9. Which part (Given or Prove) of the proof depends upon the a) hypothesis of theorem? b) conclusion of theorem? 0. Which of the following can be cited as a reason in a proof? a) Given c) efinition b) Prove d) Postulate. When can a theorem be cited as a reason for a proof?. ased upon the hypothesis of a theorem, do the drawings of different students have to be identical (same names for vertices, etc.)? For each theorem stated in xercises to 8, make a rawing. On the basis of your rawing, write a Given and a Prove for the theorem.. If two lines are perpendicular, then these lines meet to form a right angle. 4. If two lines meet to form a right angle, then these lines are perpendicular. 5. If two angles are complementary to the same angle, then 6. If two angles are supplementary to the same angle, then 7. If two lines intersect, then the vertical angles formed are congruent. 8. ny two right angles are congruent. In xercises 9 to 6, use the drawing in which intersects! at point O. 9. If m 5, find m, m, and m If m 47, find m, m, and m 4.. If m x 0 and m 4x 0, find x and m.. If m 6x 8 and m 4 7x, find x and m.. If m x and m x, find x and m. 4. If m x 5 and m x, find x and. x m 5. If m 0 and m 40, find x and m. 6. If m x 0 and m 4, find x and m 4. In xercises 7 to 5, complete the formal proof of each theorem. 7. If two angles are complementary to the same angle, then Given: is comp. to is comp. to Prove: x x O 4. is comp. to is comp. to. m m 90 m m 90. m m m m 4. m m If two angles are supplementary to the same angle, then Given: is supp. to is supp. to Prove: (HINT: See xercise 7 for help.) xercise 8 9. If two lines intersect, the vertical angles formed are congruent. 0. ny two right angles are congruent.. If the exterior sides of two adjacent acute angles form perpendicular rays, then these angles are complementary. Given:!! Prove: is comp. to.?.?.? 4.? 5.?.!!.?.?. If two rays are, then. m 90 they meet to form a rt..? 4. m m m 4.? 5. m m Substitution 6.? 6. If the sum of the measures of two angles is 90, then the angles are complementary Unless otherwise noted, all content on this page is engage Learning. opyright 04 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

7 Perspective on History 57. If two line segments are congruent, then their midpoints separate these segments into four congruent segments. Given: M is the midpoint of N is the midpoint of Prove: M M N N M N. If two angles are congruent, then their bisectors separate these angles into four congruent angles. Given:! FG bisects FH! bisects FG Prove: 4 4. The bisectors of two adjacent supplementary angles form a right angle. Given: is supp. to! bisects F! bisects Prove: F is a right angle 5. The supplement of an acute angle is an obtuse angle. (HINT: Use xercise 8 of Section.6 as a guide.) 4 F H F 4 G PRSPTIV ON HISTORY TH VLOPMNT OF GOMTRY One of the first written accounts of geometric knowledge appears in the Rhind papyrus, a collection of documents that date back to more than 000 years before hrist. In this document, hmes (an gyptian scribe) describes how north-south and east-west lines were redrawn following the overflow of the Nile River. stronomy was used to lay out the north-south line. The rest was done by people known as rope-fasteners. y tying knots in a rope, it was possible to separate the rope into segments with lengths that were in the ratio to 4 to 5. The knots were fastened at stakes in such a way that a right triangle would be formed. In Figure.69, the right angle is formed so that one side (of length 4, as shown) lies in the north-south line, and the second side (of length, as shown) lies in the east-west line. W Figure.69 N S The principle that was used by the rope-fasteners is known as the Pythagorean Theorem. However, we also know that the ancient hinese were aware of this relationship. That is, the Pythagorean Theorem was known and applied many centuries before the time of Pythagoras (the Greek mathematician for whom the theorem is named). hmes describes other facts of geometry that were known to the gyptians. Perhaps the most impressive of these facts was that their approximation of p was.604. To four decimal places of accuracy, we know today that the correct value of p is.46. Like the gyptians, the hinese treated geometry in a very practical way. In their constructions and designs, the hinese used the rule (ruler), the square, the compass, and the level. Unlike the gyptians and the hinese, the Greeks formalized and expanded the knowledge base of geometry by pursuing it as an intellectual endeavor. ccording to the Greek scribe Proclus (about 50..), Thales ( ) first established deductive proofs for several of the known theorems of geometry. Proclus also notes that it was uclid (0 75..) who collected, summarized, ordered, and verified the vast quantity of knowledge of geometry in his time. uclid s work lements was the first textbook of geometry. Much of what was found in lements is the core knowledge of geometry and thus can be found in this textbook as well. Unless otherwise noted, all content on this page is engage Learning. opyright 04 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.