Introduction to Geometric Proof

Transcription

1 5 Introduction to Geometric roof 37 4 Refer to the circle with center O a) Use a protractor to find m b) Use a protractor to find m D c) Compare results in parts (a) and (b) 44 Refer to the circle with center a) Use a protractor to find m b) Use a protractor to find m 2 c) Compare results in parts (a) and (b) R O C 2 D 42 If m 38, m UW 40, and m W 6, find m U 45 On the hanging sign, the three angles ( D, C, and DC) at vertex have the sum of measures If and! 360 m DC 90 bisects the indicated reflex angle, find m C U D C W Exercises 42, If m U x 2z, m U x z, and m W 2x z, find x if m W 60 lso, find z if m UW 3x 6 46 With 0 x 90, an acute angle has measure x Find the difference between the measure of its supplement and the measure of its complement 5 KEY CONCE Introduction to Geometric roof lgebraic roperties roof Given roblem and rove tatement Reminder dditional properties and techniques of algebra are found in ppendix o believe certain geometric principles, it is necessary to have proof his section introduces some guidelines for establishing the proof of these geometric properties everal examples are offered to help you develop your own proofs In the beginning, the form of proof will be a two-column proof, with statements in the left column and reasons in the right column ut where do the statements and reasons come from? o deal with this question, you must ask What is known (Given) and Why the conclusion (rove) should follow from this information In correctly piecing together a proof, you will usually scratch out several conclusions, discarding some and reordering the rest Each conclusion must be justified by citing the Given (hypothesis), a previously stated definition or postulate, or a theorem previously proved elected properties from algebra are often used as reasons to justify statements For instance, we use the ddition roperty to justify adding the same number to each side of an equation found in a proof often include the properties found in ables 5 and 6 on page 38

2 38 CHER LINE ND NGLE RELIONHI LE 5 roperties (a, b, and c are real numbers) ddition roperty : If a b, then a c b c ubtraction roperty : If a b, then a c b c Multiplication roperty : If a b, then a c b c Division roperty : a b If a b and c 0, then = c c s we discover in Example, some properties can be used interchangeably EXMLE Which property of equality justifies each conclusion? a) If 2x 3 7, then 2x 0 b) If 2x 0, then x 5 OLUION a) ddition roperty ; added 3 to each side of the equation b) Multiplication roperty ; multiplied each side of the equation by 2 OR Division roperty ; divided each side of the equation by 2 LE 6 Further lgebraic roperties (a, b, and c are real numbers) Reflexive roperty: a a ymmetric roperty: If a b, then b a Distributive roperty: a(b c) a b a c ubstitution roperty: If a b, then a replaces b in any equation ransitive roperty: If a b and b c, then a c efore considering geometric proof, we study algebraic proof in Examples 2 and 3 Each statement in the proof is supported by the reason why we can make that statement (claim) he first claim in the proof is the Given statement; and the sequence of steps must conclude with a final statement representing the claim to be proved (called the rove statement) In Example 2, we construct the algebraic proof of the claim, If 2x 3 7, then x 5 Where represents the statement 2x 3 7, and R represents x 5, the theorem has the form If, then R We also use letter Q to name the intermediate conclusion 2x 0 Using the letters, Q, and R, we show the logical development for the proof at the left his logical format will not be provided in future proofs EXMLE 2 GIEN: 2x 3 7 ROE: x 5

3 5 Introduction to Geometric roof 39 Logical Format 2x 3 7 If, then Q 2 2x ddition roperty Q 3 2x 0 3 ubstitution If Q, then R R 4 2x x 5 4 Division roperty 5 ubstitution EX 4 tudy Example 3 hen cover the reasons and provide the reason for each statement In turn, with statements covered, find the statement corresponding to each reason EXMLE 3 GIEN: ROE: 2(x 3) 4 0 x 6 2(x 3) x x x 2 5 x 6 2 Distributive roperty 3 ubstitution 4 ddition roperty 5 Division roperty EX 5 7 NOE : lternatively, tep 5 could use the reason Multiplication roperty (multiply by 2 ) NOE 2: he fifth step is the final step because the rove statement (x 6) has been made and justified Discover In the diagram, the wooden trim pieces are mitered (cut at an angle) to be equal and to form a right angle when placed together Use the properties of algebra to explain why the measures of and 2 are both 45 What you have done is an informal proof 2 he Discover activity at the left suggests that formal geometric proofs also exist he typical format for a problem requiring geometric proof is GIEN: ROE: DRWING Consider this problem: GIEN: -- on (Figure 54) Figure 54 ROE: First consider the Drawing (Figure 54), and relate it to any additional information described by the Given hen consider the rove statement Do you understand the claim, and does it seem reasonable? If it seems reasonable, intermediate claims can be ordered and supported to form the contents of the proof ecause a proof must begin with the Given and conclude with the rove, the proof of the preceding problem has this form: NWER m m 2 90 ecause m m 2, we see that m m 90 hus, 2 m 90, and, dividing by 2, we see that m 45 hen m 2 45 also

4 40 CHER LINE ND NGLE RELIONHI -- on? = -?? o construct the preceding proof, you must deduce from the Drawing and the Given that In turn, you may conclude (through subtraction) that he complete proof problem will have the appearance of Example 4, which follows the first of several trategy for roof features used in this textbook REGY FOR he First Line of roof General Rule: he first statement of the proof includes the Given information; also, the first reason is Given Illustration: ee the first line in the proof of Example 4 Figure 55 EX 8 0 EXMLE 4 GIEN: -- on (Figure 55) ROE: -- on 2 2 egment-ddition ostulate 3 3 ubtraction roperty ome properties of inequality (see able 7) are useful in geometric proof LE 7 roperties of Inequality (a, b, and c are real numbers) ddition roperty of Inequality: If a b, then a c b c If a b, then a c b c ubtraction roperty of Inequality: If a b, then a c b c If a b, then a c b c MLE Consider Figure 56 and this problem: GIEN: MN Q M Figure 56 N Q ROE: M NQ

5 5 Introduction to Geometric roof 4 o understand the situation, first study the Drawing (Figure 56) and the related Given hen read the rove with reference to the Drawing What may be confusing here is that the Given involves MN and Q, whereas the rove involves M and NQ However, this is easily remedied through the addition of N to each side of the inequality MN Q; see tep 2 in the proof of Example 5 EXMLE 5 M Figure 57 N Q GIEN: MN Q (Figure 57) ROE: M NQ MN Q 2 MN N N Q 3 MN N M and N Q NQ 4 M NQ 2 ddition roperty of Inequality 3 egment-ddition ostulate 4 ubstitution NOE: he final reason may come as a surprise However, the ubstitution xiom of Equality allows you to replace a quantity with its equal in any statement including an inequality! ee ppendix 3 for more information REGY FOR he Last tatement of the roof General Rule: he final statement of the proof is the rove statement Illustration: ee the last statement in the proof of Example 6 EXMLE 6 R Figure 58 U W EX, 2 tudy this proof, noting the order of the statements and reasons GIEN:! bisects! RU bisects UW (Figure 58) ROE: m R m W m! bisects RU 2 m R m U 3! bisects UW 4 m W m U 5 m R m W m U m U 6 m U m U m 7 m R m W m 2 If an angle is bisected, then the measures of the resulting angles are equal 3 ame as reason 4 ame as reason 2 5 ddition roperty (use the equations from statements 2 and 4) 6 ngle-ddition ostulate 7 ubstitution

6 42 CHER LINE ND NGLE RELIONHI Exercises 5 In Exercises to 6, which property justifies the conclusion of the statement? If 2x 2, then x 6 2 If x x 2, then 2x 2 3 If x 5 2, then x 7 4 If x 5 2, then x 7 5 If 5 3, then x 5 6 If 3x 2 3, then 3x 5 In Exercises 7 to 0, state the property or definition that justifies the conclusion (the then clause) 7 Given that s and 2 are supplementary, then m m Given that m 3 m 4 80, then s 3 and 4 are supplementary 9 Given R and! as shown, then m R m m R 0 Given that, then! m R m bisects R In Exercises to 22, use the Given information to draw a conclusion based on the stated property or definition M Exercises, 2 x : -M-; egment-ddition ostulate 2 Given: M is the midpoint of ; definition of midpoint 3 Given: m m 2; definition of angle bisector 4 Given: EG! D bisects DEF; G definition of angle bisector 5 Given: s and 2 are 2 complementary; definition E F of complementary angles Exercises Given: m m 2 90 ; definition of complementary angles 7 Given: 2x 3 7; ddition roperty 8 Given: 3x 2; Division roperty 9 Given: 7x ; ubstitution roperty 20 Given: 2 05 and 05 50% ; ransitive roperty 2: 3(2x ) 27; Distributive roperty x 5 22 Given: 4; Multiplication roperty R Exercises 9, 0 In Exercises 23 and 24, fill in the missing reasons for the algebraic proof 23 Given: rove: 3(x 5) 2 2 3x x 36 4 x 2 24 Given: rove: 2x x 6 3 x 3 In Exercises 25 and 26, fill in the missing statements for the algebraic proof 25 Given: rove: 4? 5? 26 Given: rove: x 5 3(x 5) 2 x 2 2x 9 3 x 3 2(x 3) 7 x x 30 4? 2 Distributive roperty 3 ubstitution (ddition) 4 ddition roperty 5 Division roperty 2 ubtraction roperty 3 Multiplication roperty

7 5 Introduction to Geometric roof 43 In Exercises 27 to 30, fill in the missing reasons for each geometric proof! 27 Given: D-E-F on DF D E rove: DE DF EF Exercises 27, 28! D-E-F on DF 2 DE EF DF 3 DE DF EF F 30 Given: C and D! (ee figure for Exercise 29) rove: m D m C m DC C and D! 2 m D m DC m C 3 m D m C m DC 28 Given: E is the midpoint of DF rove: DE 2 (DF) In Exercises 3 and 32, fill in the missing statements and reasons E is the midpoint of DF 2 DE EF 3 DE EF DF 4 DE DE DF 4? 5 2(DE) DF 5? 6 DE 2 (DF) 6? 29 Given: D! bisects C rove: m D 2 (m C) 3: M-N--Q on MQ M rove: MN N Q MQ 2 MN NQ MQ 3 N Q NQ 4? 4 ubstitution roperty 32 Given: W with U! and! rove: m W m U m U m W N Q D U Exercises 29, 30 D! bisects C 2 m D m DC 3 m D m DC m C 4 m D m D m C 5 2(m D) m C 6 m D 2 (m C) C 4? 5? 6? W 2 m W m U m UW 3 m UW m U m W 4? 4 ubstitution roperty 33 When the Distributive roperty is written in its symmetric form, it reads a b a c a(b c) Use this form to rewrite 5x 5y 34 nother form of the Distributive roperty (see Exercise 33) reads b a c a (b c)a Use this form to rewrite 5x 7x hen simplify

8 44 CHER LINE ND NGLE RELIONHI 35 he Multiplication roperty of Inequality requires that we reverse the inequality symbol when multiplying by a negative number Given that 7 5, form the inequality that results when we multiply each side by 2 36 he Division roperty of Inequality requires that we reverse the inequality symbol when dividing by a negative number Given that 2 4, form the inequality that results when we divide each side by 4 37 rovide reasons for this proof If a b and c d, then a c b d 2 3 a b a c b c c d 4 a c b d 4? 38 Write a proof for: If a b and c d, then a c b d (HIN: Use Exercise 37 as a guide) 6 Relationships: erpendicular Lines KEY CONCE ertical Line(s) Horizontal Line(s) erpendicular Lines Relations: Reflexive, ymmetric, and ransitive roperties Equivalence Relation erpendicular isector of a Line egment j Informally, a vertical line is one that extends up and down, like a flagpole On the other hand, a line that extends left to right is horizontal In Figure 59, is vertical and j is horizontal Where lines and j intersect, they appear to form angles of equal measure DEFINIION erpendicular lines are two lines that meet to form congruent adjacent angles Figure 59 m erpendicular lines do not have to be vertical and horizontal In Figure 60, the slanted lines m and p are perpendicular (m p) s in Figure 60, a small square is often placed in the opening of an angle formed by perpendicular lines Example provides a formal proof of the relationship between perpendicular lines and right angles tudy this proof, noting the order of the statements and reasons he numbers in parentheses to the left of the statements refer to the earlier statement(s) of the proof upon which the new statement is based p REGY FOR he Drawing for the roof General Rule: Make a drawing that accurately characterizes the Given information Illustration: For the proof of Example, see Figure 6 Figure 60 HEOREM 6 If two lines are perpendicular, then they meet to form right angles