Chapter 3. Proving Statements in Geometry

Transcription

1 3- Inductive Reasoning (pages 95 97). No; triangles may contain a right or obtuse 2. Answers will vary. Example: a. Answers will vary. b. 90 c. The sum of the measures of the acute angles of a right triangle is a. Answers will vary. b. The quadrilaterals formed are parallelograms. c. A quadrilateral with vertices at the midpoints of the sides of another quadrilateral is a parallelogram. 5. a. Answers will vary. b. The four triangles are equilateral. c. The four triangles formed by connecting midpoints of the sides of an equilateral triangle are equilateral. 6. Probably true; the experiment must include lines intersecting at acute, right, and obtuse angles. 7. Probably true; the experiment must include quadrilaterals with acute, obtuse, and right angles. 8. False; pairs of different types of numbers must be included: natural numbers, positive and negative integers, rational numbers, and irrational numbers. 9. Probably true; draw parallelograms of different sizes and angle measures. 0. False; draw quadrilaterals of different shapes: trapezoids, parallelograms, rectangles, squares, rhombuses, and others.. Probably true; draw triangles of different shapes: acute, right, obtuse, isosceles, equilateral, and scalene. 2. False; let n 40, then (40) ,68. This is not a prime number since, Applying Skills 3. No 4. Yes 5. No 6. No 7. Yes Chapter 3. Proving in Geometry Hands-On Activity a., 4, and 9 are perfect squares. b., 4, 9, 6, and 25. The numbers are perfect squares. c. The cards facing up will all be perfect squares. 3-2 Definitions as Biconditionals (pages 99 00). Doug is correct. If a container can be used to carry food, then it is a lunchbox is false. For example, a paper bag can be used to carry food. 2. a. 22,, but 2 is false. b. Positive real numbers 3. a. If a triangle is equiangular, then it has three b. If a triangle has three congruent angles, then it is equiangular. c. A triangle is equiangular if and only if it has three 4. a. If a line or a subset of a line is a bisector of a line segment, then it intersects the segment at its b. If a line or a subset of a line intersects a line segment at its midpoint, then it is a bisector of the segment. c. A line or a subset of a line is a bisector of a line segment if and only if it intersects the segment at its 5. a. If an angle is acute, then its degree measure is greater than 0 and less than 90. b. If an angle has a degree measure greater than 0 and less than 90, then it is acute. c. An angle is acute if and only if its degree measure is greater than 0 and less then a. If a triangle is obtuse, then it has one obtuse b. If a triangle has one obtuse angle, then it is obtuse. c. A triangle is obtuse if and only if it has one obtuse 7. a. If a set of points is noncollinear, then the set contains three or more points that do not all lie on the same straight line. b. If a set of points contains three or more points that do not all lie on the same straight line, then the set of points is noncollinear. c. A set of points is noncollinear if and only if the set contains three or more points that do not all lie on the same straight line. 8. a. If a part of a line is a ray, then it consists of a point on the line, called an endpoint, and all the points on one side of the endpoint. 236

2 b. If a part of a line consists of a point, called an endpoint, and all the points on one side of the endpoint, then it is a ray. c. A part of a line is a ray if and only if it consists of a point, called an endpoint, and all the points on one side of the endpoint. 9. A point B is between A and C if and only if A, B, and C are distinct collinear points and AB BC AC. 0. Two segments are congruent if and only if they have the same length.. A point is the midpoint of a line segment if and only if it divides the segment into two congruent 2. A triangle is right if and only if it has a right 3. An angle is straight if and only if it is the union of two opposite rays and its degree measure is Two rays are opposite rays if and only if they are two rays of the same line with a common endpoint and no other point in common. Applying Skills 5. A triangle is equilateral if and only if it has three congruent sides. 6. Two angles are congruent if and only if they have the same measure. 7. Two lines are perpendicular if and only if they intersect to form right angles. 3-3 Deductive Reasoning (pages 03 05). Yes. An equilateral triangle has three congruent sides, so it satisfies the definition for an isosceles triangle, which is that a triangle must have two congruent sides. 2. Yes. If B is not between A and C, then AB BC AC. 3. b. If a ray bisects an angle, then it divides the angle into two 4. b. If a triangle is scalene, then no two of its sides are congruent. If two segments are not congruent, then they do not have the same measure. 5. b. If two lines are perpendicular, then they intersect to form right angles. 6. b. If A, B, and C are distinct collinear points and AB BC AC, then B is between A and C. 7. b. If two lines are perpendicular, then they intersect to form right angles. c. LMN is a right 8. b. If a line bisects a segment, then it divides the segment into two c. DF>FE or DF FE 9. b. If points P, Q, R are collinear with PQ QR PR, then Q is between P and R. c. Q is between P and R. 0. b. If two rays are opposite rays, then they form a straight c. TSR is a straight. b. If a point is the midpoint of a line segment, then it divides the segment into two congruent c. LM>MN or LM MN 2. b. If the degree measure of an angle is greater than 0 and less than 90, then the angle is acute. c. a is acute. 3. a.. M is the midpoint. Given. of AMB. 2. AM MB 2. Definition of 3. AM MB 3. Definition of congruent b. We are given that M is the midpoint of AMB. Therefore, M divides AMB into two congruent segments, AM and MB. Since congruent segments have the same length, AM MB. 4. a.. RS ST. Given. 2. RS>ST 2. Definition of congruent 3. RST is isosceles. 3. Definition of isosceles tri b. We are given RST with RS ST. Segments with the same length are congruent, so RS>ST. An isosceles triangle is a triangle with two congruent sides. Therefore, RST is isosceles. 5. a.. CE h bisects ACB.. Given. 2. ACE ECB 2. Definition of angle 3. m ACE 3. Definition of m ECB 237

3 b. We are given that CE h bisects ACB. If a ray bisects an angle, then it divides the angle into two congruent angles, so ACE ECB. Congruent angles have the same measure. Therefore, m ACE m ECB. 6.. DE EF. Given. 2. DE>EF 2. Definition of 3. E is the midpoint 3. Definition of of DEF. 7. An obtuse angle is an angle whose degree measure is between 90 and 80. An obtuse triangle is triangle with an obtuse Since m A 90 and m B 90, A and B are not obtuse. Since ABC is an obtuse triangle, C must be obtuse. Therefore, m C In isosceles ABC with A as the vertex angle, we can only infer that legs, AB and AC, are congruent and have equal measure. The length of BC is unknown. 3-4 Direct and Indirect Proofs (pages 08 09). Yes. ABC is made up of the rays BA h and BC h, which intersect at vertex B. B is a point on both lines BA g and BC g. Therefore, BA g and BC g intersect at B. g 2. Yes. The notation ABC means that A, B, and C are collinear and that B is between A and C. 3. b.. LM MN. Given. 2. LM>MN 2. Definition of congruent c.. LM is not. Assumption. congruent to MN. 2. LM MN 2. It two segments are not congruent, then they do not have the same measure. 3. LM MN 3. Given. 4. LM>MN 4. Contradiction. 4. b.. PQR is a. Given. straight 2. m PQR A straight angle measures 80. c.. m PQR 80. Assumption. 2. PQR is not a 2. If an angle does not straight measure 80, then it is not a straight 3. PQR is a straight 3. Given. 4. m PQR Contradiction. 5. b.. PQR is a straight. Given. 2. QP h are 2. Definition of opposite rays. straight c.. QP h are. Assumption. not opposite rays. 2. PQR is not a 2. If two rays are not straight opposite rays, then their union is not a straight 3. PQR is a straight 3. Given. 4. QP h are 4. Contradiction. two opposite rays. 6. b.. QP h are. Given. opposite rays. 2. P, Q, and R are on 2. Definition of the same line. opposite rays. c.. P, Q, and R are not. Assumption. on the same line. 2. QP h are 2. If two rays are not not opposite rays. of the same line, then they are not opposite rays. 3. QP h are 3. Given. opposite rays. 4. P, Q, and R are on 4. Contradiction. the same line. 238

4 7. b.. PQR is a straight. Given. 2. QP h are 2. Definition of opposite rays. straight 3. P, Q, and R are on 3. Definition of the same line. opposite rays. c.. P, Q, and R are not. Assumption. on the same line. 2. QP h are 2. If two rays are not not opposite rays. of the same line, then they are not opposite rays. 3. PQR is not a 3. If two rays are not straight opposite rays, then their union is not a straight 4. PQR is a straight 4. Given. 5. P, Q, and R are on 5. Contradiction. the same line. 8. b.. EG h bisects DEF.. Given. 2. DEG GEF 2. Definition of angle 3. m DEG 3. Definition of m GEF c.. m DEG. Assumption. m GEF 2. DEG is not 2. If two angles do not congruent to GEF. have the same measure, then they are not congruent. 3. EG h does not 3. If a ray does not bisect DEF. divide an angle into two congruent angles, then it does not bisect the 4. EG h bisects DEF. 4. Given. 5. m DEG 5. Contradiction. m GEF 9. The indirect proofs were longer than the direct proofs because the statements to be proved followed directly from definitions in geometry. 0.. DEG GEF. Assumption. 2. EG h bisects DEF. 2. Definition of angle 3. EG h does not bisect 3. Given. DEF. 4. DEG is not 4. Contradiction. congruent to GEF. Applying Skills g. a. Given: BD h is perpendicular to ABC. b. Prove: BD h is the bisector of ABD. c. g. BD h ABC. Given. 2. ABD and CBD 2. Definition of are right angles. perpendicular lines. 3. m ABD 90 and 3. The measure of a m CBD 90 right angle is ABD CBD 4. Congruent angles have the same measure. 5. BD h is the bisector 5. Definition of angle of ABD. 2. a. Given: m EFG 80 b. Prove: FE h and FG h are not opposite rays. c.. FE h and FG h are. Assumption. opposite rays. 2. EFG is a straight 2. Definition of straight 3. m EFG The measure of a straight angle is m EFG Given. 5. FE h and FG h are 5. Contradiction. not opposite rays. 3-5 Postulates,Theorems, and Proof (pages 3 5). Yes. An angle is congruent to itself. Congruent angles can be expressed in either order. Angles congruent to the same angle are congruent to each other. 2. No. A line is not perpendicular to itself. 3. Reflexive property of equality 239

5 4. Transitive property of equality 5. Symmetric property of equality 6. Transitive property of equality Applying Skills 7.. y x 4. Given. 2. x 4 y 2. Symmetric property. 3. y 7 3. Given. 4. x Transitive property. 8.. AB BC AC. Given. 2. AC AB BC 2. Symmetric property. 3. AB BC 2 3. Given. 4. AC 2 4. Transitive property. 9.. M is the midpoint. Given. of LN. 2. LM>MN 2. Definition of 3. LM MN 3. Definition of 4. N is the midpoint 4. Given. of MP. 5. MN>NP 5. Definition of 6. MN NP 6. Definition of 7. LM NP 7. Transitive property. 0.. m FGH m JGK. Given. 2. m HGJ m JGK 2. Given. 3. m FGH m HGJ 3. Transitive property. 4. FGH HGJ 4. Definition of 5. GH h is the bisector of 5. Definition of angle FGJ.. It is not given that ADB and ADC are right angles. This may not be assumed from the diagram. Hands-On Activity a. Definition of an equilateral triangle, transitive property of congruence, and definition of congruent segments b.. ABC is. Given. equilateral. 2. BCD is 2. Given. equilateral. 3. AB>BC 3. Definition of equilateral tri 4. BC>CD 4. Definition of equilateral tri 5. AB>CD 5. Transitive property. 6. AB CD 6. Definition of 3-6 The Substitution Postulate (page 7). Yes. Segments congruent to the same segment are congruent to each other. 2. No. While PQ and RS are congruent, PQ may be located anywhere in relation to ST; that is, PQ may not be perpendicular to ST. 3.. MT 2 RT. Given. 2. RM MT 2. Given. 3. RM 2 RT 3. Substitution 4.. AD DE AE. Given. 2. AD EB 2. Given. 3. EB DE AE 3. Substitution 5.. m a m b 80. Given. 2. m a m c 2. Given. 3. m c m b Substitution 6.. y 7 2x. Given. 2. y x 5 2. Given. 3. x 5 7 2x 3. Substitution x y. Given. 2. x 8 2. Given y 3. Substitution 240

6 8. Reason. BC 2 AB 2 AC 2. Given. 2. AB DE 2. Given. 3. BC 2 DE 2 AC 2 3. Substitution 9. Reason. AB "CD. Given. 2. "CD = EF 2. Given. 3. AB EF 3. Transitive property GH = EF 4. Given. 5. AB 2 GH 5. Transitive property. 0. Reason. m Q m R. Given. m S m Q m S 2. Given. m T 3. m T m R Substitution 4. m R m T 4. Given. m U 5. m U Substitution 3-7 The Addition and Subtraction Postulates (pages 22 23). Cassie is incorrect. The definition of subtraction in terms of addition depends on the definition of negative numbers. However, there is no concept of a negative line segment, so this definition would be invalid for line 2. a. Yes. We are given that m ABC 30, m CBD 45, and m DBE 5. Then, m ABC m DBE m CBD. b. No. Since ABC and DBE are not adjacent angles, ABC DBE does not represent an 3.. AED and BFC. Given. 2. AE ED AD 2. Partition BF FC BC 3. AE BF and 3. Given. ED FC 4. BF FC AD 4. Substitution 5. AD BC 5. Transitive property. 4.. SPR QRP. Given. and RPQ PRS 2. SPR RPQ 2. Addition QRP PRS 3. SPR RPQ 3. Partition SPQ 4. QRP PRS 4. Partition QRS 5. SPQ QRS 5. Substitution 5.. AC>BC and. Given. MC>NC 2. AC 2 MC 2. Subtraction > BC 2 NC 3. AC>AMMC 3. Partition BC>BNNC 4. AM MC 2 MC 4. Substitution > BN NC2 NC or AM>BN 6.. AB>CD. Given. 2. BC>BC 2. Reflexive property. 3. AB BC 3. Addition > BC CD 4. ABCD 4. Given. 5. AC>AB BC 5. Partition BD>BC CD 6. AC>BD 6. Substitution 7.. LMN PMQ. Given. 2. NMQ NMQ 2. Reflexive property. 3. LMN NMQ 3. Addition NMQ QMP 4. LMQ 4. Partition LMN NMQ NMP NMQ QMP 5. LMQ NMP 5. Substitution 24

7 8.. m AEB 80 and. Given. m CED AEB CED 2. Definition of 3. AEB 3. Partition AEC CEB CED CEB BED 4. AEC CEB 4. Substitution CEB BED 5. CEB CEB 5. Reflexive property. 6. AEC BED 6. Subtraction 7. m AEC 7. Definition of m BED 3-8 The Multiplication and Division Postulates (pages 26 27). The word positive is needed because only positive quantities have square roots, and a number has both a positive and a negative square root. 2. Yes. The conditions of the postulate require that c d. Therefore, if one of these variables is not equal to 0, then both are not equal to 0 and the a c fractions b and d are defined. If c 0 and d 0 are both removed, it is possible that a c c 0 and d 0, and that the fractions b and d are undefined. 3.. AB 4 BC. Given Reflexive property. 3. 4AB BC 3. Multiplication 4. BC CD 4. Given. 5. 4AB CD 5. Substitution 6. CD 4AB 6. Symmetric property. 4.. m a 3m b. Given. 2. m b Given. 3. 3m b Substitution 4. m a Transitive property. 5.. MN5 2 NP. Given. 2. 2MN = NP 2. Doubles of equal 3. LM 2MN 3. Given. 4. LM NP 4. Transitive property. 5. LM>NP 5. Definition of 6.. 2(3a 4) 6. Given. 2. 3a Halves of equal Reflexive property. 4. 3a 2 4. Addition Reflexive property. 6. a 4 6. Division 7.. QR5 3 RS. Given Reflexive property. 3. 3QR RS 3. Multiplication 4. PQ 3QR 4. Given. 5. PQ RS 5. Transitive property. 6. PQ>RS 6. Definition of Alternative Proof:. PQ 3QR. Given. 2. QR5 3 RS 2. Given. 3. PQ 3A 3 RSB or 3. Substitution PQ RS 4. PQ>RS 4. Definition of Applying Skills 8. Let m Monday s distance, t Tuesday s distance, w Wednesday s distance, and f Friday s distance. 2 We are given w and w = 3 f. Then 3 m52 3 f 3 m by the transitive property of equality. By the multiplication postulate, m 2f. We are also given that m 2t, so by the transitive property of equality, 2f 2t. Since halves of equal quantities are equal, f t or the distance Melanie walked on Friday is equal to the distance she walked on Tuesday. 242

8 9. Let L library, P post office, G grocery store, and B bank. We are given LPGB, with LP 4PG and GB 3PG. By the partition postulate, PB PG GB. Using the substitution postulate, PB PG 3PG or PB 4PG. By the transitive property, LP PB or the distance from the library to the post office is equal to the distance from the post office to the bank. 0. To show that doubles of congruent segments are congruent: Given: B is the midpoint of AC, M is the midpoint of LN, and AB>LM. Prove: AC>LN. B is the midpoint. Given. of AC, M is the midpoint of LN. 2. AC 2AB, 2. Definition of LN 2LM 3. AB>LM 3. Given. 4. AB LM 4. Definition of 5. 2AB 2LM 5. Doubles of equal 6. AC LM 6. Substitution 7. AC>LN 7. Definition of To show that doubles of congruent angles are congruent: Given: BC h is the angle bisector of ABD, KL h is the angle bisector of JKM, and ABC JKL. Prove: ABD JKM. BC h is the angle. Given. bisector of ABD, h KL is the angle bisector of JKM. 2. m ABD 2. Definition of angle 2m ABC, m JKM 2m JKL 3. ABC JKL 3. Given. 4. m ABC m JKL 4. Definition of 5. 2m ABC 5. Doubles of equal 2m JKL 6. m ABD m JKM 6. Substitution 7. ABD JKM 7. Definition of To show that halves of congruent segments are congruent: Given: B is the midpoint of AC, M is the midpoint of LN, and AC>LN. Prove: AB>LM. B is the midpoint of. Given. AC, M is the midpoint of LN. 2. AB5 2 ABC, 2. Definition of LM5 2 LMN 3. AC>LN 3. Given. 4. AC LN 4. Definition of 5. 2 AC5 2 LN 5. Halves of equal 6. AB LM 6. Substitution 7. AB>LM 7. Definition of Given: BC h is the angle bisector of ABD, KL h is the angle bisector of JKM, and ABD JKM. Prove: ABC JKL. BC h is the angle. Given. bisector of ABD, h KL is the angle bisector of JKM. 2. m/abc5 2 m/abd, 2. Definition of angle m/jkl5 2 m/jkm 3. ABD JKM 3. Given. 4. m ABD m JKM 4. Definition of 5. 2 m/abd 5. Halves of equal 5 2 m/jkm 6. m ABC m JKL 6. Substitution 7. ABC JKL 7. Definition of Review Exercises (pages 29 30). a. If a triangle is obtuse, then it has one obtuse 243

9 b. If a triangle has one obtuse angle, then it is obtuse. c. A triangle is obtuse if and only if it has one obtuse 2. a. If two angles are congruent, then they have the same measure. b. If two angles have the same measure, then they are congruent. c. Two angles are congruent if and only if they have the same measure. 3. a. If two lines are perpendicular, then they intersect to form right angles. b. If two lines intersect to form right angles, then they are perpendicular. c. Two lines are perpendicular if and only if they intersect to form right angles 4. We assume a postulate to be true without proof. A theorem is a statement that has been proven. 5. Symmetric property 6. No. The reflexive property does not hold true. Example: is false. The symmetric property does not hold true. Example: If 2, then 2 is false. The transitive property holds true. Example: If 3 2 and 2, then 3 is true. 7. C A M B D. AB g bisects CD. Given. at M. 2. M is the midpoint 2. Definition of of CD. 3. CM>MD 3. Definition of 4. CM MD 4. Definition of A B C D. AC>BD. Given. 2. AC>AB BC 2. Partition BD>BC CD 3. AB BC > BC CD 3. Substitution 4. BC>BC 4. Reflexive property. 5. AB>CD 5. Subtraction S Q R P. SQ RP. Given. 2. QR QR 2. Reflexive property. 3. SQ QR 3. Addition QR RP 4. SR SQ QR 4. Partition QP QR RP 5. SR QP 5. Substitution B. BC h bisects ABD.. Given. 2. ABC CBD 2. Definition of angle 3. m ABC m CBD 3. Definition of 4. m CBD m PQR 4. Given. 5. m ABC m PQR 5. Transitive property. C A C D Q P B R 8. R M S T. RM>MS. Given. 2. MS>ST 2. Given. 3. RM>ST 3. Transitive property. E A D. CD and AB bisect. Given. each other at E. (Cont.) 244

10 2. E is the midpoint of 2. Definition of a AB and of CD. 3. AE>EB and 3. Definition of CE>ED 4. AE EB and 4. Definition of CE ED congruent 5. AE CE EB ED 5. Addition 6. CE BE 6. Given. 7. AE BE CE ED 7. Substitution 8. AB AE EB 8. Partition CD CE ED 9. AB CD 9. Substitution 3. A quantity may be substituted for its equal in any statement of equality. CD'BC is not a statement of equality; therefore, the substitution postulate is not valid. 4. a. If a sequence of numbers or letters is a palindrome, then the sequence reads the same from left to right as from right to left. b. If a sequence of numbers or letters reads the same from left to right as from right to left, then the sequence is a palindrome. c. A sequence of numbers or letters is a palindrome if and only if it reads the same from left to right as from right to left. Exploration (page 30) Step 5 is not valid. Since a b, the quantity a b is equal to 0. Thus, we cannot apply the division Cumulative Review (pages ) Part I Part II. 3x 753 Given 7 7 Reflexive property 3x 6 Addition postulate x 2 Division postulate 2.. DE'EF. Given. 2. EDF is a right 2. Definition of perpendicular lines. 3. DEF is a right 3. Definition of right tri tri Part III 3.. ABC with D a point. Given. on AB. 2. AB AD DB 2. Partition 3. AC AD DB 3. Given. 4. AC AB 4. Transitive property. 5. AC>AB 5. Definition of congruent 6. ABC is isosceles. 6. Definition of isosceles tri 4. Yes. By the partition postulate, PQ QR PR. Thus, (4a 2 3) (3a 2) 5 8a 2 6 Therefore, PQ 4(5) 3 7 and QR 3(5) 2 7. Thus, PQ>QR and Q is the midpoint of PQR. Part IV 5. Yes. By the partition postulate, 7a 2 58a a m/abd m/dbc 5 m/abc 3x 8 5x x 2 8x 2 257x 2 x524 Therefore, m ABD 3(24) 8 90 and m DBC 5(24) Thus, m ABC An angle whose degree measure is 80 is a straight 6. a. Addition postulate b. Reflexive property of equality c. Partition postulate d. Substitution postulate e. Division postulate 245