S 102 DG4PS 893 04
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DG4PSA_894_04.qxd 11/1/06 1:24 PM Page 24 Lesson 4.1 • Triangle Sum Conjecture Name Period Date In Exercises 1–9, determine the angle measures. 1. p ______, q ______ 2. x ______, y ______ 28⬚ 53⬚ x y q p 82⬚ 3. a ______, b ______ 79⬚ 17⬚ 98⬚ a 50⬚ b 23⬚ 31⬚ 4. r ______, s ______, 5. x ______, y ______ 6. y ______ t ______ 100 ⫺ x 85⬚ t s x 100⬚ r 30 ⫹ 4x y y 7x 31⬚ 7. s ______ 8. m ______ 9. mP ______ c s b 35⬚ c a P m 76⬚ a 10. Find the measure of QPT. 11. Find the sum of the measures of the marked angles. P T Q S R 12. Use the diagram to explain why A and B are complementary. 13. Use the diagram to explain why mA mB mC mD. A D B E C B C 24 CHAPTER 4 A Discovering Geometry Practice Your Skills ©2008 Kendall Hunt Publishing DG4PSA_894_04.qxd 11/1/06 1:24 PM Page 25 Lesson 4.2 • Properties of Isosceles Triangles Name Period Date In Exercises 1–3, find the angle measures. 1. mT ______ 2. mG ______ T 3. x ______ A x 110⬚ R 58⬚ I G N In Exercises 4–6, find the measures. 4. mA ______, perimeter of ABC ______ A 5. The perimeter of LMO 6. The perimeter of QRS is is 536 m. LM ______ , mM ______ 344 cm. mQ ______ , QR ______ M y⫹ 13 cm a ⫹ 7 cm 68⬚ y Q 210 m 102⬚ B R m 31 c S 39⬚ a x ⫹ 30⬚ L 163 m x O C 7. a. Name the angle(s) congruent to DAB. C B b. Name the angle(s) congruent to ADB. D A and BC ? Why? c. What can you conclude about AD 8. x _____, y _____ 4y 9. PR QR and QS RS. If mRSQ 120°, what is mQPR? 10. Use the diagram to explain why PQR is isosceles. Q Q 70⬚ T R 2x ⫹ y 79⬚ ⫺ x P R S 55⬚ P S Discovering Geometry Practice Your Skills ©2008 Kendall Hunt Publishing CHAPTER 4 25 DG4PSA_894_04.qxd 11/1/06 1:24 PM Page 26 Lesson 4.3 • Triangle Inequalities Name Period Date In Exercises 1 and 2, determine whether it is possible to draw a triangle with sides of the given measures. If it is possible, write yes. If it is not possible, write no and make a sketch demonstrating why it is not possible. 1. 16 cm, 30 cm, 45 cm 2. 9 km, 17 km, 28 km 3. If 17 and 36 are the lengths of two sides of a triangle, what is the range of possible values for the length of the third side? In Exercises 4–6, arrange the unknown measures in order from greatest to least. 4. c 5. b 18 13 71⬚ a a c 6. 32⬚ c 40⬚ d b b 61⬚ 28⬚ 20 a 7. x _____ 8. x _____ x 9. What’s wrong with this picture? x 142⬚ C 66⬚ B 160⬚ 158⬚ 120⬚ A 10. Explain why PQS is isosceles. Q P x 2x S R In Exercises 11 and 12, use a compass and straightedge to construct a triangle with the given sides. If it is not possible, explain why not. 11. A C A CHAPTER 4 Q Q C B 26 12. P B P R R Discovering Geometry Practice Your Skills ©2008 Kendall Hunt Publishing DG4PSA_894_04.qxd 11/1/06 1:24 PM Page 27 Lesson 4.4 • Are There Congruence Shortcuts? Name Period Date In Exercises 1–3, name the conjecture that leads to each congruence. 1. PAT IMT A S I D and MST AST 8 6 T P bisects MA , MT AT , 3. TS 2. SID JAN I 9 9 A 8 M N M 6 S T J A In Exercises 4–9, name a triangle congruent to the given triangle and state the congruence conjecture. If you cannot show any triangles to be congruent from the information given, write “cannot be determined” and redraw the triangles so that they are clearly not congruent. 4. M is the midpoint of AB 5. KITE is a kite with KI TI. . and PQ KIE _____ APM _____ P 6. ABC _____ Y A C T B Z E I M X B K A Q 7. MON _____ N 8. SQR _____ 9. TOP _____ y Q T U G R M 10 O D 8 S T O 6 T 4 P 2 2 4 6 8 x 10 In Exercises 10–12, use a compass and a straightedge or patty paper and a straightedge to construct a triangle with the given parts. Then, if possible, construct a different (noncongruent) triangle with the same parts. If it is not possible, explain why not. 10. S 11. T T B U U S Discovering Geometry Practice Your Skills 12. X Y X Z A B ©2008 Kendall Hunt Publishing C C X CHAPTER 4 27 DG4PSA_894_04.qxd 11/1/06 1:25 PM Page 28 Lesson 4.5 • Are There Other Congruence Shortcuts? Name Period Date In Exercises 1–6, name a triangle congruent to the given triangle and state the congruence conjecture. If you cannot show any triangles to be congruent from the information given, write “cannot be determined” and explain why. 1. PIT _____ 2. XVW _____ P E B W V O 3. ECD _____ C X I Y T is the angle bisector 4. PS 5. ACN _____ of QPR. GQ EQ. Q EQL _____ C R D 6. EFGH is a parallelogram. P PQS _____ P A Z S Q A R N 7. The perimeter of QRS is 350 cm. L 70 x 2x 125 M H Is TUV WXV? Explain. x ⫹ 55 Q L E 8. The perimeter of TUV is 95 cm. Is QRS MOL? Explain. O G K F ⫹ 15 R 2x ⫺ 10 T U x x ⫹ 25 V 40 S W X In Exercises 9 and 10, construct a triangle with the given parts. Then, if possible, construct a different (noncongruent) triangle with the same parts. If it is not possible, explain why not. 9. P Q Q P 10. A B A C 28 CHAPTER 4 Discovering Geometry Practice Your Skills ©2008 Kendall Hunt Publishing DG4PSA_894_04.qxd 11/1/06 1:25 PM Page 29 Lesson 4.6 • Corresponding Parts of Congruent Triangles Name Period Date 1. Give the shorthand name for each of the four triangle congruence conjectures. In Exercises 2–5, use the figure at right to explain why each congruence is true. WXYZ is a parallelogram. 2. WXZ YZX 3. WZX YXZ 4. WZX YXZ 5. W Y Z Y W X For Exercises 6 and 7, mark the figures with the given information. To demonstrate whether the segments or the angles indicated are congruent, determine that two triangles are congruent. Then state which conjecture proves them congruent. and 6. M is the midpoint of WX . Is YW ZX ? Why? YZ is the bisector 7. ABC is isosceles and CD BD ? Why? of the vertex angle. Is AD X C Y M Z W A In Exercises 8 and 9, use the figure at right to write a paragraph proof for each statement. CF 8. DE FD 9. EC A B D D C E F B 10. TRAP is an isosceles trapezoid with TP RA and PTR ART. RP . Write a paragraph proof explaining why TA P T A R Discovering Geometry Practice Your Skills ©2008 Kendall Hunt Publishing CHAPTER 4 29 DG4PSA_894_04.qxd 11/1/06 1:25 PM Page 30 Lesson 4.7 • Flowchart Thinking Name Period Date Complete the flowchart for each proof. SR and PQ SR 1. Given: PQ S R QR Show: SP P Flowchart Proof Q Given PQ SR __________________ __________________ PQS ______ SP QR _________________ __________________ QS ______ __________________ KI 2. Given: Kite KITE with KE I bisects EKI and ETI Show: KT K Flowchart Proof T E KE KI ETK ITK ______________ _______________ __________________ KET ______ KITE is a kite ______________ ________________ Definition of bisect __________________ ______________ 3. Given: ABCD is a parallelogram Show: A C D A C B Flowchart Proof AB CD ABCD is a parallelogram _____________ Definition of ___________ _________________________ Same segment ____________ _____________ ______________ _____________ 30 CHAPTER 4 Discovering Geometry Practice Your Skills ©2008 Kendall Hunt Publishing DG4PSA_894_04.qxd 11/1/06 1:25 PM Page 31 Lesson 4.8 • Proving Special Triangle Conjectures Name Period Date In Exercises 1–3, use the figure at right. C is a median, perimeter ABC 60, and AC 22. AD _____ 1. CD is an angle bisector, and mA 54°. mACD _____ 2. CD is an altitude, perimeter ABC 42, mACD 38°, and AD 8. 3. CD mB _____, CB _____ A 4. EQU is equilateral. D B 5. ANG is equiangular mE _____ and perimeter ANG 51. AN _____ , 6. ABC is equilateral, ACD is isosceles with base AC perimeter ABC 66, and perimeter ACD 82. Perimeter ABCD _____ C B D A 7. Complete a flowchart proof for this conjecture: In an isosceles triangle, C the altitude from the vertex angle is the median to the base. BC and altitude CD Given: Isosceles ABC with AC is a median Show: CD A Flowchart Proof D B __________________ CD is an altitude ____________________ ADC and BDC are right angles Definition of altitude AC BC Given ADC BDC ••• __________________ A ________ __________________ 8. Write a flowchart proof for this conjecture: In an isosceles triangle, the C median to the base is also the angle bisector of the vertex angle. BC and median CD Given: Isosceles ABC with AC bisects ACB Show: CD A Discovering Geometry Practice Your Skills ©2008 Kendall Hunt Publishing D CHAPTER 4 B 31 DG4PSA_894_ans.qxd 11/1/06 10:37 AM Page 99 LESSON 3.8 • The Centroid LESSON 4.2 • Properties of Isosceles Triangles 1. mT 64° 1. 2. mG 45° 3. x 125° 4. mA 39°, perimeter of ABC 46 cm C 5. LM 163 m, mM 50° 6. mQ 44°, QR 125 2. 7. a. DAB ABD BDC BCD b. ADB CBD BC by the Converse of the AIA Conjecture. c. AD G 8. x 21°, y 16° 3. CP 3.3 cm, CQ 5.7 cm, CR 4.8 cm R 9. mQPR 15° 10. mPRQ 55° by VA, which makes mP 55° by the Triangle Sum Conjecture. So, PQR is isosceles by the Converse of the Isosceles Triangle Conjecture. LESSON 4.3 • Triangle Inequalities 10 cm 6 cm 1. Yes C P 2. No 5. PC 16, CL 8, QM 15, CR 14 Incenter Circumcenter Orthocenter Centroid 9 km 28 km 4. (3, 4) 6. a. c. e. g. 17 km Q 8 cm b. Centroid d. Circumcenter f. Incenter 3. 19 x 53 4. b a c 5. b c a 6. a c d b 7. x 76° 8. x 79° 9. The interior angle at A is 60°. The interior angle at B is 20°. But now the sum of the measures of the triangle is not 180°. 3. a 78°, b 29° 10. By the Exterior Angles Conjecture, 2x x mPQS. So, mPQS x. So, by the Converse of the Isosceles Triangle Conjecture, PQS is isosceles. 4. r 40°, s 40°, t 100° 11. Not possible. AB BC AC 5. x 31°, y 64° 12. LESSON 4.1 • Triangle Sum Conjecture 1. p 67°, q 15° 7. s 28° 9. mP a 2. x 82°, y 81° 6. y 145° 1 8. m 722° 10. mQPT 135° P Q R 11. 720° 12. The sum of the measures of A and B is 90° because mC is 90° and all three angles must be 180°. So, A and B are complementary. 13. mBEA mCED because they are vertical angles. Because the measures of all three angles in each triangle add to 180°, if equal measures are subtracted from each, what remains will be equal. Discovering Geometry Practice Your Skills ©2008 Kendall Hunt Publishing LESSON 4.4 • Are There Congruence Shortcuts? 1. SAA or ASA 2. SSS 4. BQM (SAS) 5. TIE (SSS) 3. SSS ANSWERS 99 DG4PSA_894_ans.qxd 11/1/06 10:37 AM Page 100 6. Cannot be determined, as shown by the figure. 9. All triangles will be congruent by ASA. Possible triangle: Y R C A Z Q P B 10. All triangles will be congruent by SAA. Possible procedure: Use A and C to construct B and . then copy A and B at the ends of AB X 7. TNO (SAS) 8. Cannot be determined, as shown by the figure. C Q U R A B T S LESSON 4.6 • Corresponding Parts of Congruent Triangles 9. DOG (SAS) 10. Only one triangle because of SSS. 1. SSS, SAS, ASA, SAA T U S 11. Two possible triangles. A , AIA Conjecture WX 3. WZ , AIA Conjecture XY 4. ASA 5. CPCTC ZX by CPCTC. 6. YWM ZXM by SAS. YW BD by CPCTC. 7. ACD BCD by SAS. AD A B 2. YZ B C C 12. Only one triangle because of SAS. CF (see Exercise 8). 9. Possible answer: DE DEF CFE because both are right angles, FE because they are the same segment. So, EF FD by CPCTC. DEF CFE by SAS. EC Z X 8. Possible answer: DE and CF are both the distance and AB . Because the lines are parallel, between DC CF . the distances are equal. So, DE Y LESSON 4.5 • Are There Other Congruence Shortcuts? 1. Cannot be determined 10. Possible answer: It is given that TP RA and RT because they PTR ART, and TR are the same segment. So PTR ART RP by CPCTC. by SAS and TA LESSON 4.7 • Flowchart Thinking 1. (See flowchart proof at bottom of page 101.) 2. XZY (SAA) 3. ACB (ASA or SAA) 2. (See flowchart proof at bottom of page 101.) 4. PRS (ASA) 5. NRA (SAA) 3. (See flowchart proof at bottom of page 101.) 6. GQK (ASA or SAA) 7. Yes, QRS MOL by SSS. and WV are not 8. No, corresponding sides TV congruent. 100 ANSWERS LESSON 4.8 • Proving Special Triangle Conjectures 1. AD 8 2. mACD 36° 3. mB 52°, CB 13 4. mE 60° 5. AN 17 6. Perimeter ABCD 104 Discovering Geometry Practice Your Skills ©2008 Kendall Hunt Publishing DG4PSA_894_ans.qxd 11/1/06 10:37 AM Page 101 7. (See flowchart proof at bottom of page 102.) 3. 170°; 36 sides 4. 15 sides 8. Flowchart Proof 5. x 105° 6. x 18° 7. mE 150° CD is a median Given C 110° AC BC AD BD CD CD Given Definition of median Same segment B 70° D 125° 85° A 150° E ADC BDC LESSON 5.2 • Exterior Angles of a Polygon SSS Conjecture ACD BCD 1. 12 sides 2. 24 sides CPCTC 5. a 64°, b 13823° 3. 4 sides 4. 6 sides 6. a 102°, b 9° 7. a 156°, b 132°, c 108° CD bisects ACB 8. a 135°, b 40°, c 105°, d 135° Definition of bisect 9. LESSON 5.1 • Polygon Sum Conjecture 1. a 103°, b 103°, c 97°, d 83°, e 154° 2. a 92°, b 44°, c 51°, d 85°, e 44°, f 136° Lesson 4.7, Exercises 1, 2, 3 1. PQ SR Given PQ SR PQS RSQ PQS RSQ SP QR Given AIA Conjecture SAS Conjecture CPCTC QS QS Same segment 2. KE KI Given ETK ITK KITE is a kite TE TI KET KIT CPCTC KT bisects EKI and ETI Given Definition of kite SSS Conjecture EKT IKT Definition of bisect CPCTC KT KT Same segment 3. ABD CDB ABCD is a parallelogram AB CD AIA Conjecture Definition of parallelogram BD DB BDA DBC A C Same segment ASA Conjecture CPCTC Given AD CB Definition of parallelogram Discovering Geometry Practice Your Skills ©2008 Kendall Hunt Publishing ADB CBD AIA Conjecture ANSWERS 101
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