Discovering Geometry An Investigative Approach X70 Workbook Ch04
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DG4PS_893_04.qxd 11/1/06 10:20 AM 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 Key Curriculum Press DG4PS_893_04.qxd 11/1/06 10:20 AM 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 55⬚ R S P S Discovering Geometry Practice Your Skills ©2008 Key Curriculum Press CHAPTER 4 25 DG4PS_893_04.qxd 11/1/06 10:20 AM 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 Key Curriculum Press DG4PS_893_04.qxd 11/1/06 10:20 AM 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 6. ABC _____ KIE _____ APM _____ Y A C T B P 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 ©2008 Key Curriculum Press 12. X Y X Z A B Discovering Geometry Practice Your Skills C C X CHAPTER 4 27 DG4PS_893_04.qxd 11/1/06 10:20 AM 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 X W 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 Key Curriculum Press DG4PS_893_04.qxd 11/1/06 10:20 AM 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 Key Curriculum Press CHAPTER 4 29 DG4PS_893_04.qxd 11/1/06 10:20 AM 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 Key Curriculum Press DG4PS_893_04.qxd 11/1/06 10:20 AM 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 Key Curriculum Press D CHAPTER 4 B 31
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