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DG4PSA_894_05.qxd 11/1/06 1:24 PM Page 32 Lesson 5.1 • Polygon Sum Conjecture Name Period Date In Exercises 1 and 2, find each lettered angle measure. 1. a _____, b _____, c _____, d _____, e _____ 2. a _____, b _____, c _____, d _____, e _____, f _____ e d e d 97⬚ f a b b 26⬚ c 85⬚ 44⬚ c a 3. One exterior angle of a regular polygon measures 10°. What is the measure of each interior angle? How many sides does the polygon have? 4. The sum of the measures of the interior angles of a regular polygon is 2340°. How many sides does the polygon have? 5. ABCD is a square. ABE is an equilateral 6. ABCDE is a regular pentagon. ABFG triangle. is a square. x _____ x _____ D C D E x E G F C x B A A B 7. Use a protractor to draw pentagon ABCDE with mA 85°, mB 125°, mC 110°, and mD 70°. What is mE? Measure it, and check your work by calculating. 32 CHAPTER 5 Discovering Geometry Practice Your Skills ©2008 Key Curriculum Press DG4PSA_894_05.qxd 11/1/06 1:24 PM Page 33 Lesson 5.2 • Exterior Angles of a Polygon Name Period 1. How many sides does a regular polygon Date 2. How many sides does a polygon have if the have if each exterior angle measures 30°? sum of the measures of the interior angles is 3960°? 3. If the sum of the measures of the interior 4. If the sum of the measures of the interior angles of a polygon equals the sum of the measures of its exterior angles, how many sides does it have? angles of a polygon is twice the sum of its exterior angles, how many sides does it have? In Exercises 5–7, find each lettered angle measure. 5. a _____, b _____ 116⬚ 6. a _____, b _____ 7. a _____, b _____, c _____ a a b x 82⬚ 2x a c 3x b 82⬚ b 134⬚ 72⬚ 8. Find each lettered angle measure. 150⬚ a _____ c d b _____ c _____ d _____ a 95⬚ b 9. Construct an equiangular quadrilateral that is not regular. Discovering Geometry Practice Your Skills ©2008 Key Curriculum Press CHAPTER 5 33 DG4PSA_894_05.qxd 11/1/06 1:24 PM Page 34 Lesson 5.3 • Kite and Trapezoid Properties Name Period Date In Exercises 1–4, find each lettered measure. 1. Perimeter 116. x _____ 2. x _____, y _____ x y 56⬚ x 28 3. x _____, y _____ 4. x _____, y _____ 137⬚ 78⬚ x x 41⬚ 22⬚ y y 5. Perimeter PQRS 220. PS _____ S 4x ⫹ 1 6. b 2a 1. a _____ M R b a P 34 N 2x ⫺ 3 L Q T 4 K In Exercises 7 and 8, use the properties of kites and trapezoids to construct each figure. Use patty paper or a compass and a straightedge. , B, and distance 7. Construct an isosceles trapezoid given base AB between bases XY. A B X Y B , BC , and BD . 8. Construct kite ABCD with AB A B B C B D 9. Write a paragraph or flowchart proof of the Converse of the Isosceles parallel to TP with E on TR . Trapezoid Conjecture. Hint: Draw AE P A Given: Trapezoid TRAP with T R RA Show: TP 34 CHAPTER 5 T R Discovering Geometry Practice Your Skills ©2008 Key Curriculum Press DG4PSA_894_05.qxd 11/1/06 1:24 PM Page 35 Lesson 5.4 • Properties of Midsegments Name Period Date In Exercises 1–3, each figure shows a midsegment. 1. a _____, b _____, 2. x _____, y _____, c _____ 3. x _____, y _____, z _____ z _____ 41 b 16 14 z y 54⬚ a c 13 x 37⬚ y 29 z 11 x 21 4. X, Y, and Z are midpoints. Perimeter PQR 132, RQ 55, and PZ 20. Perimeter XYZ _____ R Z PQ _____ ZX _____ P is the midsegment. Find the 5. MN Y Q X 6. Explain how to find the width of the lake from coordinates of M and N. Find the and MN . slopes of AB y A to B using a tape measure, but without using a boat or getting your feet wet. C (20, 10) B A Lake M A (4, 2) N x B (9, –6) 7. M, N, and O are midpoints. What type of quadrilateral C is AMNO? How do you know? Give a flowchart proof showing that ONC MBN. N O A M B 8. Give a paragraph or flowchart proof. Given: PQR with PD DF FH HR R and QE EG GI IR H FG DE PQ Show: HI F D P Discovering Geometry Practice Your Skills ©2008 Key Curriculum Press I G E Q CHAPTER 5 35 DG4PSA_894_05.qxd 11/1/06 1:24 PM Page 36 Lesson 5.5 • Properties of Parallelograms Name Period Date In Exercises 1–7, ABCD is a parallelogram. 1. Perimeter ABCD _____ 26 cm D 2. AO 11, and BO 7. 3. Perimeter ABCD 46. AC _____, BD _____ C C D 15 cm AB _____, BC _____ C D B A 2x ⫺ 7 O A 4. a _____, b _____, 5. Perimeter ABCD 119, and c _____ 6. a _____, b _____, BC 24. AB _____ D b C D B x⫹9 A B c _____ D C b c 110⬚ 19⬚ A B A c A a 62⬚ 27⬚ 68⬚ C a 26⬚ B B 7. Perimeter ABCD 16x 12. AD _____ D 8. Ball B is struck at the same instant by two forces, F1 and F2. Show the resultant force on the ball. C 4x 3 A 63 B F2 F1 B 9. Find each lettered angle measure. a _____ g _____ G b _____ h _____ 81⬚ c _____ i _____ d _____ j _____ e _____ k _____ h 26⬚ D g A f _____ e a d i E f c 38⬚ j b B 38⬚ F k C and BD . 10. Construct a parallelogram with diagonals AC Is your parallelogram unique? If not, construct a different (noncongruent) parallelogram. A 36 C CHAPTER 5 B D Discovering Geometry Practice Your Skills ©2008 Key Curriculum Press DG4PSA_894_05.qxd 11/1/06 1:24 PM Page 37 Lesson 5.6 • Properties of Special Parallelograms Name Period 1. PQRS is a rectangle and OS 16. Date 2. KLMN is a square and 3. ABCD is a rhombus, NM 8. AD 11, and DO 6. OQ _____ mOKL _____ OB _____ mQRS _____ mMOL _____ BC _____ PR _____ Perimeter KLMN _____ mAOD _____ S N R M D O C O O Q P A K B L In Exercises 4–11, match each description with all the terms that fit it. a. Trapezoid b. Isosceles triangle c. Parallelogram d. Rhombus e. Kite f. Rectangle g. Square h. All quadrilaterals 4. _____ Diagonals bisect each other. 5. _____ Diagonals are perpendicular. 6. _____ Diagonals are congruent. 7. _____ Measures of interior angles sum to 360°. 8. _____ Opposite sides are congruent. 10. _____ Both diagonals bisect angles. 9. _____ Opposite angles are congruent. 11. _____ Diagonals are perpendicular bisectors of each other. In Exercises 12 and 13, graph the points and determine whether ABCD is a trapezoid, parallelogram, rectangle, or none of these. 12. A(4, 1), B(0, 3), C(4, 0), D(1, 5) 13. A(0, 3), B(1, 2), C(3, 4), D(2, 1) 5 –5 5 5 –5 5 –5 –5 and CAB. 14. Construct rectangle ABCD with diagonal AC A C CAB A Discovering Geometry Practice Your Skills ©2008 Key Curriculum Press CHAPTER 5 37 DG4PSA_894_05.qxd 11/1/06 1:24 PM Page 38 Lesson 5.7 • Proving Quadrilateral Properties Name Period Date Write or complete each flowchart proof. D Q C QC 1. Given: ABCD is a parallelogram and AP and PQ bisect each other Show: AC R A B P Flowchart Proof APR CQR ___________________ APR _____ DC AB Given AR _____ Given ______________ AC and QP bisect each other _____________ Definition of bisect ______________ AIA Conjecture D BC and CD AD 2. Given: Dart ABCD with AB Show: A C B C A 3. Show that the diagonals of a rhombus divide the rhombus into C D four congruent triangles. O Given: Rhombus ABCD Show: ABO CBO CDO ADO A AC , DX AC 4. Given: Parallelogram ABCD, BY B D BY Show: DX A 38 CHAPTER 5 C Y X B Discovering Geometry Practice Your Skills ©2008 Key Curriculum Press 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 Key Curriculum Press ADB CBD AIA Conjecture ANSWERS 101 DG4PSA_894_ans.qxd 11/1/06 10:37 AM Page 102 LESSON 5.3 • Kite and Trapezoid Properties B A 1. x 30 2. x 124°, y 56° 3. x 64°, y 43° 4. x 12°, y 49° 5. PS 33 6. a 11 7. D M C A 8. N P 7. AMNO is a parallelogram. By the Triangle Mid AM and MN AO . segment Conjecture, ON Flowchart Proof B D OC _1 AC 2 MN _1 AC 2 Definition of midpoint Midsegment Conjecture C A OC MN B 9. Possible answer: PT with E on TR . Paragraph proof: Draw AE TEAP is a parallelogram. T AER because they are corresponding angles of parallel lines. T R because it is given, so AER R, because both are congruent to T. Therefore, AER is isosceles by the Converse of the Isosceles EA because they are Triangle Conjecture. TP EA opposite sides of a parallelogram and AR RA because AER is isosceles. Therefore, TP . because both are congruent to EA Both congruent to _1 AC 2 MB _1 AB 2 ON _1 AB 2 Definition of midpoint Midsegment Conjecture ON MB Both congruent to _1 AB 2 CON A NMB A CA Conjecture CA Conjecture LESSON 5.4 • Properties of Midsegments CON NMB 1. a 89°, b 54°, c 91° Both congruent to A 2. x 21, y 7, z 32 ONC MBN 3. x 17, y 11, z 6.5 4. Perimeter XYZ 66, PQ 37, ZX 27.5 1.6, 5. M(12, 6), N(14.5, 2); slope AB 1.6 slope MN 6. Pick a point P from which A and B can be viewed over land. Measure AP and BP and find the midpoints M and N. AB 2MN. SAS Conjecture FG by 8. Paragraph proof: Looking at FGR, HI the Triangle Midsegment Conjecture. Looking at PQ for the same reason. Because PQR, FG PQ , quadrilateral FGQP is a trapezoid and FG is the midsegment, so it is parallel to FG DE . Therefore, HI FG DE PQ . and PQ Lesson 4.8, Exercise 7 CD CD Same segment CD is an altitude Given 102 ADC and BDC are right angles Definition of altitude ANSWERS ADC BDC ADC BDC AD BD CD is a median Both are right angles SAA Conjecture CPCTC Definition of median AC BC A B Given Converse of IT Discovering Geometry Practice Your Skills ©2008 Key Curriculum Press DG4PSA_894_ans.qxd 11/1/06 10:37 AM Page 103 LESSON 5.5 • Properties of Parallelograms 4. c, d, f, g 5. d, e, g 1. Perimeter ABCD 82 cm 6. f, g 7. h 2. AC 22, BD 14 8. c, d, f, g 9. c, d, f, g 3. AB 16, BC 7 10. d, g 11. d, g 4. a 51°, b 48°, c 70° 12. None 13. Parallelogram 5. AB 35.5 14. 6. a 41°, b 86°, c 53° D C A B 7. AD 75 8. F2 Resultant vector F1 F2 LESSON 5.7 • Proving Quadrilateral Properties F1 9. a 38°, b 142°, c 142°, d 38°, e 142°, f 38°, g 52°, h 12°, i 61°, j 81°, k 61° 1. (See flowchart proof at bottom of page.) 2. Flowchart Proof 10. No AB CB D Given C A B AD CD ABD CBD A C Given SSS Conjecture CPCTC DB DB D Same segment 3. Flowchart Proof C A AO CO Diagonals of parallelogram bisect each other B LESSON 5.6 • Properties of Special Parallelograms 1. OQ 16, mQRS 90°, PR 32 AB CB CD AD BO DO Definition of rhombus Diagonals of a parallelogram bisect each other 2. mOKL 45°, mMOL 90°, perimeter KLMN 32 ABO CBO CDO ADO SSS Conjecture 3. OB 6, BC 11, mAOD 90° Lesson 5.7, Exercise 1 APR CQR AIA Conjecture AR CR DC AB AP CQ APR CQR CPCTC AC and QP bisect each other Given Given ASA Conjecture PR QR Definition of bisect PAR QCR CPCTC AIA Conjecture Discovering Geometry Practice Your Skills ©2008 Key Curriculum Press ANSWERS 103 DG4PSA_894_ans.qxd 11/1/06 10:37 AM Page 104 3. z 45° 4. Flowchart Proof AD BC 4. w 100°, x 50°, y 110° Definition of parallelogram 5. w 49°, x 122.5°, y 65.5° DAX BCY 6. x 16 cm, y cannot be determined AIA Conjecture AD CB AXD CYB Opposite sides of parallelogram Both are 90° AXD CYB SAA Conjecture DX BY CPCTC LESSON 6.1 • Tangent Properties 1. w 126° 2. mBQX 65° 3. a. mNQP 90°, mMPQ 90° and NQ b. Trapezoid. Possible explanation: MP , so they are are both perpendicular to PQ parallel to each other. The distance from M to is MP, and the distance from N to PQ is PQ NQ. But the two circles are not congruent, so is not a constant MP NQ. Therefore, MN and they are not parallel. distance from PQ Exactly one pair of sides is parallel, so MNQP is a trapezoid. 1 4. y 3x 10 5. Possible answer: Tangent segments from a point to a PB , PB PC , and circle are congruent. So, PA PD . Therefore, PA PD . PC 6. a. 4.85 cm b. 11.55 cm 9. P(0,1), M(4, 2) 49°, mABC 253°, mBAC 156°, 10. mAB mACB 311° 11. Possible answer: Fold and crease to match the endpoints of the arc. The crease is the perpendicular bisector of the chord connecting the endpoints. Fold and crease so that one endpoint falls on any other point on the arc. The crease is the perpendicular bisector of the chord between the two matching points. The center is the intersection of the two creases. Center LESSON 6.3 • Arcs and Angles 180°, mMN 100° 1. mXNM 40°, mXN 3. a 90°, b 55°, c 35° 4. a 50°, b 60°, c 70° A T P LESSON 6.2 • Chord Properties 1. a 95°, b 85°, c 47.5° 2. v cannot be determined, w 90° ANSWERS 8. The perpendicular segment from the center of the circle bisects the chord, so the chord has length 12 units. But the diameter of the circle is 12 units, and the chord cannot be as long as the diameter because it doesn’t pass through the center of the circle. 2. x 120°, y 60°, z 120° 7. 104 ON because 7. Kite. Possible explanation: OM congruent chords AB and AC are the same distance AN because they are halves from the center. AM of congruent chords. So, AMON has two pairs of adjacent congruent sides and is a kite. 5. x 140° 72°, mC 36°, mCB 108° 6. mA 90°, mAB 140°, mD 30°, mAB 60°, 7. mAD mDAB 200° 8. p 128°, q 87°, r 58°, s 87° 9. a 50°, b 50°, c 80°, d 50°, e 130°, f 90°, g 50°, h 50°, j 90°, k 40°, m 80°, n 50° Discovering Geometry Practice Your Skills ©2008 Key Curriculum Press