UNIT 5 LEARNING TARGETS. HP/4 Highly Proficient WOW, Excellent. PR/3 Proficient Yes, Satisfactory. DP/1 Developing Proficiency Not yet, Insufficient

Transcription

1 Geometry 1-2 Properties of Polygons My academic goal for this unit is... UNIT 5 Name: Teacher: Per: heck for Understanding Key: Understanding at start of the unit Understanding after practice Understanding before unit test LERNING TRGETS How is my understanding? Test Score Retake? 5a 5b I can apply the polygon sum conjecture to determine unknown angle measures I can apply the exterior angle sum conjecture to determine unknown angles measures c I can apply the properties of kites and trapezoids to determine unknown angle measures and segment lengths d I can apply the properties of midsegments to determine unknown angle measures and segment lengths e I can apply the properties of parallelograms to determine unknown angle measures and segment lengths f I can prove the properties of quadrilaterals What is an interior angle? What is an exterior angle? P/1 eveloping Proficiency Not yet, Insufficient P/2 lose to Proficient Yes, but..., Minimal PR/3 Proficient Yes, Satisfactory HP/4 Highly Proficient WOW, Excellent I can t do it and am not able to explain process or key points I can sort of do it and am able to show process, but not able to identify/explain key math points I can do it and able to both explain process and identify/explain math points I m great at doing it and am able to explain key math points accurately in a variety of problems Unit 5 Geometry 1-2 Polygon Properties 1

2 Unit 5 onjectures Title onjecture iagram Quadrilateral Sum onjecture The sum of the measures of the four angles in any quadrilateral is Pentagon Sum onjecture The sum of the measures of the five angles of any pentagon is Polygon Sum onjecture The sum of the measures of the n interior angles of an n-gon is Exterior ngle Sum onjecture Equiangular Polygon onjecture For any polygon, the sum of the measures of a set of exterior angles is You can find the measure of each interior angle of an equiangular n-gon by using either of these formulas... Kite ngles onjecture The angles of a kite are. Kite iagonals onjecture Kite iagonal isector onjecture The diagonals of a kite are The diagonals connecting the vertex angles of a kite is the of the other diagonal. 2 Polygon Properties Geometry 1-2 Unit 5

3 Unit 5 onjectures Title onjecture iagram Kite ngle isector onjecture Trapezoid onsecutive ngles onjecture Isosceles Trapezoid onjecture Isosceles Trapezoid iagonals onjecture Three Midsegments onjecture Triangle Midsegment onjecture Trapezoid Midsegment onjecture Parallelogram Opposite ngles onjecture The angle of a kite are by a. The consecutive angles between the bases of a trapezoid are The base angles of an isosceles trapezoid are The diagonals of an isosceles trapezoid are The three midsegments of a triangle divide it into midsegment of a triangle is to the third side and the length of. The midsegment of a trapezoid is to the bases and is equal to The opposite angles of a parallelogram are Unit 5 Geometry 1-2 Polygon Properties 3

4 Unit 5 onjectures Title onjecture iagram Parallelogram onsecutive ngles onjecture Parallelogram Opposite Sides onjecture Parallelogram iagonals onjecture ouble-edged Straightedge onjecture Rhombus iagonals onjecture The consecutive angles of a parallelogram are The opposite side of a parallelogram are The diagonals of a parallelogram If two parallel lines are intersected by a second pair of parallel lines the same distance apart as the first pair, then the parallelogram formed is a The diagonals of a rhombus are and they Rhombus ngles onjecture The of a rhombus the angles of a rhombus. Rectangle iagonals onjecture Square iagonals onjecture The diagonals of a rectangle are and The diagonals of a square are,, and they 4 Polygon Properties Geometry 1-2 Unit 5

5 Notes Unit 5 Geometry 1-2 Polygon Properties 5

6 Notes 6 Polygon Properties Geometry 1-2 Unit 5

7 Geometry 1-2 Name: j a2b0w1l5k bklu]tlag MS`ovf\t_wRaWrJed HL]Lqy.v \ lil^ll Vrkiag]hltmsf srsenseeirgvpesdr. Practice: Polygon Sum onjecture ate: Period: Find the interior angle sum for each polygon. Round your answer to the nearest tenth if necessary. 1) regular 25-gon 2) regular dodecagon 3) regular 21-gon 4) regular 14-gon 5) 6) 7) 8) Find the measure of one interior angle in each regular polygon. Round your answer to the nearest tenth if necessary. 9) regular 16-gon 10) regular octagon 11) regular 14-gon 12) regular 22-gon Unit 5 Geometry 1-2 Polygon Properties 7

8 13) 14) 15) 16) Find the measure of one exterior angle in each regular polygon. Round your answer to the nearest tenth if necessary. 17) regular 15-gon 18) regular pentagon 19) regular 14-gon 20) regular dodecagon 21) 22) 23) 24) 8 Polygon Properties Geometry 1-2 Unit 5

9 Lesson 5.1 Polygon Sum onjecture Name Period ate In Exercises 1 and 2, find each lettered angle measure. 1. a, b, c, 2. a, b, c, d, e d, e, f 26 a b e c d 97 b 85 d f e 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? c 44 a 4. The sum of the measures of the interior angles of a regular polygon is How many sides does the polygon have? 5. is a square. E is an equilateral 6. E is a regular pentagon. FG triangle. is a square. x x E x E G F x 7. Use a protractor to draw pentagon E with m 85, m 125, m 110, and m 70. What is m E? Measure it, and check your work by calculating. Unit 5 Geometry 1-2 Polygon Properties 9

10 Lesson 5.2 Exterior ngles of a Polygon Name Period ate 1. How many sides does a regular polygon have if each exterior angle measures 30? 2. How many sides does a polygon have if the sum of the measures of the interior angles is 3960? 3. 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? 4. If the sum of the measures of the interior 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 6. a, b 7. a, b, a 116 c b 82 a 2x x b a 82 b c 3x Find each lettered angle measure. a b d 150 c c d a 95 b 9. onstruct an equiangular quadrilateral that is not regular. 10 Polygon Properties Geometry 1-2 Unit 5

11 Lesson 5.3 Kite and Trapezoid Properties Name Period ate In Exercises 1 4, find each lettered measure. 1. Perimeter 116. x 2. x, y x y 28 x x, y 4. x, y x x y y 5. Perimeter PQRS 220. PS 6. b 2a 1. a S 4x 1 R M 2x 3 N a 34 b L P T 4 Q 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. 7. onstruct an isosceles trapezoid given base,, and distance between bases XY. X Y 8. onstruct kite with,, and. 9. Write a paragraph or flowchart proof of the onverse of the Isosceles Trapezoid onjecture. Hint: raw E parallel to TP with E on TR. P : Trapezoid TRP with T R Show: TP R T R Unit 5 Geometry 1-2 Polygon Properties 11

12 Lesson 5.4 Properties of Midsegments Name Period ate In Exercises 1 3, each figure shows a midsegment. 1. a, b, 2. x, y, 3. x, y, c z z b 54 a c x 16 y z z 29 x y 4. X, Y, and Z are midpoints. Perimeter PQR 132, RQ 55, and PZ 20. Perimeter XYZ PQ ZX R Z Y P X Q 5. MN is the midsegment. Find the coordinates of M and N. Find the slopes of and MN. 6. Explain how to find the width of the lake from to using a tape measure, but without using a boat or getting your feet wet. y M (20, 10) Lake (4, 2) N x (9, 6) 7. M, N, and O are midpoints. What type of quadrilateral is MNO? How do you know? Give a flowchart proof showing that ON MN. O M N 8. Give a paragraph or flowchart proof. : Show: PQR with P F FH HR and QE EG GI IR HI FG E PQ H F R I G E P Q 12 Polygon Properties Geometry 1-2 Unit 5

13 Geometry 1-2 Name: p `2i0n1m7W ykvustnav aszodf]t_wmafroe] nlal^._ d xglnlw Erdi^gLhGtKsd grjeqseeyrlvlekdh. Parallelograms Find the measurement indicated in each parallelogram. ate: Period: 1) X? W 2) S? T Y 60 V 105 R U 3) X 122 U 4) F 85 W? V E? 5) T ? U 6) Q T 94? R S S V 7) 95? 41 8) S? 63 T R 58 Q Unit 5 Geometry 1-2 Polygon Properties 13

14 9) V W 10) U 11 V? 12 T? W Y X 11) T 8 U 12) J K S? V M? L ) XZ = 30 Find PZ W X 14) UK = 16 Find UW T U K P W V Z Y 15) RE = 7.4 Find RT T Q 16) F = 38 Find N G E N S R E F 14 Polygon Properties Geometry 1-2 Unit 5

15 Lesson 5.5 Properties of Parallelograms Name Period ate In Exercises 1 7, is a parallelogram. 1. Perimeter 2. O 11, and O Perimeter cm,, 15 cm 2x 7 O x 9 4. a, b, 5. Perimeter 119, and 6. a, b, c 24. c 19 b c 110 a 62 b c 26 a 7. Perimeter 16x all is struck at the same instant by two forces, F 1 and F 2. Show the resultant force on the ball. 4x 3 63 F 2 F 1 9. Find each lettered angle measure. a b c d e f g h i j k g a 26 f G 81 h e d i E 38 c b 38 j F k 10. onstruct a parallelogram with diagonals and. Is your parallelogram unique? If not, construct a different (noncongruent) parallelogram. Unit 5 Geometry 1-2 Polygon Properties 15

16 Lesson 5.6 Properties of Special Parallelograms Name Period ate 1. PQRS is a rectangle and 2. KLMN is a square and 3. is a rhombus, OS 16. NM 8. 11, and O 6. OQ m OKL O m QRS m MOL PR Perimeter KLMN m O S O R N O M O P Q K 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. ll quadrilaterals 4. iagonals bisect each other. 6. iagonals are congruent. 8. Opposite sides are congruent. 10. oth diagonals bisect angles. 5. iagonals are perpendicular. 7. Measures of interior angles sum to Opposite angles are congruent. 11. iagonals are perpendicular bisectors of each other. In Exercises 12 and 13, graph the points and determine whether is a trapezoid, parallelogram, rectangle, or none of these. 12. ( 4, 1), (0, 3), (4, 0), ( 1, 5) 13. (0, 3), ( 1, 2), ( 3, 4), ( 2, 1) onstruct rectangle with diagonal and. 16 Polygon Properties Geometry 1-2 Unit 5

17 Lesson 5.7 Proving Quadrilateral Properties Name Period ate Write or complete each flowchart proof. 1. : is a parallelogram and P Q Q R Show: and PQ bisect each other Flowchart Proof P PR QR R PR and QP bisect each other efinition of bisect I onjecture 2. : art with and Show: 3. Show that the diagonals of a rhombus divide the rhombus into four congruent triangles. : Rhombus O Show: O O O O 4. : Parallelogram, Y, X Show: X Y X Y Unit 5 Geometry 1-2 Polygon Properties 17

18 Unit 5 hallenge Problems 1. (Target 5a) On her hapter 5 test, Ms. onovan asked her students to find the measure of each interior angle of a regular 15-gon. Here are a few of the answers students gave: a. Tell which of the answers is correct and explain why it is correct. b. hoose two of the incorrect answers and explain the error the student may have made in finding the answer. 2. (Target 5a & 5b) In Lesson 5.1, you found formulas for the sum of the interior angle measures of a regular polygon and for the measure of each interior angle. Now you will consider the outside angles of a regular polygon. In the regular hexagon below, one outside angle is marked. The measure of this angle is the number of degrees in the rotation indicated by the arrow. Find a formula for the sum of the measures of the outside angles of a regular n-gon and a formula for the measure of one outside angle. Show and explain all your work and simplify the formulas as much as possible. 18 Polygon Properties Geometry 1-2 Unit 5

19 Unit 5 hallenge Problems 3. (Target 5a, 5b and 5c) classical semicircular arch is really half of a regular polygon built with blocks whose faces are congruent isosceles trapezoids. For example, the inner arch in the diagram below is half of a regular 18-gon. Keystone Voussoir butment Rise Span The angle measures of the isosceles trapezoids forming the arch depend on how many blocks are used to build the arch. a. If a semicircular arch has only three blocks, what are the angle measures of the isosceles trapezoids? If an arch has five blocks, what are the angle measures? Show all your work. b. the number of blocks, b, in the arch, find formulas for the measures of the trapezoid base angles. You should find two formulas one for the outer base angles (those whose vertices are on the outside of the arch) and one for the inner base angles. Explain the reasoning you used to find the formulas. 4. (Target 5c, 5d and 5e) In the game Find the Oddball, a player looks at four objects and determines which one doesn t belong with the others. For example, consider the four objects below..... You might say that object doesn t belong because it has four sides, while the other shapes each have three. Or, you might say that object doesn t belong because it is the only shape with a right angle. You may be able to find and explain other oddballs. reate three different groups of four quadrilaterals, each with at least one shape that can be considered an oddball. For each group you create, identify every possible oddball and explain why it doesn t belong with the other objects. Unit 5 Geometry 1-2 Polygon Properties 19

20 Unit 5 hallenge Problems 5. (Targets 5c, 5e & 5f) onsider the following points: (1, 4), (2, 12), (9, 8). a. Graph the points. dd a fourth point so that points,,, and are the vertices of a particular type of quadrilateral. Name the quadrilateral and use algebra to verify one of the properties of that type of quadrilateral. Show all your work. b. Find the coordinates of the point where the diagonals of quadrilateral intersect. Show all your work. 6. (Target 5e) The modern art section of the Museum of Geometric rt is a large rectangular room. The museum directors want to build a wall in the center of the room to create more room for displaying art. The wall will be built so that it is parallel to two of the opposite sides and its ends are equally distant from the other two sides. Once the center wall is in place, a path will be painted on the floor around it. The path will be created by connecting the midsegments of the triangles and trapezoids formed by connecting the ends of the center wall to the corners of the room. (See the diagram.) a. If the room measures 80 ft by 100 ft and the wall is 70 ft long, how long will the path be? oes your answer depend on which sides the wall is parallel to? Explain. Path enter wall b. Now generalize your results. If the room measures a feet by b feet and the center wall is x feet long, how long will the path around the wall be? 20 Polygon Properties Geometry 1-2 Unit 5

21 7. (See flowchart proof at bottom of page 102.) 8. Flowchart Proof is a median efinition of median SSS onjecture PT bisects efinition of bisect Same segment ; 36 sides sides 5. x x m E 150 LESSON 5.2 Exterior ngles of a Polygon sides sides 3. 4 sides 4. 6 sides 5. a 64, b a 102, b 9 7. a 156, b 132, c a 135, b 40, c 105, d E LESSON 5.1 Polygon Sum onjecture 1. a 103, b 103, c 97, d 83, e a 92, b 44, c 51, d 85, e 44, f 136 Lesson 4.7, Exercises 1, 2, 3 1. PQ SR PQ SR PQS RSQ PQS RSQ SP QR I onjecture SS onjecture PT QS QS Same segment 2. KE KI ETK ITK KITE is a kite TE TI efinition of kite KET KIT SSS onjecture PT EKT IKT PT KT bisects EKI and ETI efinition of bisect KT KT Same segment 3. I onjecture is a parallelogram efinition of parallelogram Same segment S onjecture PT efinition of parallelogram I onjecture Unit 5 Geometry 1-2 Polygon Properties 21

22 LESSON 5.3 Kite and Trapezoid Properties 1. x x 124, y x 64, y x 12, y PS a M N P 7. MNO is a parallelogram. y the Triangle Midsegment onjecture, ON M and MN O. Flowchart Proof 8. O 1_ 2 efinition of midpoint MN 1_ 2 Midsegment onjecture 9. Possible answer: Paragraph proof: raw E PT with E on TR. TEP is a parallelogram. T ER because they are corresponding angles of parallel lines. T R because it is given, so ER R, because both are congruent to T. Therefore, ER is isosceles by the onverse of the Isosceles Triangle onjecture. TP E because they are opposite sides of a parallelogram and R E because ER is isosceles. Therefore, TP R because both are congruent to E. LESSON 5.4 Properties of Midsegments 1. a 89, b 54, c x 21, y 7, z x 17, y 11, z Perimeter XYZ 66, PQ 37, ZX M(12, 6), N(14.5, 2); slope 1.6, slope MN Pick a point P from which and can be viewed over land. Measure P and P and find the midpoints M and N. 2MN. M 1_ 2 efinition of midpoint ON onjecture O MN oth congruent to 1_ 2 ON NM oth congruent to ON MN SS onjecture ON 1_ 2 Midsegment onjecture ON M oth congruent to 1_ 2 NM onjecture 8. Paragraph proof: Looking at FGR, HI FG by the Triangle Midsegment onjecture. Looking at PQR, FG PQ for the same reason. ecause FG PQ, quadrilateral FGQP is a trapezoid and E is the midsegment, so it is parallel to FG and PQ. Therefore, HI FG E PQ. Lesson 4.8, Exercise 7 Same segment is an altitude and are right angles efinition of altitude oth are right angles S onjecture PT is a median efinition of median onverse of IT 22 Polygon Properties Geometry 1-2 Unit 5

23 LESSON 5.5 Properties of Parallelograms 1. Perimeter 82 cm 2. 22, , 7 4. a 51, b 48, c a 41, b 86, c a 38, b 142, c 142, d 38, e 142, f 38, g 52, h 12, i 61, j 81, k No F 2 F 1 Resultant vector F 1 F 2 4. c, d, f, g 5. d, e, g 6. f, g 7. h 8. c, d, f, g 9. c, d, f, g 10. d, g 11. d, g 12. None 13. Parallelogram 14. LESSON 5.7 Proving Quadrilateral Properties 1. (See flowchart proof at bottom of page.) 2. Flowchart Proof SSS onjecture PT Same segment 3. Flowchart Proof LESSON 5.6 Properties of Special Parallelograms 1. OQ 16, m QRS 90, PR m OKL 45, m MOL 90, perimeter KLMN O 6, 11, m O 90 efinition of rhombus O O iagonals of parallelogram bisect each other O O O O SSS onjecture O O iagonals of a parallelogram bisect each other Lesson 5.7, Exercise 1 PR QR I onjecture R R P Q PR QR S onjecture PT PR QR and QP bisect each other efinition of bisect PR QR PT I onjecture Unit 5 Geometry 1-2 Polygon Properties 23

24 nswers to Practice: Polygon Sum onjecture 1) ) ) ) ) 900 6) ) ) ) ) ) ) ) ) ) ) ) 24 18) 72 19) ) 30 21) 40 22) 36 23) 45 24) 72 nswers to Parallelograms 1) 120 2) 105 3) 58 4) 95 5) 45 6) 94 7) 44 8) 59 9) 12 10) 11 11) 8 12) ) 15 14) 32 15) ) Polygon Properties Geometry 1-2 Unit 5