CHAPTER 5 PROJECT. Quadrilaterals. (page 166) Project is worth 200 points. Your score is.

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1 HPTER 5 PROJET Quadrilaterals (page 166) Name DUE: Notes - 6 points per page -worth 90 points Quizzes - 2 points per problem - worth 60 points Homework -10 points per lesson - worth 50 points Tangram Picture onus - worth 0-20 points Project is worth 200 points. Your score is. (1) The test will be taken after the projects have all been returned and reviewed. (2) Staple all the homework assignments, in order, to the end of this project. (3) Points will be deducted for incorrect spelling. No abbreviations! (4) 20 points deducted for every day this late.

2 5-1: Properties of Parallelograms (page 167) PRLLELOGRM: a quadrilateral with both pairs of opposite sides. written: D Theorem 5-1 Opposite sides of a parallelogram are. Given:!EFGH H G Prove: EF! HG FG! EH E F Key Step Proof:!HGE!!, by, then EF! HG & FG! EH, by.

3 Theorem 5-2 Opposite angles of a parallelogram are. Given:!D Prove:! "!D!D "!D D Key Step Proof:!!! and!d!!, by, then! "!D and!d "!D, by. Theorem 5-3 Diagonals of a parallelogram each other. Given:!QRST T S Prove: QS & TR bisect each other M Q R Key Step Proof:!QMR!!, by and, then QM! MS and TM! MR, by.

4 examples: Find the value of x and y in each parallelogram. (1) 12 7 x x = y = y (2) xº yº x = 105º 75º y = NOTE: The 5th property of a parallelogram: onsecutive angles of a parallelogram are. (3) 2x+5 y+17 2y-5 4x-19 x = y = ssignment: Written Exercises, page 169 to 171: 5-11 LL # s and LL # s

5 5-2: Ways to Prove that Quadrilaterals are Parallelograms (page172) Theorem 5-4 If both pairs of sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Given: TS! QR T S TQ! SR Prove: Quad. QRST is a Key Step Proof:!TSQ!!, by Q Postulate and R then!1 "!2 and!3 "!4, by PT, and opposite sides are parallel, by, and QRST is a parallelogram, by. Theorem 5-5 If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is parallelogram. Given:! D! D Prove: Quad. D is a Key Step Proof:!!!, by Postulate and D then! D, by, and D is a parallelogram, by.

6 Theorem 5-6 If both pairs of angles of a quadrilateral are congruent, then the quadrilateral is parallelogram. Given: m = m = xº D m = m D = yº Prove: Quad. D is a xº yº Proof: Statements Reasons 1. m = m = xº 1. Given m = m D = yº 2. 2 x + 2 y = 360º 2. The sum of the measures of the angles of a quadrilateral is. 3. x + y = 180º 3. Property of Equality yº xº 4.! D and D! 4. SSI Supplementary 5. Quad. D is a 5. Definition of Theorem 5-7 If the diagonals of a quadrilateral each other, then the quadrilateral is parallelogram. Given: & D bisect each other. Prove: Quad. D is a Key Step Proof: D!!! and!d!!, by Postulate, and then D! &! D, by PT, and D is a parallelogram, by both pairs of opposite sides congruent.

7 5 Ways to Prove that a Quadrilateral is a Parallelogram: (1) Show that both pairs of sides are parallel. (2) Show that both pairs of sides are congruent. (3) Show that one pair of sides are both &. (4) Show that both pairs of angles are congruent. (5) Show that the bisect each other. examples: (1) Draw a convex quadrilateral that has two pairs of congruent sides but that is not a parallelogram. [See lassroom Exercises, page 173: #10] Draw a non-convex one for bonus! (2) Draw a quadrilateral that is not a parallelogram but that has one pair of congruent sides and one pair of parallel sides. [See lassroom Exercises, page 173: #11] Draw a second one for bonus! ssignment: Written Exercises, pages 174 & 175: 1-6 LL # s, 19 & 20

8 5-3: Theorems Involving Parallel Lines (page 177) Theorem 5-8 If two lines are parallel, then all points on one line are from the other line. Given: l m l & are any points on l! m ; D! m m Prove: = D Theorem 5-9 If three parallel lines cut off segments on one transversal, then they cut off segments on every transversal. Given: Prove:! Y! Z! Y! Y Z

9 Theorem 5-10 line that contains the of one side of a triangle and is to another side passes through the midpoint of the third side. Given: M is the midpoint of D MN M N Prove: N is the midpoint of. Theorem 5-11 The segment that joins the of two sides of a triangle: (1) is to the third side. (2) is as long as the third side. Given: M is the midpoint of N is the midpoint of M N Prove: (1) MN (2) MN =

10 examples: (1) Given: MN! ST! YZ and MS = SY N If NT = x + 6 and NZ = 3x - 8, find the value of x, NT, TZ, and NZ. T Z SHOW YOUR WORK! M S Y x = NT = TZ = NZ = (2) Name the points shown that must be midpoints of the sides of the large triangle. Midpoints are Y 6 6 Z (3) Given: & are the midpoints of RS & RT. If ST = 4x + 4 and = x + 40, find x, ST, &. R SHOW YOUR WORK! x = = S T ST = ssignment: Written Exercises, page 180: 1-13 odd # s Take Quiz on Lessons 5-1 to 5-3: Parallelograms

11 Quiz on Lessons 5-1 to 5-3: Parallelograms 2 points each 40 points total Quadrilateral KLMN is parallelogram. omplete each statement. 1. KN! 2.!NML " 3. M! 4.!1 " 5.!MLN! 6.!KNM is supplementary to K 7 N L 4 3 M If it is possible to prove that a quadrilateral is a parallelogram from the given information, then NME THE PRLLELOGRM. If it is not possible, then write NONE. 7. F! E ; F! E 8. FD! ; F! D 9. F! E ; D! E 10.!1 "!3 ;!2 "!4 11. FD! ; D! E 12. FG! G ; G! GE F 4 E 3 D G 1 2

12 omplete each statement with the NUMER that makes the statement true. If insufficient information is given to determine an answer, then write NP for NOT POSSILE. 13. DE = 80º 14. m = 15. m DF = 12 D 41º E 16. = F If PQ = 5, then W = 18. If YZ = 7, then Y = 19. If Q = 12, then PW = 20. If WZ = 21, then W = T P W Q R Y S Z

13 5-4: Special Quadrilaterals (page 184) RETNGLE: a quadrilateral with right angles. * Every rectangle is a, because both pairs of opposite angles are. Theorem 5-12 The diagonals of a rectangle are. D Theorem 5-16 If an angle of a parallelogram is a angle, then the parallelogram is a rectangle. W Z Y

14 RHOMUS: a quadrilateral with congruent sides. * Every rhombus is a, because both pairs of opposite angles are. Theorem 5-13 The diagonals of a rhombus are. D Theorem 5-14 Each diagonal of a rhombus two angles of the rhombus. D Theorem 5-17 If two sides of a parallelogram are congruent, then the parallelogram is a rhombus. D

15 SQURE: a quadrilateral with right angles and congruent sides. * Every square is a, because both pairs of opposite sides or opposite are. Summary of Special Parallelograms RETNGLE: has right angles. has all the properties of a. has diagonals that are. RHOMUS: has congruent sides. has all the properties of a. has diagonals that are. has diagonals bisect its. SQURE: has right angles and congruent sides. has all the properties of a. has diagonals that are both &. has diagonals bisect its. has all the properties of a &.

16 Theorem 5-15 The midpoint of the of a right triangle is equidistant from the three vertices. Given: Right is midpoint of Prove: = = * The name given to point is the, which is the intersection of the - of each side. examples: State which kind of quadrilateral each diagram represents. ircle correct response. (1) (2) rectangle - rhombus - square rectangle - rhombus - square (3) (4) rectangle - rhombus - square rectangle - rhombus - square ssignment: Written Exercises, pages 187 & 188: 1-10 (make & complete the chart), odd # s, 24, 25, 26, 27

17 5-5: Trapezoids (page 190) TRPEZOID : a quadrilateral with exactly one pair of sides. SES of a Trapezoid : the LEGS of a Trapezoid : the sides. sides. ISOSELES TRPEZOID : a trapezoid with legs. Theorem 5-18 ase angles of an isosceles trapezoid are. Given: Prove: Trapezoid Y with Y Y

18 MEDIN of a Trapezoid: is the segment that joins the of the legs. Draw the median of the trapezoid. Draw a median of the triangle. Theorem 5-19 The median of a trapezoid: (1) is to the bases. (2) has a length equal to the of the base lengths. Given: Trapezoid PQRS with median MN S R Prove: (1) MN PQ & MN SR M N (2) MN = (PQ + SR) P Q examples: Given a trapezoid and its median, find the value of x. SHOW YOUR WORK! (1) 6 (2) 7x - 2 x 5x 12 x + 12 x = x = ssignment: Written Exercises, pages 192 & 193: 1-9 odd # s, LL # s Take Quiz on Lessons 5-4 & 5-5: Special Quadrilaterals

19 Quiz on Lessons 5-4 & 5-5: Special Quadrilaterals 2 points each 40 points total Given quadrilateral HIJK is a rhombus. lassify each statement as must be true (MUST) or not necessarily true (NOT). 21.!HIJ "!JKH 22.!IHJ "!KHJ 23. IK! HJ H M I 24. IM! KM K J Given quadrilateral RSTV is a rectangle. lassify each statement as must be true (MUST) or not necessarily true (NOT). 25. RT! VS V T 26. RT! VS 27. TR bisects!vts 28. R! T R S Fill in the blank with the correct numerical answer. 29. In!, M = m!m = 8 M 53º 6

20 omplete each statement given MN is the median of trapezoid D. 31. If DN = 5, then D = 32. If m D = 100º, then m DNM = 33. If D = 12 and = 20, then MN = 34. If MN = 8 and = 9, then D = M N 35. If trapezoid D is isosceles & m D = 100º, then m = D 36. If trapezoid D is isosceles & DN = x, then = Using the given information, tell whether parallelogram QRST is EST described as a rectangle, rhombus, or square. 37. QS! RT 38. QR! RS Q T 8 7 R 39. QS! RT ; QS " RT 6 5 S 40.!1 "!2 ;!3 "!4

21 TNGRM ONUS (possible 20 ONUS points) Directions: ut out the seven (7) figures of the tangrams to create a unique picture. Please include color and be sure all seven (7) figures are able to be identified The picture is to be placed on an 8.5 by 11 sheet of paper. SUMIT THIS SEPRTELY FROM THE PROJET!