age of 8 6.2 roperties of arallelograms What you should learn GOL Use some properties of parallelograms. GOL 2 Use properties of parallelograms in real-life situations, such as the drafting table shown in Example 6. Why you should learn it You can use properties of parallelograms to understand how a scissors lift works in Exs. 5 54. EL LIE GOL OETIE O LLELOG In this lesson and in the rest of the chapter you will study special uadrilaterals. parallelogram is a uadrilateral with both pairs of opposite sides parallel. When you mark diagrams of uadrilaterals, use matching arrowheads to indicate which sides are parallel. or example, in the diagram at the right, Q and Q. The symbol Q is read parallelogram Q. THEOE OUT LLELOG THEOE 6.2 If a uadrilateral is a parallelogram, then its opposite sides are congruent. Q and Q THEOE 6. If a uadrilateral is a parallelogram, then its opposite angles are congruent. and Q THEOE 6.4 If a uadrilateral is a parallelogram, then its consecutive angles are supplementary. m + m Q = 80, m Q + m = 80, m + m = 80, m + m = 80 THEOE 6.5 If a uadrilateral is a parallelogram, then its diagonals bisect each other. Q and 0 hapter 6 Quadrilaterals Theorem 6.2 is proved in Example 5. You are asked to prove Theorem 6., Theorem 6.4, and Theorem 6.5 in Exercises 8 44.THEOE OUT LLELOG
age 2 of 8 EXLE Using roperties of arallelograms GH is a parallelogram. ind the unknown length. Explain your reasoning. 5 G a. H b. H OLUTION a. H = G Opposite sides of a are. H = 5 ubstitute 5 for G. b. = G iagonals of a bisect each other. = ubstitute for G. EXLE 2 Using roperties of arallelograms Q is a parallelogram. ind the angle measure. a. m b. m Q 70 OLUTION a. m = m Opposite angles of a are. m = 70 ubstitute 70 for m. b. m Q + m = 80 onsecutive of a are supplementary. m Q + 70 = 80 ubstitute 70 for m. m Q = 0 ubtract 70 from each side. EXLE Using lgebra with arallelograms Q is a parallelogram. ind the value of x. x 20 OLUTION m + m = 80 x + 20 = 80 x = 60 onsecutive angles of a are supplementary. ubstitute x for m and 20 for m. ubtract 20 from each side. x = 20 ivide each side by. 6.2 roperties of arallelograms
age of 8 GOL 2 EONING OUT LLELOG EXLE 4 roving acts about arallelograms GIVEN and EG are parallelograms. OVE lan how that both angles are congruent to 2. Then use the Transitive roperty of ongruence. E 2 G OLUTION ethod Write a two-column proof. tatements easons. is a. EG is a.. Given 2. 2, 2 2. Opposite angles of a are... Transitive roperty of ongruence ethod 2 Write a paragraph proof. is a parallelogram, so 2 because opposite angles of a parallelogram are congruent. EG is a parallelogram, so 2. y the Transitive roperty of ongruence,. EXLE 5 roving Theorem 6.2 GIVEN is a parallelogram. OVE, OLUTION tatements. is a. 2. raw.., 4., 5. 6. 7., easons. Given 2. Through any two points there exists exactly one line.. efinition of parallelogram 4. lternate Interior ngles Theorem 5. eflexive roperty of ongruence 6. ongruence ostulate 7. orresponding parts of are. 2 hapter 6 Quadrilaterals
age 4 of 8 OU ON EE EXLE 6 Using arallelograms in eal Life UNITUE EIGN drafting table is made so that the legs can be joined in different ways to change the slope of the drawing surface. In the arrangement below, the legs and do not bisect each other. Is a parallelogram? UNITUE EIGN urniture designers use geometry, trigonometry, and other skills to create designs for furniture. EE LIN www.mcdougallittell.com EL INTENET LIE OLUTION No. If were a parallelogram, then by Theorem 6.5 would bisect and would bisect. GUIE TIE Vocabulary heck oncept heck. Write a definition of parallelogram. ecide whether the figure is a parallelogram. If it is not, explain why not. 2.. kill heck IENTIYING ONGUENT T Use the diagram of parallelogram L at the right. omplete the statement, and give a reason for your answer. 4.? 5. N? 6. L? 7. L? 8. N? 9. L? 0. NL?. L? ind the measure in parallelogram LNQ. Explain your reasoning. 2. L. L L 8.2 4. LQ 5. Q 00 7 6. m LN 7. m NQL 29 8. m NQ 9. m LQ N L N 8 6.2 roperties of arallelograms
age 5 of 8 TIE N LITION TUENT HEL Extra ractice to help you master skills is on p. 8. INING EUE ind the measure in parallelogram. Explain your reasoning. 20. E 2. 22. 2. m 20 0 E 2 24. m 25. m xy UING LGE ind the value of each variable in the parallelogram. 26. 4 27. a b 28. y x 0 0 6.5 s r 29. 6 0. 70. p 5 2m n m k 4 8 xy UING LGE ind the value of each variable in the parallelogram. 2. 9. 6 4. 8 y 2x 4 2u 2 5u 0 v 4w 2z 4z 5 w 5. d c 6. 2f 5 7. f 2 g (b 0) (b 0) 5f 7 4r (t 5) (2t 0) s TUENT HEL HOEWO HEL Example : Exs. 20 22 Example 2: Exs. 2 25 Example : Exs. 26 7 Example 4: Exs. 55 58 Example 5: Exs. 8 44 Example 6: Exs. 45 54 8. OVING THEOE 6. opy and complete the proof of Theorem 6.: If a uadrilateral is a parallelogram, then its opposite angles are congruent. GIVEN is a. OVE, aragraph roof Opposite sides of a parallelogram are congruent, so a. and b.. y the eflexive roperty of ongruence, c.. because of the d. ongruence ostulate. ecause e. parts of congruent triangles are congruent,. To prove that, draw f. and use the same reasoning. 4 hapter 6 Quadrilaterals
age 6 of 8 9. OVING THEOE 6.4 opy and complete the two-column proof of Theorem 6.4: If a uadrilateral is a parallelogram, then its consecutive angles are supplementary. GIVEN L is a. OVE and are supplementary. L tatements.? 2. m = m L, m = m. m + m L + m + m =? 4. m + m + m + m = 60 5. 2(? +? ) = 60 6. m + m = 80 7. and are supplementary. easons. Given 2.?. um of measures of int. of a uad. is 60. 4.? 5. istributive property 6.? prop. of euality 7.? You can use the same reasoning to prove any other pair of consecutive angles in L are supplementary. EVELOING OOINTE OO opy and complete the coordinate proof of Theorem 6.5. GIVEN O is a. OVE and O bisect each other. y (a, b) (?,?) lan for roof ind the coordinates of the midpoints of the diagonals of O and show that they are the same. TUENT HEL HOEWO HEL Visit our Web site www.mcdougallittell.com for help with the coordinate proof in Exs. 40 44. INTENET 40. oint is on the x-axis, and the length of O is c units. What are the coordinates of point? 4. The length of is also c units, and is horizontal. What are the coordinates of point? 42. What are the coordinates of the midpoint of? O(0, 0) (c,?) x 4. What are the coordinates of the midpoint of O? 44. Writing How do you know that and O bisect each other? ING In Exercises 45 and 46, use the following information. In a recipe for baklava, the pastry should be cut into triangles that form congruent parallelograms, as shown. Write a paragraph proof to prove the statement. 45. is supplementary to 6. 46. 4 is supplementary to 5. 6.2 roperties of arallelograms 5
age 7 of 8 TI LUTE In Exercises 47 50, use the following information. In the diagram at the right, the slope of the handrail is eual to the slope of the stairs. The balusters (vertical posts) support the handrail. 2 6 47. Which angle in the red parallelogram is congruent to? 48. Which angles in the blue parallelogram are supplementary to 6? 49. Which postulate can be used to prove that 5? 50. Writing Is the red parallelogram congruent to the blue parallelogram? Explain your reasoning. IO LIT hotographers can use scissors lifts for overhead shots, as shown at the left. The crossing beams of the lift form parallelograms that move together to raise and lower the platform. In Exercises 5 54, use the diagram of parallelogram at the right. 5. What is m when m = 20? 52. uppose you decrease m. What happens to m? 5. uppose you decrease m. What happens to? 54. uppose you decrease m. What happens to the overall height of the scissors lift? TWO-OLUN OO Write a two-column proof. 55. GIVEN and E are s. 56. GIVEN Q and TUV are s. OVE E OVE 5 7 4 8 E U T 2 V 57. GIVEN WXYZ is a. 58. GIVEN, EG, H are s. OVE WZ YX OVE 2 W Z X Y H 4 E 2 G 6 hapter 6 Quadrilaterals
age 8 of 8 59. Writing In the diagram, G, EG, and GE are parallelograms. opy the diagram and add as many other angle measures as you can. Then describe how you know the angle measures you added are correct. 45 G 20 E Test reparation 60. ULTILE HOIE In LN, what is the value of s? 5 20 40 52 E 70 L (2s 0) (s 50) N hallenge EXT HLLENGE www.mcdougallittell.com 6. ULTILE HOIE In, point E is the intersection of the diagonals. Which of the following is not necessarily true? = = E = E = E E = E xy UING LGE uppose points (, 2), (, 6), and (6, 4) are three vertices of a parallelogram. y 62. Give the coordinates of a point that could be the fourth vertex. ketch the parallelogram in a coordinate plane. 6. Explain how to check to make sure the figure you drew in Exercise 62 is a parallelogram. 64. How many different parallelograms can be formed using,, and as vertices? ketch each parallelogram and label the coordinates of the fourth vertex. x IXE EVIEW xy UING LGE Use the istance ormula to find. (eview. for 6.) 65. (2, ), (6, 9) 66. (º4, 2), (2, º) 67. (º8, º4), (º, º) xy UING LGE ind the slope of. (eview.6 for 6.) 68. (2, ), (6, 9) 69. (º4, 2), (2, º) 70. (º8, º4), (º, º) 7. ING In a parking lot, two guidelines are painted so that they are both perpendicular to the line along the curb. re the guidelines parallel? Explain why or why not. (eview.5) Name the shortest and longest sides of the triangle. Explain. (eview 5.5) 72. 7. E 74. H 45 65 5 55 G 60 6.2 roperties of arallelograms 7