8.2 Investigate Parallelograms

Transcription

1 Investigating g Geometry TIVITY 8.2 Investigate arallelograms T E I graphing calculator or computer Use before esson 8.2 classzone.com Keystrokes Q U E T I O N What are some of the properties of a parallelogram? You can use geometry drawing software to investigate relationships in special quadrilaterals. E X O E raw a quadrilateral TE 1 raw parallel lines onstruct ] and a line parallel to ] through point. Then construct ] and a line parallel to ] through point. Finally, construct a point at the intersection of the line drawn parallel to ] and the line drawn parallel to ]. TE 2 raw quadrilateral onstruct segments to form the sides of quadrilateral. fter you construct }, }, }, and }, hide the parallel lines that you drew in tep 1. TE 3 easure side lengths easure the side lengths,,, and. rag point or point to change the side lengths of. What do you notice about the side lengths? TE 4 easure angles Find the measures of,,, and. rag point or point to change the angle measures of. What do you notice about the angle measures? TE TE 2 N W O N U I O N Use your observations to complete these eercises 1. The quadrilateral you drew in the Eplore is called a parallelogram. Why do you think this type of quadrilateral has this name? 2. ased on your observations, make a conjecture about the side lengths of a parallelogram and a conjecture about the angle measures of a parallelogram. 3. EONING raw a parallelogram and its diagonals. easure the distance from the intersection of the diagonals to each verte of the parallelogram. ake and test a conjecture about the diagonals of a parallelogram. 514 hapter 8 Quadrilaterals

2 8.2 Use roperties of arallelograms efore You used a property of polygons to find angle measures. Now You will find angle and side measures in parallelograms. Why? o you can solve a problem about airplanes, as in E. 38. Key Vocabulary parallelogram parallelogram is a quadrilateral with both pairs of opposite sides parallel. The term parallelogram Q can be written as ~Q. In ~Q, } Q i } and }Q i } by definition. The theorems below describe other properties of parallelograms. THEOE For Your Notebook THEOE 8.3 If a quadrilateral is a parallelogram, then its opposite sides are congruent. If Q is a parallelogram, then } Q > } and }Q > }. roof: p. 516 THEOE 8.4 If a quadrilateral is a parallelogram, then its opposite angles are congruent. If Q is a parallelogram, then > and Q >. roof: E. 42, p. 520 E X E 1 Use properties of parallelograms GE Find the values of and y. is a parallelogram by the definition of a parallelogram. Use Theorem 8.3 to find the value of. 5 Opposite sides of a ~ are > ubstitute 1 4 for and 12 for. ubtract 4 from each side. y Theorem 8.4, >, or m 5 m. o, c In ~, 5 8 and y Use roperties of arallelograms 515

3 O O F Theorem 8.3 If a quadrilateral is a parallelogram, then its opposite sides are congruent. GIVEN c Q is a parallelogram. OVE c } Q > }, } Q > } lan for roof a. raw diagonal } Q to form n Q and nq. b. Use the ongruence ostulate to show that nq > nq. c. Use congruent triangles to show that } Q > } and } Q > }. lan in ction TTEENT a. 1. Q is a ~. 2. raw } Q. 3. } Q i }, } Q i } b. 4. Q > Q, Q > Q 5. } Q > } Q 6. nq > nq c. 7. } Q > }, } Q > } EON 1. Given 2. Through any 2 points there eists eactly 1 line. 3. efinition of parallelogram 4. lternate Interior ngles Theorem 5. efleive roperty of ongruence 6. ongruence ostulate 7. orresp. parts of > ns are >. GUIE TIE for Eample 1 1. Find FG and m G. 2. Find the values of and y. G F H E J K 508 y 1 3 INTEIO NGE The onsecutive Interior ngles Theorem (page 155) states that if two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary. pair of consecutive angles in a parallelogram are like a pair of consecutive interior angles between parallel lines. This similarity suggests Theorem 8.5. THEOE For Your Notebook THEOE 8.5 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. 8 If Q is a parallelogram, then roof: E. 43, p hapter 8 Quadrilaterals

4 E X E 2 Use properties of a parallelogram EK s shown, part of the etending arm of a desk lamp is a parallelogram. The angles of the parallelogram change as the lamp is raised and lowered. Find m when m olution y Theorem 8.5, the consecutive angle pairs in ~ are supplementary. o, m 1 m ecause m , m THEOE For Your Notebook THEOE 8.6 If a quadrilateral is a parallelogram, then its diagonals bisect each other. roof: E. 44, p. 521 }Q > } and } > } E X E 3 tandardized Test ractice The diagonals of ~NO intersect at point. What are the coordinates of? y 1 7 } 2, , 7 } } 2, , 5 } O 1 N IIFY UTION In Eample 3, you can use either diagonal to find the coordinates of. Using } O simplifies calculations because one endpoint is (0, 0). olution y Theorem 8.6, the diagonals of a parallelogram bisect each other. o, is the midpoint of diagonals } N and } O. Use the idpoint Formula. oordinates of midpoint of } O } 2, } } 2, 22 c The correct answer is. GUIE TIE for Eamples 2 and 3 Find the indicated measure in ~JK. 3. N 4. K 5. m J 6. m K K J N Use roperties of arallelograms 517

5 8.2 EXEIE KI TIE HOEWOK KEY 5 WOKE-OUT OUTION on p. W1 for Es. 9, 13, and 39 5 TNIZE TET TIE Es. 2, 16, 29, 35, and VOUY What property of a parallelogram is included in the definition of a parallelogram? What properties are described by the theorems in this lesson? 2. WITING In parallelogram, m Eplain how you would find the other angle measures of ~. EXE 1 on p. 515 for Es GE Find the value of each variable in the parallelogram. y 4. m a8 9 n p (d 2 21)8 20 z (g 1 4) h EXE 2 on p. 517 for Es FINING NGE EUE Find the measure of the indicated angle in the parallelogram. 9. Find m. 10. Find m. 11. Find m Y. 958 N W 1198 X 518 Z Y 12. KETHING In ~Q, m is 24 degrees more than m. ketch ~Q. Find the measure of each interior angle. Then label each angle with its measure. EXE 3 on p. 517 for Es GE Find the value of each variable in the parallelogram n 5y 12 b 1 5a 4m n 4y UTIE HOIE The diagonals of parallelogram OQ intersect at point. What are the coordinates of point? 6 y Q 1 1, 5 } , 5 } , 3 } , 3 } 2 2 O hapter 8 Quadrilaterals

6 EONING Use the photo to copy and complete the statement. Eplain. 17. } >? 18. >? 19. >? 20. m 5? 21. m 5? 22. m 5? UING IG Find the indicated measure in ~EFGH. Eplain. 23. m EJF 24. m EGF E m HFG 26. m GEF J m HGF 28. m EHG 458 H at classzone.com G F 29. UTIE HOIE In parallelogram, 5 14 inches and 5 20 inches. What is the perimeter (in inches) of ~? GE The measure of one interior angle of a parallelogram is 0.25 times the measure of another angle. Find the measure of each angle. 31. GE The measure of one interior angle of a parallelogram is 50 degrees more than 4 times the measure of another angle. Find the measure of each angle. 32. EO NYI In ~, m student says that m Eplain why this statement is incorrect. 33. UING IG In the diagram, QT and TUV are parallelograms. Find the values of and y. Eplain your reasoning. 20 U V T FINING EIETE The sides of ~NQ are represented by the epressions below. ketch ~NQ and find its perimeter. Q Q 5 y 1 14 N N 5 4y HOT EONE In, m , m 5 668, and m Eplain why cannot be a parallelogram. 36. FINING NGE EUE In ~N shown at the right, m N 5 328, m N 5 ( 2 )8, m N 5 128, and N is an acute angle. Find m N. N 37. HENGE oints (1, 2), (3, 6), and (6, 4) are three vertices of ~. Find the coordinates of each point that could be verte. ketch each possible parallelogram in a separate coordinate plane. Justify your answers. 8.2 Use roperties of arallelograms 519

7 OE OVING EXE 2 on p. 517 for E INE The diagram shows the mechanism for opening the canopy on a small airplane. Two pivot arms attach at four pivot points,,, and. These points form the vertices of a parallelogram. Find m when m Eplain your reasoning. 39. IO The mirror shown is attached to the wall by an arm that can etend away from the wall. In the figure, points, Q,, and are the vertices of a parallelogram. This parallelogram is one of several that change shape as the mirror is etended. a. If Q 5 3 inches, find. b. If m Q 5 708, what is m? c. What happens to m as m Q increases? What happens to Q as m Q decreases? Eplain. 40. UING TIO In ~NO, the ratio of to N is 4 : 3. Find if the perimeter of NO is OEN-ENE TH raw a triangle. opy the triangle and combine the two triangles to form a quadrilateral. how that the quadrilateral is a parallelogram. Then show how you can make additional copies of the triangle to form a larger parallelogram that is similar to the first parallelogram. Justify your method. 42. OVING THEOE 8.4 Use the diagram of quadrilateral with the auiliary line segment drawn to write a two-column proof of Theorem 8.4. GIVEN c is a parallelogram. OVE c >, > 43. OVING THEOE 8.5 Use properties of parallel lines to prove Theorem GIVEN c Q is a parallelogram. OVE c WOKE-OUT OUTION on p. W1 5 TNIZE TET TIE

8 44. OVING THEOE 8.6 Theorem 8.6 states that if a quadrilateral is a parallelogram, then its diagonals bisect each other. Write a two-column proof of Theorem HENGE uppose you choose a point on the base of an isosceles triangle. You draw segments from that point perpendicular to the legs of the triangle. rove that the sum of the lengths of those segments is equal to the length of the altitude drawn to one leg. F GIVEN c n is isosceles with base }, }F is the altitude drawn to }, }E }, } G } E G OVE c For anywhere on }, E 1 G 5 F. IXE EVIEW EVIEW repare for esson 8.3 in Es Tell whether the lines through the given points are parallel, perpendicular, or neither. Justify your answer. (p. 171) 46. ine 1: (2, 4), (4, 1) 47. ine 1: (26, 7), (22, 3) 48. ine 1: (23, 0), (26, 5) ine 2: (5, 7), (9, 0) ine 2: (9, 21), (2, 6) ine 2: (3, 25), (5, 210) ecide if the side lengths form a triangle. If so, would the triangle be acute, right, or obtuse? (p. 441) 49. 9, 13, and , 12, and , 9, and Ï } , 12, and , 10, and , 10, and 11 Find the value of. Write your answer in simplest radical form. (p. 457) QUIZ for essons Find the value of. (p. 507) Find the value of each variable in the parallelogram. (p. 515) b8 a8 15 7y 2 6 2y (a 2 10) EXT TIE for esson 8.2, p. 910 ONINE QUIZ at classzone.com 521