Objectives To use relationships among sides and angles of parallelograms To use relationships among diagonals of parallelograms

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1 6-2 roperties of arallelograms ontent tandards.o.11 rove theorems about parallelogram. s include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other... lso..5 Objectives o use relationships among sides and angles of parallelograms o use relationships among diagonals of parallelograms on t settle for your answer too soon! e sure to find all pairs of congruent triangles. Use the information given in the diagram. Which triangles are congruent? How do you know? Lesson Vocabulary parallelogram opposite sides opposite angles consecutive angles MHMIL I parallelogram is a quadrilateral with both pairs of opposite sides parallel. In the olve It, you made some conjectures about the characteristics of a parallelogram. In this lesson, you will verify whether your conjectures are correct. ssential Understanding arallelograms have special properties regarding their sides, angles, and diagonals. In a quadrilateral, opposite sides do not share a vertex and opposite angles do not share a side. and are opposite sides. and are opposite angles. You can abbreviate parallelogram with the symbol and parallelograms with the symbol. You can use what you know about parallel lines and transversals to prove some theorems about parallelograms. 6-3 If a quadrilateral is a parallelogram, then its opposite sides are congruent. is a and Lesson 6-2 roperties of arallelograms 359

2 roof roof of 6-3 iven: rove: and tatements easons 1) is a parallelogram. 1) iven 2) and 2) efinition of parallelogram 3) 1 4 and 3 2 3) If lines are, then alt. int. are. 4) 4) eflexive roperty of 5) 5) 6) and 6) orresp. parts of are. ngles of a polygon that share a side are consecutive angles. In the diagram, and are consecutive angles because they share side. he theorem below uses the fact that consecutive angles of a parallelogram are same-side interior angles of parallel lines. and are also consecutive angles. 6-4 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. is a m m 180 m m 180 m m 180 m m 180 You will prove 6-4 in xercise 32. What information from the diagram helps you get started? rom the diagram, you know m and that and are consecutive angles. o, you can write an equation and solve for m. roblem 1 Using onsecutive ngles Multiple hoice What is m in? m m 180 onsecutive angles of a are suppl. m ubstitute. m 116 ubtract 64 from each side. 64 he correct answer is. ot It? 1. uppose you adjust the lamp so that m 86. What is m in? 360 hapter 6 olygons and uadrilaterals

3 6-5 If a quadrilateral is a parallelogram, then its opposite angles are congruent. is a. and proof of 6-5 in roblem 2 uses the consecutive angles of a parallelogram and the fact that supplements of the same angle are congruent. roof roblem 2 Using roperties of arallelograms in a roof iven: rove: and is a. iven and are consecutive. ef. of consecutive and are consecutive. ef. of consecutive and are consecutive. ef. of consecutive Why is a flow proof useful here? flow proof allows you to see how the pairing of two statements leads to a conclusion. and are supplementary. onsecutive are supplementary. upplements of the same are. and are supplementary. onsecutive are supplementary. upplements of the same are. and are supplementary. onsecutive are supplementary. ot It? 2. Use the diagram at the right. iven:, K MK rove: M K M Lesson 6-2 roperties of arallelograms 361

4 he diagonals of parallelograms have a special property. 6-6 If a quadrilateral is a parallelogram, then its diagonals bisect each other. is a and You will prove 6-6 in xercise 13. You can use 6-6 to find unknown lengths in parallelograms. roblem 3 Using lgebra to ind Lengths olve a system of linear equations to find the values of x and y in KLMN. What are KM and LN? L x y 10 2x 8 y 2 M K N he diagonals of a parallelogram bisect each other. et up a system of linear equations by substituting the algebraic expressions for each segment length. ubstitute (y 2) for x in equation. hen solve for y. ubstitute 14 for y in equation. hen solve for x. Use the values of x and y to find KM and LN. K M L N y 10 2x 8 x y 2 y 10 2(y 2) 8 y 10 2y 4 8 y 10 2y 4 10 y 4 14 y x KM 2(K) LN 2(L) 2(y 10) 2(x) 2(1 4 10) 2(1 6) ot It? 3. a. ind the values of x and y in at the right. What are and? b. easoning In roblem 3, does it matter which variable you solve for first? xplain. 3y 7 x 1 y 2x 362 hapter 6 olygons and uadrilaterals

5 You will use parallelograms to prove the following theorem. 6-7 If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. and You will prove 6-7 in xercise 43. roblem 4 Using arallel Lines and ransversals What information do you need? You know the length of. o find H, you need the lengths of and H. In the figure, H, 2, and What is H? H H H H ince lines divide into equal parts, they also divide H into equal parts. egment ddition ostulate ubstitute. H ot It? 4. Use the figure in roblem 4. If H 6 and 15, what is? Lesson heck o you know HOW? Use the diagram of to find each value. 1. m 2. m 3. x What are and in the figure at the right? x x 3 o you UNN? 6. easoning If you know one angle measure of a parallelogram, how do you find the other three angle measures? xplain. 7. ompare and ontrast What is the difference between a quadrilateral and a parallelogram? 8. rror nalysis Your classmate says that V 10. xplain why the statement may not be correct. MHMIL I V 5 cm Lesson 6-2 roperties of arallelograms 363

6 ractice and roblem-olving xercises MHMIL I ractice lgebra ind the value of x in each parallelogram. ee roblem x x x 113 x eveloping roof omplete this two-column proof of 6-6. iven: 2 rove: and bisect each other at. 1 tatements easons 4 3 ee roblem 2. 1) is a parallelogram. 1) iven 2) 2) a. 3) 1 4; 2 3 3) b. 4) 4) c. 5) d. 5) 6) ; 6) e. 7) f. 7) efinition of bisector lgebra ind the values of x and y in. ee roblem x, y 4, x 2, y 15. x 2, y, 2x, y y, x 3, 2y, 3x 1 In the figure,. ind each length. 17. ZU 18. XZ 19. U 20. XV 21. YX 22. YV 23. WX 24. WV W U 2.25 Y Z 3 X V ee roblem 4. pply lgebra ind the value(s) of the variable(s) in each parallelogram (6a 10) (4a 4) 6y 1 4y 9 x (2a 30) 2x hapter 6 olygons and uadrilaterals

7 28. hink bout a lan What are the values of x and y in the parallelogram? How are the angles related? Which variable should you solve for first? y 3y 3x lgebra ind the value of a. hen find each side length or angle measure. 29. a a 20.4 a 1.6 H 18.5 (20a 30) 5a K (17a 48) J 31. tudio Lighting pantograph is an expandable device shown at the right. antographs are used in the television industry in positioning lighting and other equipment. In the photo, points,,, and are the vertices of a parallelogram. is one of many parallelograms that change shape as the pantograph extends and retracts. a. If 2.5 ft, what is? b. If m 129, what is m? c. What happens to m as m increases or decreases? xplain. 32. rove 6-4. roof iven: rove: is supplementary to. is supplementary to. Use the diagram at the right for each proof. roof 33. iven: LN and NH rove: L 34. iven: LN and NH rove: L 35. iven: LN and NH rove: is supplementary to. Use the diagram at the right for each proof. roof 36. iven: W and XYZ rove: X Y L N H 37. iven: W and XYZ rove: XY X Z W ind the measures of the numbered angles for each parallelogram lgebra he perimeter of is 92 cm. is 7 cm more than twice. ind the lengths of all four sides of. Lesson 6-2 roperties of arallelograms 365

8 hallenge 42. Writing Is there an congruence theorem for parallelograms? xplain. 43. rove 6-7. Use the diagram at the right. roof iven:, rove: (Hint: raw lines through and parallel to and intersecting at and at H.) H Measurement xplain how to separate a blank card into three strips that are the same height by using lined paper, a straightedge, and 6-7. tandardized est rep / 45. is a parallelogram with m 4x and m x 10. Which statement explains why you can use the equation 4x (x 10) 180 to solve for x? he measures of the interior angles of a quadrilateral have a sum of 360. Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. onsecutive angles of a parallelogram are supplementary. 46. In the figure of at the right,. Which statement must be true? m m 180 m m n obtuse triangle has side lengths of 5 cm, 9 cm, and 12 cm. What is the length of the side opposite the obtuse angle? 5 cm 9 cm 12 cm not enough information hort esponse 48. ind the measure of one exterior angle of a regular hexagon. xplain your method. Mixed eview ind the sum of the measures of the interior angles of each polygon. ee Lesson decagon gon gon gon 53. What additional information do you need to prove by the HL? et eady! o prepare for Lesson 6-3, do xercise wo consecutive angles in a parallelogram have measures x 5 and 4x 10. ind the measure of the smaller angle. ee Lesson 4-6. ee Lesson hapter 6 olygons and uadrilaterals