Essential Question What are the properties of parallelograms?

Transcription

1 7. roperties of arallelograms ssential uestion What are the properties of parallelograms? iscovering roperties of arallelograms Work with a partner. Use dynamic geometry software. a. onstruct any parallelogram and label it. xplain your process. ample b. ind the angle measures of the parallelogram. What do you observe? c. ind the side lengths of the parallelogram. What do you observe? d. epeat parts (a) (c) for several other parallelograms. Use your results to write conjectures about the angle measures and side lengths of a parallelogram. iscovering a roperty of arallelograms Work with a partner. Use dynamic geometry software. a. onstruct any parallelogram and label it. b. raw the two diagonals of the parallelogram. abel the point of intersection. ample KIN N O O To be proficient in math, you need to analyze givens, constraints, relationships, and goals. c. ind the segment lengths,,, and. What do you observe? d. epeat parts (a) (c) for several other parallelograms. Use your results to write a conjecture about the diagonals of a parallelogram. ommunicate Your nswer 3. What are the properties of parallelograms? ection 7. roperties of arallelograms 367

2 7. esson What You Will earn ore Vocabulary parallelogram, p. 368 revious quadrilateral diagonal interior angles segment bisector Use properties to find side lengths and angles of parallelograms. Use parallelograms in the coordinate plane. Using roperties of arallelograms parallelogram is a quadrilateral with both pairs of opposite sides parallel. In, and by definition. The theorems below describe other properties of parallelograms. Theorems Theorem 7.3 arallelogram Opposite ides Theorem If a quadrilateral is a parallelogram, then its opposite sides are congruent. If is a parallelogram, then and. roof p. 368 Theorem 7.4 arallelogram Opposite ngles Theorem If a quadrilateral is a parallelogram, then its opposite angles are congruent. If is a parallelogram, then and. roof x. 37, p. 373 arallelogram Opposite ides Theorem iven is a parallelogram. rove, lan for roof a. raw diagonal to form and. b. Use the ongruence Theorem (Thm. 5.10) to show that. c. Use congruent triangles to show that and. lan in ction TTNT ON 1. is a parallelogram. 1. iven a.. raw. 3., b. 4., 5.. Through any two points, there exists exactly one line. 3. efinition of parallelogram 4. lternate Interior ngles Theorem (Thm. 3.) 5. eflexive roperty of ongruence (Thm..1) ongruence Theorem (Thm. 5.10) c. 7., 7. orresponding parts of congruent triangles are congruent. 368 hapter 7 uadrilaterals and Other olygons

3 ind the values of x and y. OUTION Using roperties of arallelograms is a parallelogram by the definition of a parallelogram. Use the arallelogram Opposite ides Theorem to find the value of x. = x + 4 = 1 x = 8 Opposite sides of a parallelogram are congruent. ubstitute x + 4 for and 1 for. ubtract 4 from each side. y the arallelogram Opposite ngles Theorem,, or m = m. o, y = 65. In, x = 8 and y = 65. y x onitoring rogress Help in nglish and panish at igideasath.com 1. ind and m.. ind the values of x and y. H 8 60 J K 18 x 50 y + 3 The onsecutive Interior ngles Theorem (Theorem 3.4) 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 is like a pair of consecutive interior angles between parallel lines. This similarity suggests the arallelogram onsecutive ngles Theorem. x y Theorems Theorem 7.5 arallelogram onsecutive ngles Theorem If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. x y If is a parallelogram, then x + y = 180. y x roof x. 38, p. 373 Theorem 7.6 arallelogram iagonals Theorem If a quadrilateral is a parallelogram, then its diagonals bisect each other. If is a parallelogram, then and. roof p. 370 ection 7. roperties of arallelograms 369

4 arallelogram iagonals Theorem iven is a parallelogram. iagonals and intersect at point. rove bisects and. TTNT ON 1. is a parallelogram. 1. iven.. efinition of a parallelogram 3., 3. lternate Interior ngles Theorem (Thm. 3.) arallelogram Opposite ides Theorem ongruence Theorem (Thm. 5.10) 6., 6. orresponding parts of congruent triangles are congruent. 7. bisects and. 7. efinition of segment bisector Using roperties of a arallelogram s shown, part of the extending arm of a desk lamp is a parallelogram. The angles of the parallelogram change as the lamp is raised and lowered. ind m when m = 110. OUTION y the arallelogram onsecutive ngles Theorem, the consecutive angle pairs in are supplementary. o, m + m = 180. ecause m = 110, m = = 70. Writing a Two-olumn roof Write a two-column proof. iven and are parallelograms. rove TTNT 1. and are parallelograms. ON 1. iven.,. If a quadrilateral is a parallelogram, then its opposite angles are congruent Vertical ngles ongruence Theorem (Thm..6) Transitive roperty of ongruence (Thm..) onitoring rogress 370 hapter 7 uadrilaterals and Other olygons Help in nglish and panish at igideasath.com 3. WHT I? In xample, find m when m is twice the measure of. 4. Using the figure and the given statement in xample 3, prove that and are supplementary angles.

5 Using arallelograms in the oordinate lane JUTIYIN T In xample 4, you can use either diagonal to find the coordinates of the intersection. Using diagonal O helps simplify the calculation because one endpoint is (0, 0). Using arallelograms in the oordinate lane ind the coordinates of the intersection of the diagonals of NO with vertices (1, 4), (7, 4), N(6, 0), and O(0, 0). OUTION y the arallelogram iagonals Theorem, the diagonals of a parallelogram bisect each other. o, the coordinates of the intersection are the midpoints of diagonals N and O. coordinates of midpoint of O = ( 7 + 0, The coordinates of the intersection of the diagonals are ( 7, ). You can check your answer by graphing NO and drawing the diagonals. The point of intersection appears to be correct. ) = ( 7, ) 4 y idpoint ormula O 4 N 8 x When graphing a polygon in the coordinate plane, the name of the polygon gives the order of the vertices. Using arallelograms in the oordinate lane Three vertices of WXYZ are W( 1, 3), X( 3, ), and Z(4, 4). ind the coordinates of vertex Y. OUTION tep 1 raph the vertices W, X, and Z. tep ind the slope of WX. slope of WX = ( 3) 3 ( 1) = 5 = 5 X tep 3 tart at Z(4, 4). Use the rise and run 4 5 from tep to find vertex Y. rise of 5 represents a change of 5 units W up. run of represents a change of 4 units left. o, plot the point that is 5 units up and units left from Z(4, 4). The point is (, 1). abel it as vertex Y. tep 4 ind the slopes of XY and WZ to verify that they are parallel. slope of XY 1 = ( 3) = 1 5 = 1 slope of WZ 4 ( 3) = 5 4 ( 1) y Y(, 1) x 5 Z = 1 5 = 1 5 o, the coordinates of vertex Y are (, 1). onitoring rogress Help in nglish and panish at igideasath.com 5. ind the coordinates of the intersection of the diagonals of TUV with vertices (, 3), T(1, 5), U(6, 3), and V(3, 1). 6. Three vertices of are (, 4), (5, ), and (3, 1). ind the coordinates of vertex. ection 7. roperties of arallelograms 371

6 7. xercises ynamic olutions available at igideasath.com Vocabulary and ore oncept heck 1. VOUY Why is a parallelogram always a quadrilateral, but a quadrilateral is only sometimes a parallelogram?. WITIN You are given one angle measure of a parallelogram. xplain how you can find the other angle measures of the parallelogram. onitoring rogress and odeling with athematics In xercises 3 6, find the value of each variable in the parallelogram. (ee xample 1.) 3. x 15 y n m In xercises 17 0, find the value of each variable in the parallelogram m n 18. d c (b 10) (b + 10) 5. 0 (d 1) 105 z (g + 4) 16 h k + 4 m 8 11 In xercises 7 and 8, find the measure of the indicated angle in the parallelogram. (ee xample.) 7. ind m. 8. ind m N In xercises 9 16, find the indicated measure in N. xplain your reasoning m N 14. m N 15. m N 16. m N 8 N 0. u + 6 v 3 5u 10 O NYI In xercises 1 and, describe and correct the error in using properties of parallelograms T ecause quadrilateral TUV is a parallelogram, V. o, m V = 50. K V J U ecause quadrilateral HJK is a parallelogram, H. H 37 hapter 7 uadrilaterals and Other olygons

7 OO In xercises 3 and 4, write a two-column proof. (ee xample 3.) 3. iven and are parallelograms. rove 4. iven,, and HJK are parallelograms. rove 3 H 4 J 3 K 1 In xercises 5 and 6, find the coordinates of the intersection of the diagonals of the parallelogram with the given vertices. (ee xample 4.) 5. W(, 5), X(, 5), Y(4, 0), Z(0, 0) 6. ( 1, 3), (5, ), (1, ), T( 5, 1) In xercises 7 30, three vertices of are given. ind the coordinates of the remaining vertex. (ee xample 5.) 7. (0, ), ( 1, 5), (4, 0) 8. (, 4), (0, 7), (1, 0) 9. ( 4, ), ( 3, 1), (3, 3) 30. ( 1, 4), (5, 6), (8, 0) THTI ONNTION In xercises 31 and 3, find the measure of each angle. 31. The measure of one interior angle of a parallelogram is 0.5 times the measure of another angle. 3. The measure of one interior angle of a parallelogram is 50 degrees more than 4 times the measure of another angle. 33. KIN N UNT In quadrilateral, m = 14, m = 56, and m = 14. Your friend claims quadrilateral could be a parallelogram. Is your friend correct? xplain your reasoning. 34. TTNIN TO IION J and K are consecutive angles in a parallelogram, m J = (3x + 7), and m K = (5x 11). ind the measure of each angle. 35. ONTUTION onstruct any parallelogram and label it. raw diagonals and. xplain how to use paper folding to verify the arallelogram iagonals Theorem (Theorem 7.6) for. 36. OIN WITH THTI The feathers on an arrow form two congruent parallelograms. The parallelograms are reflections of each other over the line that contains their shared side. how that m = m OVIN THO Use the diagram to write a two-column proof of the arallelogram Opposite ngles Theorem (Theorem 7.4). iven is a parallelogram. rove, 38. OVIN THO Use the diagram to write a two-column proof of the arallelogram onsecutive ngles Theorem (Theorem 7.5). y x y x iven is a parallelogram. rove x + y = O OVIN The sides of N are represented by the expressions below. ketch N and find its perimeter. = x + 37 = y + 14 N = x 5 N = 4y O OVIN In N, the ratio of to N is 4 : 3. ind when the perimeter of N is 8. ection 7. roperties of arallelograms 373

8 41. TT ONIN an you prove that two parallelograms are congruent by proving that all their corresponding sides are congruent? xplain your reasoning. 44. THOUHT OVOKIN Is it possible that any triangle can be partitioned into four congruent triangles that can be rearranged to form a parallelogram? xplain your reasoning. 4. HOW O YOU IT? The mirror shown is attached to the wall by an arm that can extend away from the wall. In the figure, points,,, and are the vertices of a parallelogram. This parallelogram is one of several that change shape as the mirror is extended. 45. ITI THINKIN oints W(1, ), X(3, 6), and Y(6, 4) are three vertices of a parallelogram. How many parallelograms can be created using these three vertices? ind the coordinates of each point that could be the fourth vertex. 46. OO In the diagram, K bisects H, and J bisects. rove that K J. (Hint: Write equations using the angle measures of the triangles and quadrilaterals formed.) a. What happens to m as m increases? xplain. b. What happens to as m decreases? xplain. c. What happens to the overall distance between the mirror and the wall when m decreases? xplain. 43. THTI ONNTION In TUV, m TU = 3, m UV = (x ), m TUV = 1x, and TUV is an acute angle. ind m UV. H J K 47. OO rove the ongruent arts of arallel ines orollary: If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. J H K T V U iven H JK, J J rove HK K (Hint: raw K and such that quadrilateral KJ and quadrilateral J are parallelograms.) aintaining athematical roficiency etermine whether lines l and m are parallel. xplain your reasoning. (ection 3.3) 48. m 49. eviewing what you learned in previous grades and lessons m m 374 hapter 7 uadrilaterals and Other olygons