Essential Question How can you prove that a quadrilateral is a parallelogram? Work with a partner. Use dynamic geometry software.

Transcription

1 OMMON OR Learning Standards HSG-O..11 HSG-SRT..5 HSG-MG..1 RSONING STRTLY 7.3 To be proficient in math, you need to know and flexibly use different properties of objects. Proving That a Quadrilateral Is a Parallelogram ssential Question How can you prove that a quadrilateral is a parallelogram? Work with a partner. Use dynamic geometry software. Proving That a Quadrilateral Is a Parallelogram 1 a. onstruct any quadrilateral whose opposite sides are congruent. b. Is the quadrilateral a parallelogram? Justify your answer c. Repeat parts (a) and (b) for several other quadrilaterals. Then write a conjecture based on your results. d. Write the converse of your conjecture. Is the converse true? xplain. 3 5 Sample Points (1, 1) (0, ) (, ) (5, 1) Segments = 3.1 =.7 = 3.1 =.7 Proving That a Quadrilateral Is a Parallelogram Work with a partner. Use dynamic geometry software. a. onstruct any quadrilateral whose opposite angles are congruent. b. Is the quadrilateral a parallelogram? Justify your answer. Sample Points (0, 0) (1, 3) (, ) (5, 1) ngles = 0. = = 0. = c. Repeat parts (a) and (b) for several other quadrilaterals. Then write a conjecture based on your results. d. Write the converse of your conjecture. Is the converse true? xplain. ommunicate Your nswer How can you prove that a quadrilateral is a parallelogram?. Is the quadrilateral at the left a parallelogram? xplain your reasoning. Section 7.3 Proving That a Quadrilateral Is a Parallelogram 375

2 7.3 Lesson ore Vocabulary Previous diagonal parallelogram What You Will Learn Identify and verify parallelograms. Show that a quadrilateral is a parallelogram in the coordinate plane. Identifying and Verifying Parallelograms Given a parallelogram, you can use the Parallelogram Opposite Sides Theorem (Theorem 7.3) and the Parallelogram Opposite ngles Theorem (Theorem 7.) to prove statements about the sides and angles of the parallelogram. The converses of the theorems are stated below. You can use these and other theorems in this lesson to prove that a quadrilateral with certain properties is a parallelogram. Theorems Theorem 7.7 Parallelogram Opposite Sides onverse If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If and, then is a parallelogram. Theorem 7.8 Parallelogram Opposite ngles onverse If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If and, then is a parallelogram. Proof x. 39, p. 383 Given, Prove Plan for Proof Parallelogram Opposite Sides onverse is a parallelogram. a. raw diagonal to form and. b. Use the SSS ongruence Theorem (Thm. 5.8) to show that. c. Use the lternate Interior ngles onverse (Thm. 3.) to show that opposite sides are parallel. Plan in ction STTMNTS a. 1.,. raw. 3. RSONS 1. Given. Through any two points, there exists exactly one line. 3. Reflexive Property of ongruence (Thm..1) b... SSS ongruence Theorem (Thm. 5.8) c. 5.,., 5. orresponding parts of congruent triangles are congruent.. lternate Interior ngles onverse (Thm. 3.) 7. is a parallelogram. 7. efinition of parallelogram 37 hapter 7 Quadrilaterals and Other Polygons

3 Identifying a Parallelogram n amusement park ride has a moving platform attached to four swinging arms. The platform swings back and forth, higher and higher, until it goes over the top and around in a circular motion. In the diagram below, and represent two of the swinging arms, and is parallel to the ground (line ). xplain why the moving platform is always parallel to the ground. 38 ft 1 ft 38 ft 1 ft SOLUTION The shape of quadrilateral changes as the moving platform swings around, but its side lengths do not change. oth pairs of opposite sides are congruent, so is a parallelogram by the Parallelogram Opposite Sides onverse. y the definition of a parallelogram,. ecause is parallel to line, is also parallel to line by the Transitive Property of Parallel Lines (Theorem 3.9). So, the moving platform is parallel to the ground. Monitoring Progress Help in nglish and Spanish at igideasmath.com 1. In quadrilateral WXYZ, m W =, m X = 138, and m Y =. ind m Z. Is WXYZ a parallelogram? xplain your reasoning. P x + 9 Q inding Side Lengths of a Parallelogram or what values of x and y is quadrilateral PQRS a parallelogram? S y x 1 R x + 7 SOLUTION y the Parallelogram Opposite Sides onverse, if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ind x so that PQ SR. PQ = SR Set the segment lengths equal. x + 9 = x 1 Substitute x + 9 for PQ and x 1 for SR. 10 = x Solve for x. When x = 10, PQ = = 19 and SR = (10) 1 = 19. ind y so that PS QR. PS = QR Set the segment lengths equal. y = x + 7 Substitute y for PS and x + 7 for QR. y = Substitute 10 for x. y = 17 dd. When x = 10 and y = 17, PS = 17 and QR = = 17. Quadrilateral PQRS is a parallelogram when x = 10 and y = 17. Section 7.3 Proving That a Quadrilateral Is a Parallelogram 377

4 Theorems Theorem 7.9 Opposite Sides Parallel and ongruent Theorem If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. If and, then is a parallelogram. Proof x. 0, p. 383 Theorem 7.10 Parallelogram iagonals onverse If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. If and bisect each other, then is a parallelogram. Proof x. 1, p. 383 Identifying a Parallelogram The doorway shown is part of a building in ngland. Over time, the building has leaned sideways. xplain how you know that SV = TU. S T SOLUTION In the photograph, ST UV and ST UV. y the Opposite Sides Parallel and ongruent Theorem, quadrilateral STUV is a parallelogram. y the Parallelogram Opposite Sides Theorem (Theorem 7.3), you know that opposite sides of a parallelogram are congruent. So, SV = TU. V U inding iagonal Lengths of a Parallelogram or what value of x is quadrilateral a parallelogram? SOLUTION y the Parallelogram iagonals onverse, if the diagonals of bisect each other, then it is a parallelogram. You are given that N N. ind x so that N N. N = N 5x 8 = 3x x 8 = 0 x = 8 Set the segment lengths equal. Substitute 5x 8 for N and 3x for N. Subtract 3x from each side. dd 8 to each side. x = ivide each side by. When x =, N = 5() 8 = 1 and N = 3() = 1. Quadrilateral is a parallelogram when x =. 5x 8 N 3x 378 hapter 7 Quadrilaterals and Other Polygons

5 Monitoring Progress Help in nglish and Spanish at igideasmath.com. or what values of x and y is quadrilateral a parallelogram? xplain your reasoning. (3x 3) y (y + 87) x State the theorem you can use to show that the quadrilateral is a parallelogram m. 7 in m 5 in. 7 in. 5 in or what value of x is quadrilateral MNPQ a parallelogram? xplain your reasoning. M x N 10 3x Q P oncept Summary Ways to Prove a Quadrilateral Is a Parallelogram 1. Show that both pairs of opposite sides are parallel. (efi nition). Show that both pairs of opposite sides are congruent. (Parallelogram Opposite Sides onverse) 3. Show that both pairs of opposite angles are congruent. (Parallelogram Opposite ngles onverse). Show that one pair of opposite sides are congruent and parallel. (Opposite Sides Parallel and ongruent Theorem) 5. Show that the diagonals bisect each other. (Parallelogram iagonals onverse) Section 7.3 Proving That a Quadrilateral Is a Parallelogram 379

6 Using oordinate Geometry Identifying a Parallelogram in the oordinate Plane Show that quadrilateral is a parallelogram. y (, 5) ( 3, 3) (5, ) SOLUTION Method hapter 7 Quadrilaterals and Other Polygons Show that a pair of sides are congruent and parallel. Then apply the Opposite Sides Parallel and ongruent Theorem. irst, use the istance ormula to show that and are congruent. = [ ( 3)] + (5 3) = 9 = (5 0) + ( 0) = 9 ecause = = 9,. Then, use the slope formula to show that. slope of 5 3 = ( 3) = 5 slope of = = 5 ecause and have the same slope, they are parallel. and are congruent and parallel. So, is a parallelogram by the Opposite Sides Parallel and ongruent Theorem. Method Show that opposite sides are congruent. Then apply the Parallelogram Opposite Sides onverse. In Method 1, you already have shown that because = = 9,. Now find and. = ( 3 0) + (3 0) = 3 = ( 5) + (5 ) = 3 ecause = = 3,. and. So, is a parallelogram by the Parallelogram Opposite Sides onverse. Monitoring Progress 7. Show that quadrilateral JKLM is a parallelogram. 8. Refer to the oncept Summary on page 379. xplain two other methods you can use to show that quadrilateral in xample 5 is a parallelogram. Help in nglish and Spanish at igideasmath.com K (0, 0) J y L M x x

7 7.3 xercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck 1. WRITING quadrilateral has four congruent sides. Is the quadrilateral a parallelogram? Justify your answer.. IRNT WORS, SM QUSTION Which is different? ind both answers. onstruct a quadrilateral with opposite sides congruent. onstruct a quadrilateral with one pair of parallel sides. onstruct a quadrilateral with opposite angles congruent. onstruct a quadrilateral with one pair of opposite sides congruent and parallel. Monitoring Progress and Modeling with Mathematics In xercises 3 8, state which theorem you can use to show that the quadrilateral is a parallelogram. (See xamples 1 and 3.) x y + 1 y 3 (x + 13) (y + 7) x 3 (3x 8) (5x 1) In xercises 13 1, find the value of x that makes the quadrilateral a parallelogram. (See xample.) x + 3 x x 15. 3x x x + 7 3x + In xercises 9 1, find the values of x and y that make the quadrilateral a parallelogram. (See xample.) x y 1 9 y x 5x 9 In xercises 17 0, graph the quadrilateral with the given vertices in a coordinate plane. Then show that the quadrilateral is a parallelogram. (See xample 5.) 17. (0, 1), (, ), (1, ), (8, 1) 18. ( 3, 0), ( 3, ), G(3, 1), H(3, 5) 19. J(, 3), K( 5, 7), L(3, ), M(, ) 0. N( 5, 0), P(0, ), Q(3, 0), R(, ) Section 7.3 Proving That a Quadrilateral Is a Parallelogram 381

8 RROR NLYSIS In xercises 1 and, describe and correct the error in identifying a parallelogram. 1.. G 7 7 G is a parallelogram by the Parallelogram Opposite Sides onverse (Theorem 7.7). J 13 K Parallelogram iagonals Theorem (Theorem 7.) and Parallelogram iagonals onverse (Theorem 7.10) 8. ONSTRUTION escribe a method that uses the Opposite Sides Parallel and ongruent Theorem (Theorem 7.9) to construct a parallelogram. Then construct a parallelogram using your method. 9. RSONING ollow the steps below to construct a parallelogram. xplain why this method works. State a theorem to support your answer. Step 1 Use a ruler to draw two segments that intersect at their midpoints. M JKLM is a parallelogram by the Opposite Sides Parallel and ongruent Theorem (Theorem 7.9). L 3. MTHMTIL ONNTIONS What value of x makes the quadrilateral a parallelogram? xplain how you found your answer. 3x + 1 x 5x 3x + 10 x + 1 x + 3x + 3 x. MKING N RGUMNT Your friend says you can show that quadrilateral WXYZ is a parallelogram by using the onsecutive Interior ngles onverse (Theorem 3.8) and the Opposite Sides Parallel and ongruent Theorem (Theorem 7.9). Is your friend correct? xplain your reasoning. W 5 X Step onnect the endpoints of the segments to form a parallelogram. 30. MKING N RGUMNT Your brother says to show that quadrilateral QRST is a parallelogram, you must show that QR TS and QT RS. Your sister says that you must show that QR TS and QT RS. Who is correct? xplain your reasoning. Q y R x Z 115 Y T S NLYZING RLTIONSHIPS In xercises 5 7, write the indicated theorems as a biconditional statement. 5. Parallelogram Opposite Sides Theorem (Theorem 7.3) and Parallelogram Opposite Sides onverse (Theorem 7.7). Parallelogram Opposite ngles Theorem (Theorem 7.) and Parallelogram Opposite ngles onverse (Theorem 7.8) RSONING In xercises 31 and 3, your classmate incorrectly claims that the marked information can be used to show that the figure is a parallelogram. raw a quadrilateral with the same marked properties that is clearly not a parallelogram hapter 7 Quadrilaterals and Other Polygons

9 33. MOLING WITH MTHMTIS You shoot a pool ball, and it rolls back to where it started, as shown in the diagram. The ball bounces off each wall at the same angle at which it hits the wall. H G RSONING Quadrilateral JKLM is a parallelogram. escribe how to prove that MGJ KHL. G J K a. The ball hits the first wall at an angle of 3. So m = m H = 3. What is m? xplain your reasoning. b. xplain why m G = 3. c. What is m GH? m H? d. Is quadrilateral GH a parallelogram? xplain your reasoning. 3. MOLING WITH MTHMTIS In the diagram of the parking lot shown, m JKL = 0, JK = LM = 1 feet, and KL = JM = 9 feet. M L H 39. PROVING THORM Prove the Parallelogram Opposite ngles onverse (Theorem 7.8). (Hint: Let x represent m and m. Let y represent m and m. Write and simplify an equation involving x and y.) Given, Prove is a parallelogram. J M O Q K L N a. xplain how to show that parking space JKLM is a parallelogram. b. ind m JML, m KJM, and m KLM. c. LM NO and NO PQ. Which theorem could you use to show that JK PQ? RSONING In xercises 35 37, describe how to prove that is a parallelogram. 35. P 0. PROVING THORM Use the diagram of PQRS with the auxiliary line segment drawn to prove the Opposite Sides Parallel and ongruent Theorem (Theorem 7.9). Given QR PS, QR PS Prove PQRS is a parallelogram. P Q 1. PROVING THORM Prove the Parallelogram iagonals onverse (Theorem 7.10). Given iagonals JL and KM bisect each other. Prove JKLM is a parallelogram. S R K L J P M Section 7.3 Proving That a Quadrilateral Is a Parallelogram 383

10 . PROO Write a proof. Given is a parallelogram. = Prove is a parallelogram. 3. RSONING Three interior angle measures of a quadrilateral are 7, 7, and 113. Is this enough information to conclude that the quadrilateral is a parallelogram? xplain your reasoning.. HOW O YOU S IT? music stand can be folded up, as shown. In the diagrams, and are parallelograms. Which labeled segments remain parallel as the stand is folded? 7. WRITING The Parallelogram onsecutive ngles Theorem (Theorem 7.5) says that if a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Write the converse of this theorem. Then write a plan for proving the converse. Include a diagram. 8. PROO Write a proof. Given is a parallelogram. is a right angle. Prove,, and are right angles. 9. STRT RSONING The midpoints of the sides of a quadrilateral have been joined to form what looks like a parallelogram. Show that a quadrilateral formed by connecting the midpoints of the sides of any quadrilateral is always a parallelogram. (Hint: raw a diagram. Include a diagonal of the larger quadrilateral. Show how two sides of the smaller quadrilateral relate to the diagonal.) 5. RITIL THINKING In the diagram, is a parallelogram, = = 1, and = 8. ind. xplain your reasoning. 50. RITIL THINKING Show that if is a parallelogram with its diagonals intersecting at, then you can connect the midpoints, G, H, and J of,,, and, respectively, to form another parallelogram, GHJ.. THOUGHT PROVOKING reate a regular hexagon using congruent parallelograms. J H G Maintaining Mathematical Proficiency lassify the quadrilateral. (Skills Review Handbook) Reviewing what you learned in previous grades and lessons hapter 7 Quadrilaterals and Other Polygons