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LESSON 24.3 Properties of Rectangles, Rhombuses, and Squares Name Class Date 24.3 Properties of Rectangles, Rhombuses, and Squares HARDBOUND SE PAGE 987 BEGINS HERE Essential Question: What are the properties of rectangles, rhombuses, and squares? Resource Locker Explore Common Core Math Standards Exploring Sides, Angles, and Diagonals of a Rectangle A rectangle is a quadrilateral with four right angles. The figure shows rectangle ABCD. The student is expected to: Investigate properties of rectangles. G-CO.11 A Prove theorems about parallelograms. Also G-SRT.5 Mathematical Practices A B D C Use a tile or pattern block and the following method to draw three different rectangles on a separate sheet of paper. MP.6 Precision A B Language Objective Explain to a partner how to classify different types of quadrilaterals as rectangles, rhombuses, or squares. D ENGAGE A rectangle is a parallelogram with congruent diagonals; a rhombus is a parallelogram with perpendicular diagonals, each of which bisects a pair of opposite angles; a square is both a rectangle and a rhombus and has the properties of each. © Houghton Mifflin Harcourt Publishing Company Essential Question: What are the properties of rectangles, rhombuses, and squares? B View the Engage section online. Discuss the photo, drawing attention to the geometrical shapes that form the design of the flag. Then preview the Lesson Performance Task. Use a ruler to measure the sides and diagonals of each rectangle. Keep track of the measurements and compare your results to other students. Reflect 1. Why does this method produce a rectangle? What must you assume about the tile? You must assume that the corners of the tile are right angles. Therefore, each stage of the drawing produces a line that meets the previous line at a right angle. The completed quadrilateral has four right angles, so it is a rectangle. 2. Discussion Is every rectangle also a parallelogram? Make a conjecture based upon your measurements and explain your thinking. Yes; in every case, opposite sides are congruent. All rectangles are parallelograms because a quadrilateral in which both pairs of opposite sides are congruent is a parallelogram. 3. PREVIEW: LESSON PERFORMANCE TASK Use your measurements to make two conjectures about the diagonals of a rectangle. Conjecture: The diagonals of a rectangle are congruent. Conjecture: The diagonals of a rectangle bisect each other. Module 24 be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction Lesson 3 1217 gh “File info” made throu Date Class , Rectangles res perties of 24.3 Pro buses, and Squa Rhom Name uses, and gles, rhomb rties of rectan ion: What Quest Essential G-CO.11 Prove IN1_MNLESE389762_U9M24L3 1217 Explore are the prope lograms. Also Resource Locker squares? G-SRT.5 HARDCOVER PAGES 987994 about paralle ctors icular Bise g Perpend Constructin icular Lines B end and Perp A theorems four right lateral with is a quadri . gle ABCD A rectangle shows rectan The figure gles. of rectan properties Investigate block and or pattern Use a tile of paper. separate sheet angles. the follow D ing metho d to draw C nt rectangles three differe Turn to these pages to find this lesson in the hardcover student edition. on a A D B C of the Keep track rectangle. als of each ts. and diagon re the sides to other studen to measu re your results Use a ruler ts and compa measuremen tile? e about the stage of you assum fore, each What must angles. There Reflect rectangle? a right completed ce are angle. The method produ corners of the tile at a right does this the ous line 1. Why assume that s the previ You must that meet ts ces a line a rectangle. measuremen ing produ s, so it is the draw upon your angle based right ture ral has four Make a conjec because elogram? quadrilate lelograms also a parall paral gle are every rectan rectangles ssion Is lelogram. ruent. All ng. is a paral 2. Discu are cong n your thinki congruent site sides and explai sides are case, oppo of opposite gle. Yes; in every both pairs of a rectan als which diagon ral in about the a quadrilate conjectures ruent. make two are cong rements to rectangle your measu other. nals of a Lesson 3 3. Use bisect each The diago rectangle nals of a Conjecture: The diago 1217 Conjecture: © Houghto n Mifflin Harcour t Publishin y g Compan Module 24 4L3.indd 62_U9M2 ESE3897 IN1_MNL 1217 Lesson 24.3 C 1217 4/18/14 8:02 PM 10/16/14 2:57 PM Explain 1 HARDBOUND SE Proving Diagonals of a Rectangle are Congruent PAGE 988 BEGINS HERE You can use the definition of a rectangle to prove the following theorems. Properties of Rectangles Exploring Sides, Angles, and Diagonals of a Rectangle If a quadrilateral is a rectangle, then it is a parallelogram. If a parallelogram is a rectangle, then its diagonals are congruent. Use a rectangle to prove the Properties of Rectangles Theorems. Example 1 A B A B INTEGRATE TECHNOLOGY Given: ABCD is a rectangle. _ _ Prove: ABCD is a parallelogram; AC ≅ BD. D Statements C D C D Students have the option of doing the rectangle activity either in the book or online. C Reasons 1. ABCD is a rectangle. 1. Given 2. ∠A and ∠C are right angles. 2. Definition of rectangle 3. ∠A ≅ ∠C QUESTIONING STRATEGIES 3. All right angles are congruent. 4. ∠B and ∠D are right angles. 4. Definition of rectangle 5. ∠B ≅ ∠D 5. All right angles are congruent. Is a rectangle a parallelogram? How do you know? Yes; since the opposite sides are congruent, the Opposite Sides Criterion applies. 6. ABCD is a parallelogram. 6. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ― ― 7. AD ≅ CB 7. If a quadrilateral is a parallelogram, then its opposite sides are congruent. 8. DC ≅ DC 8. Reflexive Property of Congruence 9. ∠D and ∠C are right angles. 9. Definition of rectangle ― ― 10. ∠D ≅ ∠C 11. △ADC ≅ △BCD ― ― 12. AC ≅ BD Proving Diagonals of a Rectangle Are Congruent 11. SAS Triangle Congruence Theorem 12. CPCTC A B Find each measure. 6. AD = 7.5 cm and DC = 10 cm. Find DB. D By the Pythagorean Theorem, AC 2 = 7.5 2 + 10 2, so AC = 12.5 cm. Diagonals of a rectangle are congruent, so DB = AC = 12.5 cm. AB = 17 cm and BC = 12.75 cm. Find DB. Opposite sides of the rectangle are congruent, so DC = AB = 17 cm. By the Pythagorean Theorem, DB 2 = 17 2 + 12.75 2, so DB = 21.25 cm. Module 24 1218 © Houghton Mifflin Harcourt Publishing Company Discussion A student says you can also prove the diagonals are congruent in Example 1 by using the SSS Triangle Congruence Theorem to show that △ADC ≅ △BCD. Do you agree? Explain. No; in order to use the SSS Triangle Congruence Theorem, you would have to know that _ _ AC ≅ BD, which is what you are trying to prove. Your Turn 5. EXPLAIN 1 10. All right angles are congruent. Reflect 4. EXPLORE QUESTIONING STRATEGIES What two things do you have to prove in order to prove the theorem? Prove the figure is a parallelogram and then prove triangles that contain the diagonals of the rectangle are congruent. Does it matter which pair of triangles you prove congruent for this proof? Explain. Yes; you need to choose a pair of triangles that contain the diagonals of the rectangle as sides. C Lesson 3 AVOID COMMON ERRORS PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U9M24L3.indd 1218 4/18/14 8:04 PM Learning Progressions In this lesson, students extend their earlier work with parallelograms to explore the properties of three special quadrilaterals: the rectangle, a quadrilateral with four congruent (right) angles; the rhombus, a quadrilateral with four congruent sides; and the square, a quadrilateral with four congruent (right) angles and four congruent sides. The proofs demonstrate that each of these quadrilaterals is a parallelogram; as such, they inherit all of the properties of parallelograms. Proving these theorems is an application of the Law of Detachment (if p → q is true and p is true, then q is true), which students will use as they further their study of mathematics. Students may try to prove that a rectangle is a parallelogram by stating that the opposite sides are congruent and using the Opposite Sides Criterion for a parallelogram. However, the given information states only that the figure is a rectangle (that is, it has four right angles). The fact that a rectangle has opposite sides that are congruent is a consequence of the fact that a rectangle is a parallelogram. In order to avoid circular reasoning, congruent opposite sides cannot be used as part of this proof. Properties of Rectangles, Rhombuses, and Squares 1218 HARDBOUND SE EXPLAIN 2 Explain 2 PAGE 989 BEGINS HERE Proving Diagonals of a Rhombus are Perpendicular J A rhombus is a quadrilateral with four congruent sides. The figure shows rhombus JKLM. Proving Diagonals of a Rhombus Are Perpendicular M Properties of Rhombuses If a quadrilateral is a rhombus, then it is a parallelogram. If a parallelogram is a rhombus, then its diagonals are perpendicular. If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. QUESTIONING STRATEGIES Example 2 Why do the diagonals of a rhombus bisect each other? Because a rhombus is a parallelogram, the diagonals bisect each other. Prove that the diagonals of a rhombus are perpendicular. Given: JKLM is a rhombus. ― ― Prove: JL ⊥ MK M J N K L _ To prove the diagonals of a rhombus are perpendicular, do you need to show that each of the four angles formed by the intersecting diagonals is a right angle? Why or why not? No; you need to show only that one angle is a right angle. If one angle is a right angle, it’s easy to see that the others must also be right angles by the Vertical Angles Theorem and the Linear Pair Theorem. ― ― ― Since JKLM is a rhombus, JM ≅ JK . Because JKLM is also a parallelogram, MN ≅ KN because ― ≅ JN ―, diagonals of a parallelogram bisect each other . By the Reflexive Property of Congruence, JN so △JNM ≅ △JNK by the SSS Triangle Congruence Theorem . So, ∠JNM ≅ ∠JNK by CPCTC. By the Linear Pair Theorem, ∠JNM and ∠JNK are supplementary. This means that m∠JNM + m∠JNK = 180° . Since the angles are congruent, m∠JNM = m∠JNK so by substitution , m∠JNM + m∠JNK = 180° or _ ― 2m∠JNK = 180°. Therefore, m∠JNK = 90° and JL ⊥ MK. Reflect © Houghton Mifflin Harcourt Publishing Company CONNECT VOCABULARY Have students complete a quadrilateral chart, with pictures and explanations for each type of quadrilateral they have encountered in this unit (rhombus, square, rectangle, other parallelograms, and so on). K L ― 7. What can you say about the image _of J in the proof after a reflection across _ MK? Why? That its image is L because MK is the perpendicular bisector of JL. 8. What property about the diagonals of a rhombus is the same as a property of all parallelograms? What special property do the diagonals of a rhombus have? The diagonals of rhombuses bisect each other, which is a property of all parallelograms. The diagonals of rhombuses are perpendicular, which is a property unique to rhombuses. Your Turn 9. Prove that if a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. M J N K Given: JKLM is a rhombus. ― L Prove: MK bisects ∠JML and ∠JKL; ― JL bisects ∠MJK and ∠MLK. _ _ _ _ _ _ Since JKLM is a rhombus, JM ≅ LM and JK ≅ LK. MK ≅ MK by the Reflexive Property of Congruence. So, △MJK ≅ △MLK by the SSS Triangle _ Congruence Theorem. Therefore, MK bisects ∠JML and ∠JKL. A similar ∠JMK ≅ ∠LMK and ∠JKM ≅ ∠LKM by CPCTC, so _ argument shows that JL bisects ∠MJK and ∠MLK. Module 24 1219 Lesson 3 COLLABORATIVE LEARNING IN1_MNLESE389762_U9M24L3.indd 1219 Whole Class Activity As a class, come up with a list of the properties common to some of the quadrilaterals studied in the lesson, such as “Opposite sides congruent” and “Diagonals perpendicular.” Use the list to construct a table summarizing which quadrilateral (parallelogram, rectangle, rhombus, square) has each property. Save the table as a class review resource. 1219 Lesson 24.3 4/18/14 8:04 PM Explain 3 Example 3 Use rhombus VWXY to find each measure. V Y PAGE 990 BEGINS HERE 6m - 12 (9n + 4)° W Z HARDBOUND SE Using Properties of Rhombuses to Find Measures Using Properties of Rhombuses to Find Measures 4m + 4 X (3n2 - 0.75)° Find XY. QUESTIONING STRATEGIES _ _ All sides of a rhombus are congruent, so VW ≅ WX and VW = WX. What type of angles do the diagonals of a rhombus form? right angles 6m - 12 = 4m + 4 Substitute values for VW and WX. m=8 Solve for m. EXPLAIN 3 VW = 6(8) - 12 = 36 _ _ Because all sides of the rhombus are congruent, then VW ≅ XY, and XY = 36. Sustitute the value of m to find VW. AVOID COMMON ERRORS Find ∠YVW. Students may be confused when asked to solve equations to find segments of special quadrilaterals. Remind students that the sides and angles marked in the figure are not necessarily the ones asked for in the problem. Have students, when they read a problem, copy the figure and circle the length or measure they are asked to find. The diagonals of a rhombus are perpendicular , so ∠WZX is a right angle and m∠WZX = 90° . Since m∠WZX = (3n 2 - 0.75)°, then (3n 2 - 0.75°) = 90° . 3n 2 - 0.75 = 90 Solve for n. n = 5.5 Substitute the value of n to find m∠WVZ. m∠WVZ = 53.5° _ Since VX bisects ∠YVW, then ∠YVZ ≅ ∠WVZ m∠YVW = 2(53.5°) = 107° Substitute 53.5° for m∠WVZ. © Houghton Mifflin Harcourt Publishing Company Your Turn Use the rhombus VWXY from Example 3 to find each measure. 10. Find m∠VYX. 11. Find m∠XYZ. 1 (m∠VYX) m∠XYZ = __ 2 From Part B, m∠YVW = 107° 1( m∠XYZ = _ 73°) 2 107° + m∠VYX = 180° m∠VYX = 73° Module 24 m∠XYZ = 36. 5° 1220 Lesson 3 DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U9M24L3 1220 9/30/14 12:46 PM Modeling Divide students into groups. Have each group cut out models of a rectangle, a rhombus, and a square from construction paper. Then have students use a protractor and a ruler to verify the properties of rectangles and rhombuses from the theorems in this lesson. Properties of Rectangles, Rhombuses, and Squares 1220 Explain 4 EXPLAIN 4 Investigating the Properties of a Square A square is a quadrilateral with four sides congruent and four right angles. Investigating the Properties of a Square Example 4 If a quadrilateral is a square, then it is a parallelogram. By definition, a square is a quadrilateral with four congruent sides. Any quadrilateral with both pairs of opposite sides congruent is a parallelogram, so a square is a parallelogram. QUESTIONING STRATEGIES Why is a square also a rhombus? A square is a quadrilateral with 4 congruent sides, so a square is a rhombus. Explain why each conditional statement is true. HARDBOUND SE PAGE 991 If a quadrilateral is a square, then it is a rectangle. By definition, a square is a quadrilateral with four right angles . BEGINS HERE By definition, a rectangle is also a quadrilateral with four right angles . Therefore, a square is a rectangle. AVOID COMMON ERRORS Your Turn When working with shapes they are familiar with, like squares, some students have trouble distinguishing between properties that belong to the shape by definition and properties that must be proven. When discussing these definitions, point out which properties are included and which are not. 12. Explain why this conditional statement is true: If a quadrilateral is a square, then it is a rhombus. By definition, a rhombus is a quadrilateral with four congruent sides. Since a square is also a quadrilateral that has four congruent sides, then a square is a rhombus. 13. Look at Part A. Use a different way to explain why this conditional statement is true: If a quadrilateral is a square, then it is a parallelogram. Possible answer: All four angles of a square are right angles. All right angles are congruent. Any quadrilateral with both pairs of opposite angles congruent is a parallelogram, so a square is a parallelogram. QUESTIONING STRATEGIES How do the definitions of the special quadrilaterals in this lesson differ from the theorems in this lesson? The definitions only identify the figures as quadrilaterals. Proving the properties as theorems shows they are also parallelograms. © Houghton Mifflin Harcourt Publishing Company Elaborate ELABORATE Quadrilateral 14. Discussion The Venn diagram shows how quadrilaterals, parallelograms, rectangles, rhombuses, and squares are related to each other. From this lesson, what do you notice about the definitions and theorems regarding these figures? A rectangle, a rhombus, and a square are all Parallelogram Square Rectangle Rhombus defined as quadrilaterals, but they can only be shown to be parallelograms by theorems that prove their properties. 15. Essential Question Check-In What are the properties of rectangles and rhombuses? How does a square relate to rectangles and rhombuses? A rectangle is a parallelogram with four right angles and congruent diagonals. A rhombus is a parallelogram with four congruent sides, diagonals that are perpendicular, and each of SUMMARIZE THE LESSON Have students make a graphic organizer or chart to summarize properties of rectangles and rhombuses. Sample: its diagonals bisects a pair of opposite angles. A square is both a rectangle and a rhombus, so it has the properties of both. Module 24 1221 Lesson 3 LANGUAGE SUPPORT IN1_MNLESE389762_U9M24L3.indd 1221 Rectangle Diagonals are congruent. Is a parallelogram 1221 Lesson 24.3 Rhombus Diagonals are perpendicular. Diagonals bisect a pair of opposite angles. Connect Vocabulary Students may have difficulty distinguishing the special quadrilaterals in this lesson. Have them write the definitions on a poster, and look through magazines or books to find pictures of rectangles, rhombuses, and squares that they can add to the poster. Have them group the pictures based on the type of figure, show the pictures to the class, and then describe the identifying characteristics of each type of figure. 4/18/14 8:04 PM Evaluate: Homework and Practice 1. EVALUATE Complete the paragraph proof of the Properties of Rectangles Theorems. Given: ABCD is a rectangle. _ _ Prove: ABCD is a parallelogram; AC ≅ BD. A B D C • Online Homework • Hints and Help • Extra Practice Proof that ABCD is a parallelogram : Since ABCD is a rectangle, ∠A and ∠C are right angles. So ∠A ≅ ∠C because all right angles are congruent. ASSIGNMENT GUIDE By similar reasoning, ∠B ≅ ∠D. Therefore, ABCD is a parallelogram because it is a quadrilateral that has congruent opposite angles. Proof that the diagonals are congruent: Since ABCD is a parallelogram, ― ― AD ≅ BC because opposite sides of a parallelogram are congruent _ _ Also, DC ≅ DC by the Reflexive Property of Congruence. By the definition of a . rectangle, ∠D and ∠C are right angles, and so ∠D ≅ ∠C because all right angles are congruent . Therefore, △ADC ≅ △BCD by the _ SAS Triangle Congruence Theorem and _ AC ≅ BD by CPCTC. Find the lengths using rectangle ABCD. AB = 21; AD = 28. What is the value of AC + BD? BD = 21 + 28 , so BD = 35. Then, AC = 35, so AC + BD = 70. 2 3. 2 2 BC = 40; CD = 30. What is the value of BD - AC? The diagonals of the rectangle are congruent, so BD = AC. Then, BD - AC = 0. 4. A D An artist connects stained glass pieces with lead strips. In this rectangular window, the strips are cut so that FH = 34 in. Find JG. Explain. F G J E HARDBOUND SE C © Houghton Mifflin Harcourt Publishing Company • Image Credits: Courtesy of Wimberley Stain Glass/Houghton Mifflin Harcourt Photo by Pet 2. B PAGE 992 BEGINS HERE EG also measures 34 inches. Since diagonals of a parallelogram bisect each other, 1 1( ) JG = _ EG. So, JG = _ 34 = 17 in. 2 2 Exercise IN1_MNLESE389762_U9M24L3.indd 1222 Lesson 3 Depth of Knowledge (D.O.K.) Mathematical Practices 1–3 2 Skills/Concepts MP.2 Reasoning 4–6 2 Skills/Concepts MP.4 Modeling 7–19 2 Skills/Concepts MP.2 Reasoning 20 3 Strategic Thinking MP.3 Logic 21 3 Strategic Thinking MP.3 Logic Explore Exploring Properties of Rectangles Exercise 1 Example 1 Proving Diagonals of a Rectangle Are Congruent Exercises 2–6 Example 2 Proving Diagonals of a Rhombus Are Perpendicular Exercise 8 Example 3 Using Properties of Rhombuses to Find Measures Exercises 9–12 Example 4 Investigating the Properties of a Square Exercise 19 approach for exploring special quadrilaterals. Have students make a physical model of a parallelogram with paper strips and brads. Ask them to adjust the model until it is a rectangle, as shown in an exercise, and then measure its diagonals. Then have them adjust the model to form a rhombus and measure the angles made by the intersecting diagonals. H 1222 Practice INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Some students may benefit from a hands-on _ _ EG ≅ FH because the diagonals of a rectangle are congruent. Since FH measures 34 inches, Module 24 Concepts and Skills 2/26/16 4:18 AM Properties of Rectangles, Rhombuses, and Squares 1222 The rectangular gate has diagonal braces. Find each length. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 You may want to review the properties of 5. J H L 30.8 in. 7. HK = JG = 2(JL) = 2(30.8 in.) = 48 in. = 61.6 in. Find the measure of each numbered angle in the rectangle. ∠3 and ∠5 are right angles, so they each measure 90°. ∠2 ≅ to the give angle, so m∠2 = 61°. m∠1 = 90° - 61° = 29°; ∠4 ≅ ∠1, so m∠4 = m∠1 = 29°. Complete the two-column proof that the diagonals of a rhombus are perpendicular. M Given: JKLM is a rhombus. ― ― Prove: JL ⊥ MK © Houghton Mifflin Harcourt Publishing Company _ _ 1. JM ≅ JK _ _ 2. MN ≅ KN 61° 4 2 3 K L Statements Reasons 1. Definition of rhombus 2. Diagonals of a parallelogram bisect each other. _ _ 3. JN ≅ JN 3. Reflexive Property of Congruence 4. △JNM ≅ △JNK 4. SSS Triangle Congruence Theorem 5. ∠JNM ≅ ∠JNK 5. CPCTC 6. ∠JNM and ∠JNK are supplementary. 6. Linear Pair Theorem 7. m∠JNM + m∠JNK = 180° 7. Definition of supplementary 8. ∠JNM = ∠JNK 8. Definition of congruence 9. m∠JNK + ∠JNK = 180° 9. Substitution Property of Equality 10. 2m∠JNK = 180° 10. Addition 11. m∠JNK = 90° 11. Division Property of Equality 12. JL ⊥ MK 12. Definition of perpendicular lines _ Exercise IN1_MNLESE389762_U9M24L3 1223 J N ge07se_c06l04003a AB _ Module 24 Lesson 24.3 1 5 So, m∠1 = 29°, m∠2 = 61°, m∠3 = 90°, m∠4 = 29°, and m∠5 = 90°. 8. 1223 Find HK. 6. HJ = KG G 48 in. K parallelograms before assigning the exercises. Then summarize the properties of rectangles and rhombuses and point out that these figures are also parallelograms. Explain that a square has all the properties of a parallelogram, a rectangle, and a rhombus. Find HJ. _ _ HJ ≅ KG Lesson 3 1223 Depth of Knowledge (D.O.K.) Mathematical Practices 4/25/14 8:44 PM 22 3 Strategic Thinking MP.3 Logic 23 3 Strategic Thinking MP.3 Logic ABCD is a rhombus. Find each measure. 9. B F 12y° Find AB. Possible answer: 4x + 15 PAGE 993 BEGINS HERE 7x + 2 A 4x + 15 = 7x + 2 HARDBOUND SE C (4y - 1)° D 13 = 3x 1 4_ =x 3 _ _ 1 1 AB ≅ CD, so AB = CD = 7x + 2 = 7 4_ + 2 = 32_ 3 3 10. Find m∠ABC. AVOID COMMON ERRORS Students might expect to use only the current lesson’s properties and theorems when writing proofs. Point out that proofs often build on previous knowledge and thus use concepts learned in earlier lessons.v ( ) Possible answer: _ AC ⊥ ¯ BD, so m∠AFB = 90°; 12y = 90, so y = 7.5. m∠ABC = 180° - 2(4y -1)° = 180° - 2(4 ⋅ 7.5 -1)° = 122° Find the measure of each numbered angle in the rhombus. 11. 12. 1 27° 5 2 3 2 3 4 70° 4 1 5 ∠2 ≅ ∠3 ≅ ∠5 ≅ to the given angle, so they each measure 27°. ∠4 ≅ to the given angle, so m∠4 = 70°. m∠1 + m∠2 + 27° = 180° m∠1 + m∠2 + 70° = 180° ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠5 m∠1 + 27° + 27° = 180° m∠1 + m∠1 + 70° = 180° ∠4 ≅ ∠1, so m∠4 = 126° So, m∠1 = 55°, m∠2 = 55°, m∠3 = 55°, m∠4 = 70° and m∠5 = 55°. m∠1 = 126° m∠1 = 55° 13. Select the word that best describes when each of the following statements are true. Select the correct answer for each lettered part. A. A rectangle is a parallelogram. always sometimes never B. A parallelogram is a rhombus. always sometimes never C. A square is a rhombus. always sometimes never D. A rhombus is a square. always sometimes never E. A rhombus is a rectangle. always sometimes never Module 24 IN1_MNLESE389762_U9M24L3 1224 1224 © Houghton Mifflin Harcourt Publishing Company So, m∠1 = 126°, m∠2 = 27°, m∠3 = 27°, m∠4 = 126°, and m∠5 = 27°. Lesson 3 9/30/14 12:49 PM Properties of Rectangles, Rhombuses, and Squares 1224 14. Use properties of special parallelograms to complete the proof. _ Given: EFGH is a rectangle. J is the midpoint of EH. Prove: △FJG is isosceles. Statements F E G H J Reasons 1. EFGH is a rectangle. J is the _ midpoint of EH. 1. Given 2. ∠E and ∠H are right angles. 2. Definition of rectangle 3. ∠E ≅ ∠H 3. All right angles are congruent. 4. EFGH is a parallelogram. 4. A rectangle is a parallelogram. 5. EF ≅ GH 5. In a parallelogram, opposite _ _ _ _ sides are congruent. 6. EJ ≅ HJ 6. Definition of midpoint 7. △FJE ≅ △GJH 7. SAS Triangle Congruence Theorem _ _ 8. FJ ≅ GJ 9. △FJE is isosceles. 8. CPCTC 9. Definition of isosceles triangle 15. Explain the Error Find and explain the error in this paragraph proof. Then describe a way to correct the proof. K © Houghton Mifflin Harcourt Publishing Company Given: JKLM is a rhombus. J Prove: JKLM is a parallelogram. Proof: It is given that JLKM is a rhombus. So, by the definition of a rhombus, ― ― ― ― JK ≅ LM, and KL ≅ MJ. If a quadrilateral is a parallelogram, then its opposite sides are congruent. So JKLM is a parallelogram. IN1_MNLESE389762_U9M24L3.indd 1225 Lesson 24.3 M You cannot use a theorem that assumes the quadrilateral is a parallelogram to justify the final statement because you do not know that JKLM is a parallelogram. That is what you are trying to prove. Instead, use the converse, which states that if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. So therefore, JKLM is a parallelogram. Module 24 1225 L 1225 Lesson 3 4/18/14 8:03 PM HARDBOUND SE The opening of a soccer goal is shaped like a rectangle. PAGE 994 BEGINS HERE 16. Draw a rectangle to represent a soccer goal. Label the rectangle ABCD _ to show that the distance between the goalposts, BC, is three times the distance from the top of the goalpost to the ground. If_ the perimeter of ABCD is 64 feet, what is the length of BC? Possible drawing: 3x B x A C D Perimeter = AB + BC + CD + DA 64 = x + 3x + x + 3x 64 = 8x 8=x BC = 3x = 3(8) = 24 feet PEERTOPEER DISCUSSION Ask students to work with a partner to write conditional statements in the form “If a quadrilateral is a [figure], then it is a [figure].” An example would be “If a quadrilateral is a rectangle, then it is a parallelogram.” Also have them write conditional statements in the form “If a parallelogram is a [figure], then it is a [figure],” for example, “If a parallelogram is a square, then it is a rhombus.” Have them write as many of these statements as possible and then compare their statements with other student pairs. 17. In your rectangle from Evaluate 16, suppose the distance from B to D is (y + 10) feet, _ and the distance from A to C is (2y - 5.3) feet. What is the approximate length of AC? _ _ The diagonals of a rectangle are congruent, so AC ≅ BD. AC = BD 2y - 5.3 = y + 10 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©belterz/ iStockPhoto.com y = 15.3 So, AC = 2(15.3) - 5.3 = 30.6 - 5.3 = 25.3 feet. (7b - 5) meters and QR_ = (2b - 0.5) 18. PQRS is a rhombus, with PQ =_ meters. If S is the midpoint of RT, what is the length of RT? Possible answer: Q P R _ _ By the midpoint, TS ≅ RS. _ _ _ _ By the definition of a rhombus, PQ ≅ QR ≅ RS ≅ SP. PQ = QR S T 7b - 5 = 2b − 0.5 b = 0.9 RT = TS + RS = 2(RS) = 2(QR) = 2(2b − 0.5) = 2(2(0.9) − 0.5) = 2(1.8 − 0.5) = 2(1.3) = 2.6 _ So, the length of RT is 2.6 meters. Module 24 IN1_MNLESE389762_U9M24L3.indd 1226 1226 Lesson 3 4/18/14 8:03 PM Properties of Rectangles, Rhombuses, and Squares 1226 19. Communicate Mathematical Ideas List the properties that a square “inherits” because it is each of the following quadrilaterals. JOURNAL Have students describe the properties of a rectangle and a rhombus, then have them draw and label examples of each type of figure. a. a parallelogram • Both pairs of opposite sides are parallel. • Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • One angle is supplementary to both of its consecutive angles. • The diagonals bisect each other. b. a rectangle • The diagonals are congruent. c. a rhombus • The diagonals are perpendicular. • Each diagonal bisects a pair of opposite angles. H.O.T. Focus on Higher Order Thinking Justify Reasoning For the given figure, describe any rotations or reflections that would carry the figure onto itself. Explain. 20. A rhombus that is not a square 180° rotation around its center; reflectional symmetry across a line that contains opposite vertices (two lines) 21. A rectangle that is not a square © Houghton Mifflin Harcourt Publishing Company 180° rotation around its center; reflectional symmetry across a line that contains the midpoints of opposite sides (two lines) 22. A square 90° rotation around its center; reflectional symmetry across a line that contains opposite vertices (two lines), or a line that contains the midpoints of opposite sides (two lines) 23. Analyze Relationships Look at your answers for Exercises 20–22. How does your answer to Exercise 22 relate to your answers to Exercises 20 and 21? Explain. A square has the reflectional properties of both a rhombus that is not a square and a rectangle that is not a square. Because squares have all angles congruent and all sides congruent, as opposed to only all sides congruent (rhombuses) or only all angles congruent (rectangles), a square has 90° rotational symmetry. Module 24 IN1_MNLESE389762_U9M24L3.indd 1227 1227 Lesson 24.3 1227 Lesson 3 4/18/14 8:03 PM Lesson Performance Task INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Have students look at the rhombus on the The portion of the Arkansas state flag that is not red is a rhombus. On one flag, the diagonals of the rhombus measure 24 inches and 36 inches. Find the area of the rhombus. Justify your reasoning. Arkansas flag with diagonals measuring 24 inches and 36 inches. Pose these questions: • Suppose you cut the rhombus along its diagonals to make four shapes, and then rearranged the shapes to form another type of quadrilateral that you studied in this lesson. Name the quadrilateral and its dimensions. a rectangle measuring 18 in. × 24 in. or 12 in. × 36 in. 432 in. 2 Possible answer: Draw the diagonals of the rhombus. Since they are perpendicular, the four triangles into which you have divided the rhombus 1 1 ( )( ) are right triangles. The area of each is _ bh = _ 18 12 = 108 in 2. There are 2 2 four such triangles with a total area of 4 ∙ 108 = 432 in 2. 12 in. □ Module 24 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©David H. Lewis/E+/Getty Images 18 in. 1228 • Find the area of the new quadrilateral. Then propose a method for finding the area of a rhombus if you know the lengths of its diagonals. Divide the product of the diagonals by 2. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 What is the relationship between the combined areas of the four red triangles on an Arkansas flag, and the area of the rest of the flag? Answer without referring to the actual dimensions of the flag. They are equal. Sample answer: The four red triangles can be rearranged to form the rhombus that makes up the rest of the flag. So, the two are equal in area. Lesson 3 EXTENSION ACTIVITY IN1_MNLESE389762_U9M24L3.indd 1228 The design of the Arkansas flag is composed of a number of geometrical shapes. Ask students to write and solve at least four problems involving the names, perimeters, areas, and/or other information relating to the shapes on the flag. Students should continue to use 24 inches and 36 inches as the lengths of the diagonals of the larger rhombus. 4/18/14 8:03 PM Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain. 0 points: Student does not demonstrate understanding of the problem. Properties of Rectangles, Rhombuses, and Squares 1228