Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Line (geometry) wikipedia , lookup
Rational trigonometry wikipedia , lookup
Multilateration wikipedia , lookup
Four color theorem wikipedia , lookup
Integer triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Pythagorean theorem wikipedia , lookup
LESSON 24.1 Name Properties of Parallelograms Class Date 24.1 Properties of Parallelograms Essential Question: What can you conclude about the sides, angles, and diagonals of a parallelogram? Common Core Math Standards Explore The student is expected to: Resource Locker Investigating Parallelograms A quadrilateral is a polygon with four sides. A parallelogram is a quadrilateral that has two pairs of parallel sides. You can use geometry software to investigate properties of parallelograms. G-CO.11 Prove theorems about parallelograms. Also G-SRT.5 Mathematical Practices MP.3 Logic Language Objective Explain to a partner why pictures of quadrilaterals are or are not parallelograms. ENGAGE Draw a straight line. Then plot a point that is not on the line. Construct a line through the point that is parallel to the line. This gives you a pair of parallel lines. Repeat Step A to construct a second pair of parallel lines that intersect those from Step A. The intersections of the parallel lines create a parallelogram. Plot points at these intersections. Label the points A, B, C, and D. Opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo, asking students to identify the geometric figures formed by a scissor lift. Then preview the Lesson Performance Task. © Houghton Mifflin Harcourt Publishing Company Essential Question: What can you conclude about the sides, angles, and diagonals of a parallelogram? Identify the opposite sides and opposite angles of the parallelogram. _ _ _ _ Opposite sides: Side AB is opposite side DC. Side AD is opposite side BC. Opposite angles: ∠A is opposite ∠C. ∠B is opposite ∠D. Module 24 be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction Lesson 1 1189 gh "File info" made throu Date Class Name perties 24.1 Pro ograms of Parallel s, and sides, angle about the conclude can you ion: What elogram? G-SRT.5 of a parall lograms. Also about paralle theorems diagonals Resource Locker Quest Essential G-CO.11 Prove HARDCOVER PAGES 965974 g Para Investigatin lateral that is a quadri ties elogram gate proper sides. A parallsoftware to investi try n with four l is a polygo can use geome A quadrilateraof parallel sides. You has two pairs s. logram of paralle Explore IN1_MNLESE389762_U9M24L1.indd 1189 s llelogram a line Construct lines. on the line. of parallel that is not you a pair plot a point This gives t line. Then to the line. Draw a straigh is parallel point that through the n Mifflin Harcour t Publishin y g Compan Turn to these pages to find this lesson in the hardcover student edition. © Houghto A to Repeat Step Step A. construct a second el lines that pair of parall intersect those from m. Plot a parallelogra el lines create A, B, C, and D. the parall the points ctions of _ The interse ctions. Label elogram. these interse of the parall side BC. _ points at te angles_ is opposite and opposi Side AD opposite sides side DC. _ Identify the opposite Side AB is site ∠D. sides: ∠B is oppo site ∠C. Opposite oppo is ∠A angles: Opposite Lesson 1 1189 Module 24 4L1.indd 62_U9M2 ESE3897 IN1_MNL 1189 Lesson 24.1 1189 4/18/14 7:44 PM 4/18/14 7:47 PM Measure each angle of the parallelogram. EXPLORE Measure the length of each side of the parallelogram. You can do this by measuring the distance between consecutive vertices. Investigating Parallelograms INTEGRATE TECHNOLOGY Students have the option of doing the parallelogram activity either in the book or online. Then drag the points and lines in your construction to change the shape of the parallelogram. As you do so, look for relationships in the measurements. Make a conjecture about the sides and angles of a parallelogram. QUESTIONING STRATEGIES As you drag points, does the quadrilateral remain a parallelogram? Yes, the lines that form opposite sides remain parallel. Conjecture: Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. A segment that connects two nonconsecutive vertices of a polygon is a diagonal. _ ¯ and BD. Plot a point at the intersection of the diagonals and Construct diagonals AC label it E. ¯, BE ¯, and DE ¯, CE ¯. Measure the length of AE Drag the points and lines in your construction to change the shape of the parallelogram. As you do so, look for relationships in the measurements in Step G. Make a conjecture about the diagonals of a parallelogram. What do you notice about consecutive angles in the parallelogram? Why does this make sense? Consecutive angles are supplementary. This makes sense because opposite sides are parallel, so consecutive angles are same-side interior angles. By the Same-Side Interior Angles Postulate, these angles are supplementary. © Houghton Mifflin Harcourt Publishing Company Conjecture: The diagonals of a parallelogram bisect each other. Reflect 1. Consecutive angles are the angles at consecutive vertices, such as ∠A and ∠B, or ∠A and ∠D. Use your construction to make a conjecture about consecutive angles of a parallelogram. Conjecture: Consecutive angles of a parallelogram are supplementary. Module 24 1190 Lesson 1 PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U9M24L1.indd 1190 4/18/14 7:47 PM Math Background In this lesson, students extend their earlier work with triangle congruence criteria and triangle properties to prove facts about parallelograms. A parallelogram is a quadrilateral whose opposite sides are parallel. Like every polygon, parallelograms are named by listing consecutive vertices. Because of this convention, the pairs of parallel sides can be identified from the parallelogram’s name. For example, _ _ _ _ _ _ _ _ ▱JKLM has sides JK , KL, LM, and MJ with JK || LM and KL || MJ . This relationship is easily verified by sketching a parallelogram with consecutive vertices J, K, L, and M. Properties of Parallelograms 1190 2. EXPLAIN 1 Proving Opposite Sides Are Congruent Critique Reasoning A student claims that the perimeter of △AEB in the construction is always equal to the perimeter of △CED. Without doing any further measurements in your construction, explain whether or not you agree with the student’s statement. Agree; AE = CE, BE = DE, and BA = DC, since the diagonals of the parallelogram bisect each other and the opposite sides are congruent. So AE + EB + BA = CE + ED + DC. Explain 1 INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 The proof that the opposite sides of a The conjecture you made in the Explore about opposite sides of a parallelogram can be stated as a theorem. The proof involves drawing an auxiliary line in the figure. Theorem If a quadrilateral is a parallelogram, then its opposite sides are congruent. parallelogram are congruent depends on students understanding that the opposite sides of a quadrilateral do not share a vertex (that is, they do not intersect). Consecutive sides of a quadrilateral do share a vertex (that is, they intersect). You may want to help students draw and label a quadrilateral for reference. Example 1 Prove that the opposite sides of a parallelogram are congruent. Given: ABCD is a parallelogram. _ _ _ _ Prove: AB ≅ CD and AD ≅ CB A B D C Statements 1. ABCD is a parallelogram. _ 2. Draw DB. _ _ _ _ 3. AB∥DC, AD∥BC QUESTIONING STRATEGIES 4. ∠ADB ≅ ∠CBD ∠ABD ≅ ∠CDB _ _ 5. DB ≅ DB © Houghton Mifflin Harcourt Publishing Company Why do you think the proof is based on drawing a diagonal of the parallelogram? Drawing the diagonal creates two triangles, which lets you use triangle congruence criteria and the fact that corresponding parts of congruent triangles are congruent. Proving Opposite Sides Are Congruent 6. △ABD ≅ △CDB _ _ _ _ 7. AB ≅ CD and AD ≅ CB Reasons 1. Given 2. Through any two points, there is exactly one line. 3. Definition of parallelogram 4. Alternate Interior Angles Theorem 5. Reflexive Property of Congruence 6. ASA Triangle Congruence Theorem 7. CPCTC Reflect 3. Explain how you can use the rotational symmetry of a parallelogram to give an argument that supports the above theorem. Under a 180° rotation around the center, each side is mapped to its opposite side. Since rotations preserve distance, this shows that opposite sides are congruent. Module 24 1191 Lesson 1 COLLABORATIVE LEARNING IN1_MNLESE389762_U9M24L1 1191 Small Group Activity Using geometry software, have one student construct a parallelogram. Ask a second student to measure both the opposite angles and the opposite sides of the parallelogram to verify the corresponding theorems in this lesson. Ask a third student to add the diagonals to the parallelogram and use the measuring features to verify that the diagonals of a parallelogram bisect each other. Ask a fourth student to verify that the consecutive angles of a parallelogram are supplementary by verifying this property. As students change the parallelogram, opposite sides and angles remain congruent. 1191 Lesson 24.1 9/30/14 12:02 PM Explain 2 Proving Opposite Angles Are Congruent EXPLAIN 2 The conjecture from the Explore about opposite angles of a parallelogram can also be proven and stated as a theorem. Proving Opposite Angles Are Congruent Theorem If a quadrilateral is a parallelogram, then its opposite angles are congruent. Prove that the opposite angles of a parallelogram are congruent. Example 2 A Given: ABCD is a parallelogram. Prove: ∠A ≅ ∠C (A similar proof shows that ∠B ≅ ∠D.) Statements D C The proof that the opposite angles of a parallelogram are congruent depends on students understanding the difference between opposite angles and consecutive angles. The opposite angles of a quadrilateral do not share a side, while the consecutive angles of a quadrilateral do share a side. You may want to help students draw and label a parallelogram for reference. Reasons 1. ABCD is a parallelogram. _ 2. Draw DB. 2. Through any two points, there is exactly _ _ _ _ 3. AB∥DC, AD∥BC 3. Definition of parallelogram 4. ∠ADB ≅ ∠CBD, 4. Alternate Interior Angles Theorem ∠ABD ≅ ∠CDB _ _ 5. DB ≅ DB CONNECT VOCABULARY B 1. Given one line. 5. Reflexive Property of Congruence 6. △ABD ≅ △CDB 6. ASA Triangle Congruence Theorem 7. ∠A ≅ ∠C 7. CPCTC QUESTIONING STRATEGIES How are the opposite angles of a parallelogram related? They are congruent. Reflect 4. Explain how the proof would change in order to prove ∠B ≅ ∠D. _ In the second step, draw the diagonal AC. The remaining steps are all How is the proof of this theorem similar to the proof that the opposite sides of a parallelogram are congruent? They both start with drawing a diagonal and then using a triangle congruence criterion and the fact that corresponding parts of congruent traingles are congruent. similar to those in the above proof. © Houghton Mifflin Harcourt Publishing Company 5. In Reflect 1, you noticed that the consecutive angles of a parallelogram are supplementary. This can be stated as the theorem, If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Explain why this theorem is true. Each pair of consecutive angles of a parallelogram are same-side interior angles for a pair of parallel lines (the opposite sides of the parallelogram), so the angles are supplementary by the Same-Side Interior Angles Postulate. Explain 3 Proving Diagonals Bisect Each Other EXPLAIN 3 The conjecture from the Explore about diagonals of a parallelogram can also be proven and stated as a theorem. One proof is shown on the facing page. Proving Diagonals Bisect Each Other Theorem If a quadrilateral is a parallelogram, then its diagonals bisect each other. Module 24 1192 Lesson 1 DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U9M24L1.indd 1192 Modeling Give each group a different model parallelogram made with straws. Ask each student in the group to use measuring tools to investigate the opposite angles and the opposite sides of their parallelogram and then record the results in a table. Since a parallelogram is not a rigid figure, ask students to change the angle measures of their parallelogram, redo the measurements and record their results in the table. After several parallelograms are explored, have the group make a conjecture about the opposite angles and opposite sides of a parallelogram, and then present their data and conjecture to the class. 4/18/14 7:47 PM INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 As students work on the proof in this lesson, ask them to think about how the format of the proof makes it easier to understand the underlying structure of the argument. Students should recognize that a flow proof shows how one statement connects to the next, which may not be as apparent in a two-column format. A two-column format, on the other hand, may make the justification for each step clearer than a flow proof. Properties of Parallelograms 1192 Example 3 QUESTIONING STRATEGIES Why do you think this theorem was introduced after the theorems about the sides and angles of a parallelogram? The proof of this theorem depends upon the fact that opposite sides of a parallelogram are congruent. Complete the flow proof that the diagonals of a parallelogram bisect each other. Given: ABCD is a parallelogram. _ _ _ _ Prove: AE ≅ CE and BE ≅ DE A B E D C ABCD is a parallelogram. Given AB ‖ DC AVOID COMMON ERRORS AB ≅ DC Definition of parallelogram Students may be confused when asked to label the congruent parts of the parallelogram in this proof. Have students use one color pencil to mark the pairs of congruent sides, a second color pencil to mark the pairs of congruent angles on the parallelogram, and a third and fourth color pencil to mark the congruent segments on the diagonals of the parallelogram. Opposite sides of a parallelogram are congruent. ∠ABE ≅ ∠CDE ∠BAE ≅ ∠DCE Alternate Interior Angles Theorem Alternate Interior Angles Theorem △ABE ≅ △CDE ASA Triangle Congruence Theorem AE ≅ CE and BE ≅ DE CPCTC Reflect © Houghton Mifflin Harcourt Publishing Company 6. Discussion Is it possible to prove the theorem using a different triangle congruence theorem? Explain. Yes; ∠AEB ≅ ∠CED because they are vertical angles. Together with ¯, you can prove the theorem using the ¯ ≅ CD ∠ABE ≅ ∠CDE and AB AAS Triangle Congruence Theorem. Module 24 1193 Lesson 1 LANGUAGE SUPPORT IN1_MNLESE389762_U9M24L1 1193 Communicate Math Provide each pair of students with pictures of different quadrilaterals, including rectangles, squares, and other parallelograms, and rulers and protractors. The first student chooses a picture, measures angles and side lengths, and tells the second student why this picture is or is not a parallelogram. The second student writes notes about the picture as the first student explains. Students switch roles and repeat the process. 1193 Lesson 24.1 9/30/14 12:05 PM Explain 4 Using Properties of Parallelograms EXPLAIN 4 You can use the properties of parallelograms to find unknown lengths or angle measures in a figure. Example 4 ABCD is a parallelogram. Find each measure. B AD Use_ the fact _that opposite sides of a parallelogram are congruent, so AD ≅ CB and therefore AD = CB. 5x + 19 (6y + 5)° A (8y − 17)° 7x = 5x + 19 Write an equation. Using Properties of Parallelograms C 7x AVOID COMMON ERRORS D As students prepare to solve these types of problems, they may get confused about which segments or angles correspond. Ask them to mark the congruent parts carefully and then think about which property applies to the information that is marked. x = 9.5 Solve for x. AD = 7x = 7(9.5) = 66.5 m∠B Use the fact that opposite angles of a parallelogram are congruent, so ∠B ≅ ∠ D and therefore m∠B = m∠ D . Solve for y. QUESTIONING STRATEGIES 6y + 5 = 8y - 17 Write an equation. (( ) ) m∠B = (6y + 5)° = 6 11 + 5 ° How are the opposite sides of a parallelogram related? How are the diagonals of a parallelogram related? The opposite sides are congruent and the diagonals bisect each other. 11 = y = 71 ° Reflect 7. Suppose you wanted to find the measures of the other angles of parallelogram ABCD. Explain your steps. Possible answer (using congruent opposite angles; can also be done using supplementary consecutive angles): Since the sum of the measures of a © Houghton Mifflin Harcourt Publishing Company quadrilateral is 360°, m∠A + 71° + m∠C + 71° = 360°. Since opposite angles of a parallelogram are congruent, ∠A ≅ ∠C, so m∠A + 71° + m∠A + 71° = 360°. Solving shows that m∠A = 109°. Therefore, m∠A = m∠C = 109°. Module 24 IN1_MNLESE389762_U9M24L1 1194 1194 Lesson 1 9/30/14 12:06 PM Properties of Parallelograms 1194 Your Turn ELABORATE PQRS is a parallelogram. Find each measure. 8. QUESTIONING STRATEGIES QR _ _ PS ≅ QR, so PS = QR P 2z + 4 = 3z - 4 What do we have to know first to apply any of these theorems? The quadrilateral is a parallelogram. 2z + 4 8= z 9. PR S QR = 3z - 4 = 3(8) - 4 = 20 x+ 9 Q T 4x - 6 3z - 4 R _ _ PT ≅ RT, so PT = RT x + 9 = 4x - 6 5=x SUMMARIZE THE LESSON PT = x + 9 = 5 + 9 = 14 PR = 2PT, so PR = 2(14) = 28 Have students make a graphic organizer to summarize what they know about the sides, angles, and diagonals of a parallelogram. Elaborate 10. What do you need to know first in order to apply any of the theorems of this lesson? You must know that the given quadrilateral is a parallelogram. Sample: _ 11. In parallelogram ABCD, point P lies on DC, as shown in the figure. Explain why it must be the case that DC = 2AD. Use what you know about base angles of an isosceles triangle. Parallelogram A D B A x° x° C D AB ≅ DC AD ≅ BC Opposite angles are congruent ∠A ≅ ∠C ∠B ≅ ∠D The diagonals bisect each other If E is the point where diagonals AC and BD intersect, then AE ≅ CE and BE ≅ DE sides of a parallelogram. So, DC = DP + PC = AD + BC = AD + AD = 2AD. 12. Essential Question Check-In JKLM is a parallelogram. J N Name all of the congruent segments and angles in the figure. M __ __ __ _ ¯≅ LM, JM ≅ LK, JN ≅ LN, MN ≅ KN, ∠KJM ≅ ∠MLK, ∠JKL ≅ ∠LMJ JK IN1_MNLESE389762_U9M24L1 1195 Lesson 24.1 C Alternate Interior Angles Theorem. Therefore, ∠DAP ≅ ∠DPA. This means _ _ _ _ _ _ △DAP is isosceles, with AD ≅ DP. Similarly, BC ≅ PC. Also, BC ≅ AD as opposite Module 24 1195 z° P B _ ∠DAP ≅ ∠BAP since AP is an angle bisector. Also, ∠DPA ≅ ∠BAP by the © Houghton Mifflin Harcourt Publishing Company Opposite sides are congruent y° y° 1195 K L Lesson 1 9/30/14 12:08 PM Evaluate: Homework and Practice 1. 2. Sabina has tiles in the shape of a parallelogram. She labels the angles of each tile as ∠A, ∠B, ∠C, and ∠D. Then she arranges the tiles to make the pattern shown here and uses the pattern to make a conjecture about opposite angles of a parallelogram. What conjecture does she make? How does the pattern help her make the conjecture? EVALUATE • Online Homework • Hints and Help • Extra Practice Pablo traced along both edges of a ruler to draw two pairs of parallel lines, as shown. Explain the next steps he could take in order to make a conjecture about the diagonals of a parallelogram. _ Possible _ answer: He can use the ruler to draw JL and KM, label their intersection as point N, and use the ruler to find that JN = LN and KN = MN. His conjecture would be that the diagonals of a parallelogram bisect each other. K J N L M ASSIGNMENT GUIDE A B A B A B D C D C D C A B A B A B D C D C D C Possible conjecture: Opposite angles of a parallelogram are congruent. In the pattern, vertical angles ∠A and ∠C, are congruent, as are vertical angles and ∠B and ∠D. 3. Complete the flow proof that the opposite sides of a parallelogram are congruent. Given: ABCD is a parallelogram. _ _ _ _ Prove: AB ≅ CD and AD ≅ CB A B D C ABCD is a parallelogram. Given Draw DB. Definition of parallelogram Through any two points, there is exactly one line. ∠ADB ≅ ∠CBD, ∠ABD ≅ ∠CDB DB ≅ DB © Houghton Mifflin Harcourt Publishing Company AB || DC , AD || BC Reflex. Prop. of Cong. Alt. Int. Angles Thm. ∆ABD ≅ ∆CDB CPCTC Exercise IN1_MNLESE389762_U9M24L1.indd 1196 Lesson 1 Depth of Knowledge (D.O.K.) Mathematical Practices 1–2 2 Skills/Concepts MP.5 Using Tools 3–5 2 Skills/Concepts MP.3 Logic 6–9 2 Skills/Concepts MP.4 Modeling 10–13 2 Skills/Concepts MP.4 Modeling 14–16 2 Skills/Concepts MP.3 Logic 17–24 2 Skills/Concepts MP.2 Reasoning 3 Strategic Thinking MP.4 Modeling 25 Explore Investigating Parallelograms Exercises 1–2 Example 1 Proving Opposite Sides Are Congruent Exercise 3 Example 2 Proving Opposite Angles Are Congruent Exercise 4 Example 3 Proving Diagonals Bisect Each Other Exercise 5 Example 4 Using Properties of Parallelograms Exercises 6–13 approach for finding the properties of parallelograms. Have students copy a larger version of the parallelogram in an exercise onto a sheet of paper and then cut it out. Make sure the diagonals are drawn in. Have them use a ruler to find the segment lengths of the diagonals and verify that the diagonals bisect each other. Students can then cut the parallelogram along one diagonal and place the opposite sides on top of each other to confirm that they are coincident, and therefore congruent. AB ≅ CD and AD ≅ CB 1196 Practice INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Some students may benefit from a hands-on ASA Cong. Thm. Module 24 Concepts and Skills 4/18/14 7:47 PM Properties of Parallelograms 1196 4. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Encourage students to copy the A Write the proof that the opposite angles of a parallelogram are congruent as a paragraph proof. B D Given: ABCD is a parallelogram. C Prove: ∠A ≅ ∠C (A similar proof shows that ∠B ≅ ∠D.) parallelograms in the exercises onto their papers and use colored pencils to keep track of the congruent angles and segments. 5. _ _ Possible answer: It is given that ABCD is a parallelogram, so AB∥DC __ and AD∥BC by the def. of a parallelogram. By the Alt. Int. Angles Thm., ∠ADB ≅ ∠CBD and _ ∠ABD ≅ ∠CDB. Since two points determine a line, you __ can draw DB. DB ≅ DB by the Reflex. Prop. of Cong. You can conclude that △ABD ≅ △CDB by the ASA Cong. Thm., and so, ∠A ≅ ∠C by CPCTC. Write the proof that the diagonals of a parallelogram A B E bisect each other as a two-column proof. D Given: ABCD is a parallelogram. _ _ _ _ Prove: AE ≅ CE and BE ≅ DE C Statements Reasons 1. ABCD is a parallelogram. 1. Given _ _ 2. AB ∥ DC 2. Definition of parallelogram 3. ∠ABE ≅ ∠CDE , ∠BAE ≅ ∠DCE _ _ 4. AB ≅ DC 3. Alt. Int. Angles Thm. 4. Opposite sides of a parallelogram are congruent. 5. △ABE ≅ △CDE _ _ _ _ 6. AE ≅ CE and BE ≅ DE 5. ASA Triangle Cong. Thm. 6. CPCTC EFGH is a parallelogram. Find each measure. © Houghton Mifflin Harcourt Publishing Company 6. 7. FG _ _ HE ≅ FG; 5z - 16 = 3z + 8; z = 12; FG = 44 EG H ― ― EJ ≅ GJ; 4w + 4 = 2w + 22; w = 9; EJ = 40; EG = 2EJ, 80 m∠B AD 2w + 22 3y - 1 (9x - 5)° B D (10x - 19)° Exercise IN1_MNLESE389762_U9M24L1.indd 1197 1197 Lesson 24.1 G y + 15 ― ― AD ≅ CB; 3y - 1 = y + 15; y = 8; AD = 23 Module 24 3z + 8 A ∠B ≅ ∠D; 9x - 5 = 10x - 19; 14 = x; m∠B = 121° 9. F J 5z - 16 ABCD is a parallelogram. Find each measure. 8. 4w + 4 E Lesson 1 1197 Depth of Knowledge (D.O.K.) C Mathematical Practices 26 3 Strategic Thinking MP.3 Logic 27 3 Strategic Thinking MP.2 Reasoning 4/18/14 7:47 PM A staircase handrail is made from congruent parallelograms. In ▱PQRS, PQ = 17.5, ST = 18, and m∠QRS = 110°. Find each measure. Explain. Q 10. RS R T Opp. sides of PRQS are congruent, so RS = PQ = 17.5. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 When working with proofs related to P properties of parallelograms, encourage students to focus on the accuracy of the hypothesis and conclusion of each statement. In this lesson, the hypothesis of each proof includes a statement that a quadrilateral is a parallelogram. The conclusion of the proof will be a property of the parallelogram. S 11. QT The diag. of PRQS bisect each other, so QT = ST = 18. 12. m∠PQR Consec. angles of PRQS are supplementary, so m∠PQR = 70°. 13. m∠SPQ Opp. angles of PRQS are congruent, so m∠SPQ = m∠QRS = 110°. Write each proof as a two-column proof. 14. Given: GHJN and JKLM are parallelograms. H Prove: ∠G ≅ ∠L J G K L N Statements 1. GHJN and JKLM are parallelograms. 3. ∠HJN ≅ ∠KJM 4. ∠G ≅ ∠L Reasons 1. Given 2. Opp. angles of a ▱ are congruent. 3. Vertical angles are congruent. 4. Transitive Property of Congruence _ _ 15. Given: PSTV is a parallelogram. PQ ≅ RQ Q Prove: ∠STV ≅ ∠R S P Statements T V R Reasons 1. PSTV is a parallelogram. 1. Given 2. ∠STV ≅ ∠P 2. Opp. angles of a ▱ are congruent. ― ― 3. PQ ≅ RQ 4. △PQR is isosceles. 5. ∠P ≅ ∠R 6. ∠STV ≅ ∠R Module 24 IN1_MNLESE389762_U9M24L1.indd 1198 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Byjeng/ Shutterstock 2. ∠G ≅ ∠HJN, ∠KJM ≅ ∠L M 3. Given 4. Definition of isosceles triangle 5. Isosceles Triangle Theorem 6. Transitive Property of Congruence 1198 Lesson 1 4/18/14 7:46 PM Properties of Parallelograms 1198 16. Given: ABCD and AFGH are parallelograms. AVOID COMMON ERRORS B Prove: ∠C ≅ ∠G Advise students to pay close attention to the markings on a diagram, especially when writing a proof. Explain that a quadrilateral with only one set of parallel lines is not a parallelogram. C F G A Statements 1. ABCD and AFGH are H D Reasons 1. Given parallelograms. 2. ∠C ≅ ∠A, ∠A ≅ ∠G 2. Opposite angles of a parallelogram are congruent. 3. ∠C ≅ ∠G 3. Trans. Prop. of Cong. Justify Reasoning Determine whether each statement is always, sometimes, or never true. Explain your reasoning. _ _ 17. If quadrilateral RSTU is a parallelogram, then RS ≅ ST. Sometimes; opposite sides of a parallelogram are congruent, but ― ― consecutive sides, such as RS and ST , may or may not be congruent. 18. If a parallelogram has a 30° angle, then it also has a 150° angle. Always; consecutive angles of a parallelogram are supplementary, so the angle that is a consecutive angle to the 30° angle must measure 150°. _ _ 19. If quadrilateral GHJK is a parallelogram, then GH is congruent to JK . Always; opposite sides of a parallelogram are congruent. 20. In parallelogram ABCD, ∠A is acute and ∠C is obtuse. © Houghton Mifflin Harcourt Publishing Company Never; opposite angles of a parallelogram are congruent; ∠A and ∠C are opposite angles, so they must have the same measure. _ _ 21. In parallelogram MNPQ, the diagonals MP and NQ meet at R with MR = 7 cm and RP = 5 cm. Never; diagonals of a parallelogram bisect each other. Module 24 IN1_MNLESE389762_U9M24L1.indd 1199 1199 Lesson 24.1 1199 Lesson 1 4/18/14 7:46 PM 22. Communicate Mathematical Ideas Explain how you can use the rotational symmetry of a parallelogram to give an argument that supports the fact that opposite angles of a parallelogram are congruent. PEERTOPEER DISCUSSION Ask students to discuss with a partner how to create a graphic organizer that will display all the properties of parallelograms that they have learned in the lesson. Then have them create the graphic organizer, making sure they include the following properties: Under a 180° rotation around the center of the parallelogram, each angle is mapped to its opposite angle. Since rotations preserve angle measure, this shows that opposite angles are congruent. 23. To repair a large truck or bus, a mechanic might use a parallelogram lift. The figure shows a side view of the lift. FGKL, GHJK, and FHJL are parallelograms. F G 1 2 5 L 3 6 H 4 7 8 K • Opposite sides are parallel. J • Opposite sides are congruent. a. Which angles are congruent to ∠1? Explain. _ _ ∠3, ∠6, ∠8; ∠3 ≅ ∠1 since FL ǁ GK and ∠3 and ∠1 are corresponding angles; ∠6 ≅ ∠1 since opposite angles of a parallelogram are congruent; ∠8 ≅ ∠1 since opposite angles of a parallelogram are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. b. What is the relationship between ∠1 and each of the remaining labeled angles? Explain. ∠1 is supplementary to ∠2, ∠4, ∠5, and ∠7; ∠1 is supplementary to ∠2, ∠4, and ∠5 since consecutive angles of a parallelogram are supplementary. Because opposite angles of a parallelogram are congruent, ∠7 ≅ ∠4. So if ∠1 is supplementary to ∠4, then ∠1 is also supplementary to ∠7. 24. Justify Reasoning ABCD is a parallelogram. Determine A B whether each statement must be true. Select the correct E answer for each lettered part. Explain your reasoning. D D. ∠DAC ≅ ∠BCA E. △AED ≅ △CEB F. ∠DAC ≅ ∠BAC Yes No Yes No Yes No Yes No Yes No Yes No C a. Yes; opposite sides of a parallelogram are congruent so AB = DC and AD = BC. The perimeter of ABCD is AB + BC + DC + AD = AB + BC + AB + BC = 2AB + 2BC. 1 DB. b. Yes; the diag. of a parallelogram bisect each other, so DE = EB and DE = _ 2 c. No; opp. sides are cong., but consecutive sides may or may not be cong. _ _ d. Yes; AD ǁ BC , so ∠DAC ≅ ∠BCA because they are alternate interior angles. _ _ _ _ e. Yes; the diag. of a parallelogram bisect each other, so AE ≅ CE and DE ≅ BE ; ∠AED ≅ ∠CEB because they are vert. angles; △AED ≅ △CEB by the SAS Cong. Thm. © Houghton Mifflin Harcourt Publishing Company A. The perimeter of ABCD is 2AB + 2BC. 1 DB B. DE = _ _ 2_ C. BC ≅ DC f. No; the diag. of a parallelogram bisect each other, but they may or may not bisect opp. angles of the parallelogram. Module 24 IN1_MNLESE389762_U9M24L1 1200 1200 Lesson 1 9/30/14 12:16 PM Properties of Parallelograms 1200 JOURNAL H.O.T. Focus on Higher Order Thinking 25. Represent Real-World Problems A store sells tiles in the shape of a parallelogram. The perimeter of each tile is 29 inches. One side of each tile is 2.5 inches longer than another side. What are the side lengths of the tile? Explain your steps. Have students summarize the relationships they discovered about parallelograms. Possible answer: Represent the side lengths as x and (x + 2.5). Since opposite sides of a parallelogram are congruent, the perimeter can be written as x + (x + 2.5) + x + (x + 2.5), or 4x + 5. Write an equation to solve for x. 4x + 5 = 29 So, x = 6, and (x + 2.5) = 8.5. So, the side lengths of the tile are 6 inches and 8.5 inches. 26. Critique Reasoning A student claims that there is an SSSS congruence criterion for parallelograms. That is, if all four sides of one parallelogram are congruent to the four sides of another parallelogram, then the parallelograms are congruent. Do you agree? If so, explain why. If not, give a counterexample. Hint: Draw a picture. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Tashatuvango/iStockPhoto.com No; two parallelograms may have four congruent sides, but their four angles do not also have to be congruent. The figures show a counterexample. 27. Analyze Relationships The figure shows two congruent parallelograms. How are x and y related? Write an equation that expresses the relationship. Explain your reasoning. x° x° x° x° IN1_MNLESE389762_U9M24L1.indd 1201 Lesson 24.1 y° y = 2x; Possible explanation: Since opposite sides of a parallelogram are parallel, if the common side of the parallelograms is extended, the angles formed by the extended side each measure x° because alternate interior angles are congruent. Module 24 1201 x° 1201 Lesson 1 4/18/14 7:46 PM Lesson Performance Task INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 An inventor considered building a scissor lift The principle that allows a scissor lift to raise the platform on top of it to a considerable height can be illustrated with four freezer pop sticks attached at the corners. A A B D B from triangles rather than parallelograms. The inventor believed that simpler and cheaper 3-sided shapes would appeal to budget-minded customers. What advice would you give to the inventor and why? Sample answer: Triangles are rigid, meaning their shapes can’t change. A triangular scissor lift wouldn’t work because the triangular sections could not be made to expand or contract. D C C Answer these questions about what happens to parallelogram ABCD when you change its shape as in the illustration. a. Is it still a parallelogram? Explain. b. Is its area the same? Explain. c. Compare the lengths of the diagonals in the two figures as you change them. INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 One angle of a parallelogram is a right angle. d. Describe a process that might be used to raise the platform on a scissor lift. a. Yes. Its opposite sides are still parallel and congruent. b. No. As you move points B and D closer to each other, the area decreases. _ _ c. Horizontal diagonal BD decreases in length. Vertical diagonal AC increases in length. Describe the other angles. Explain your reasoning. The other angles are also right angles; Sample answer: Since the opposite angles of a parallelogram are congruent, the angle opposite the right angle must also be a right angle. Since consecutive angles of a parallelogram are supplementary, the measures of the two remaining angles must each be 180° – 90° = 90°. So, both are also right angles. d. Possible answer: Apply an inward horizontal force from both sides on the bottom parallelogram. © Houghton Mifflin Harcourt Publishing Company Module 24 1202 Lesson 1 EXTENSION ACTIVITY IN1_MNLESE389762_U9M24L1 1202 Supply students with sheets of light cardboard or photo-print paper, and some brads. Instruct students to cut out twelve 5-inch-by-0.5-inch strips and to attach the ends with brads in such a way as to make a model of a 3-parallelogram scissor lift. Have students demonstrate the operation of the lift, describing how the elements of the parallelogram (angle measures, side length, shape, area) change as the lift increases and decreases in total height. 9/30/14 12:15 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 Parallelograms 1202