Common core geometry homework answers

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Download common core 6.2 homework answers

6-2 Properties of Parallelograms Focus on Reasoning Essential question: What can you conclude about the sides, angles, and diagonals of a parallelogram? TEACH Standards for Mathematical Content 1 G-CO.3.11 Prove theorems about parallelograms. G-SRT.2.5 Use congruence ... criteria for triangles to solve problems and to prove relationships in geometric figures. Materials: geometry software Questioning Strategies • As you use the software to drag points A, B, C, and/or D, does the quadrilateral remain a parallelogram? Why? Yes; the lines Vocabulary diagonal that form opposite sides remain parallel. • What do you notice about consecutive angles in the parallelogram? Why does this make sense? Consecutive angles are supplementary. Prerequisites Theorems about parallel lines cut by a transversal Triangle congruence criteria 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. Math Background In this lesson, students extend their earlier work with triangle congruence criteria and triangle properties to prove facts about parallelograms. This lesson gives students a chance to use inductive and deductive reasoning to investigate properties of the sides, angles, and diagonals of parallelograms. Students have encountered parallelograms in earlier grades. Ask a volunteer to define parallelogram. Students may have only an informal idea of what a parallelogram is (e.g., “a slanted rectangle”), so be sure they understand that the mathematical definition of a parallelogram is a quadrilateral with two pairs of parallel sides. You may want to show students how they can make a parallelogram by drawing lines on either side of a ruler, changing the position of the ruler, and drawing another pair of lines. Ask students to explain why this method creates a parallelogram. 2 Prove that opposite sides of a parallelogram are congruent. Questioning Strategies • Why do you think the proof is based on drawing ___ the diagonal ​DB​ ? Drawing the diagonal creates two triangles; then you can use triangle congruence criteria and CPCTC. 245 Lesson 2 © Houghton Mifflin Harcourt Publishing Company Teaching Strategy Some students may have difficulty with terms like opposite sides or consecutive angles. Remind students that 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). Opposite angles of a quadrilateral do not share a side. Consecutive angles of a quadrilateral do share a side. You may want to help students draw and label a quadrilateral for reference. INTR O D U C E Chapter 6 Investigate parallelograms. Name Class Notes 6-2 Date Properties of Parallelograms Focus on Reasoning Essential question: What can you conclude about the sides, angles, and diagonals of a parallelogram? Recall that a parallelogram is a quadrilateral that has two pairs of parallel sides. You use the symbol to name a parallelogram. For example, the figure shows ABCD. A D 1 G-CO.3.11, G-SRT.2.5 B C © Houghton Mifflin Harcourt Publishing Company Investigate parallelograms. A Use the straightedge tool of your geometry software to draw a straight line. Then plot a point that is not on the line. Select the point and line, go to the Construct menu, and construct a line through the point that is parallel to the line. This will give you a pair of parallel lines, as shown. B Repeat Step A to construct a second pair of parallel lines that intersect those from Step A. C The intersections of the parallel lines create a parallelogram. Plot points at these intersections. Label the points A, B, C, and D. D Use the Measure menu to measure each angle of the parallelogram. E Use the Measure menu to measure the length of each side of the parallelogram. (You can do this by measuring the distance between consecutive vertices.) F B A AB = 2.60 cm BC = 1.74 cm CD = 2.60 cm DA = 1.74 cm C D m∠DAB = 116.72˚ m∠ABC = 63.28˚ m∠BCD = 116.72˚ m∠CDA = 63.28˚ 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. REFLECT 1a. Make a conjecture about the sides and angles of a parallelogram. Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. 245 Chapter 6 Lesson 2 G_MFLBESE200852_C06L02.indd 245 03/05/14 4:56 PM You may have discovered the following theorem about parallelograms. Theorem 2 Prove that opposite sides of a parallelogram are congruent. A B Complete the proof. Given: ABCD is a parallelogram. ___ ____ ___ D ___ Prove: AB ≅ CD and AD ≅ BC Statements 1. ABCD is a parallelogram. ___ ___ Reasons 1. Given 2. Through any two points there exists exactly one line. 2. Draw DB​. ___ C ___ ___ 3. AB​​ǁ DC​​;​​AD​​ ​ǁ​​BC​​ 3. Definition of parallelogram 4. ∠ADB​≅​∠CBD;​∠ABD​≅​∠CDB 4. Alternate Interior Angles Theorem 5. Reflexive Property of Congruence ___ ___ 5. DB​​≅​​DB​​ 6. △ABD ≅ △CDB 7. AB​≅​CD;​AD​≅​BC 6. ASA Congruence Criterion 7. CPCTC REFLECT 2a. Explain how you can use the rotational symmetry of a parallelogram to give an argument that supports the above theorem. Under a 180° rotation about the center of the parallelogram, each side is mapped to its opposite side. Since rotations preserve distance, this shows that opposite sides are congruent. 2b. One side of a parallelogram is twice as long as another side. The perimeter of the parallelogram is 24 inches. Is it possible to find all the side lengths of the parallelogram? If so, find the lengths. If not, explain why not. Yes; consecutive sides have lengths x, 2x, x, and 2x, so x + 2x + x + 2x = 24, or © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company If a quadrilateral is a parallelogram, then opposite sides are congruent. 6x = 24. Therefore x = 4 and the side lengths are 4 in., 8 in., 4 in., and 8 in. Chapter 6 G_MFLESE200852_C06L02.indd 246 Chapter 6 246 Lesson 2 08/03/13 10:09 PM 246 Lesson 2 T EACH 3 Highlighting the Standards Investigate diagonals of parallelograms. Materials: geometry software 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. This addresses elements of Standard 3 (Construct viable arguments and critique the reasoning of others). Students should recognize that a flow proof shows how one statement connects to the next. This may not be as apparent in a two-column format. You may want to have students rewrite the proof in a two-column format as a way of exploring this further. Questioning Strategies • How many diagonals does a parallelogram have? Is this true for every quadrilateral? Two; yes • If a quadrilateral is ___ named PQRS, what are the ___ diagonals? PR and QS • Are the diagonals of a parallelogram ever congruent? If so, when does this appear to happen? Yes; when the parallelogram is a rectangle 4 Prove diagonals of a parallelogram bisect each other. 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. © Houghton Mifflin Harcourt Publishing Company Chapter 6 247 Lesson 2 Notes Essential question: What can you conclude about the diagonals of a parallelogram? A segment that connects any two nonconsecutive vertices of a polygon is a diagonal. A parallelogram ___ ___ has two diagonals. In the figure, AC and BD are diagonals of ABCD. 3 A B D C IInvestigate diagonals of parallelograms. Use geometry software to construct a parallelogram. (See Lesson 4-2 for detailed instructions.) Label the vertices of the parallelogram A, B, C, and D. A B Use the segment ___ tool ___to construct the diagonals, AC and BD . C Plot a point at the intersection of the diagonals. Label this point E. D Use the Measure the ___ to measure ___ ___ menu ___ length of AE , BE , CE , and DE . (You can do this by measuring the distance between the relevant endpoints.) A B D C B A AE = 1.52 cm BE = 2.60 cm CE = 1.52 cm DE = 2.60 cm E 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. © Houghton Mifflin Harcourt Publishing Company E C D REFLECT 3a. Make a conjecture about the diagonals of a parallelogram. The diagonals of a parallelogram bisect each other. 3b. A student claims that the perimeter of AEB 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 AB = DC since opposite sides of a parallelogram are congruent. So, AE + BE + AB = CE + DE + DC. Chapter 6 Lesson 2 247 You may have discovered the following theorem about parallelograms. Theorem T 4 Prove diagonals of a parallelogram bisect each other. Complete the proof. C A B Given: ABCD is a parallelogram. ___ ___ ___ E ___ Prove: AE CE and BE DE . C D ABCD is a parallelogram. Given ___ ___ ___ ___ AB DC AB DC Opposite sides of a parallelogram are congruent. Definition of parallelogram ∠ABE ∠CDE ∠BAE ∠DCE Alternate Interior Angles Theorem Alternate Interior Angles Theorem © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company If a quadrilateral is a parallelogram, then the diagonals bisect each other. ABE CDE ASA Congruence Criterion ___ ___ ___ ___ AE CE and BE DE . CPCTC REFLECT 4a. Explain how you can prove the theorem using a different congruence criterion. ∠AEB ∠CED because they are vertical angles. Using this fact plus the fact ___ ___ that ∠ABE ∠CDE and AB DC , it is possible to prove the theorem using the AAS Congruence Criterion. Chapter 6 Chapter 6 248 Lesson 2 248 Lesson 2 CLOS E Teaching Strategy The lesson concludes with the theorem that states that opposite angles of a parallelogram are congruent. The proof of this theorem is left as an exercise (Exercise 1). Be sure students recognize that the proof of this theorem is similar to the proof that opposite sides of a parallelogram are congruent. Noticing such similarities is an important problem-solving skill. Essential Question What can you conclude about the sides, angles, and diagonals of a parallelogram? Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. The diagonals of a parallelogram bisect each other. Summarize Have students make a graphic organizer to summarize what they know about the sides, angles, and diagonals of a parallelogram. A sample is shown below. Highlighting the Standards PR ACTICE Exercise 4 is a multi-part exercise that includes opportunities for mathematical modeling, reasoning, and communication. It is a good opportunity to address Standard 4 (Model with mathematics). Draw students’ attention to the way they interpret their mathematical results in the context of the real-world situation. Specifically, ask students to explain what their mathematical findings tell them about the appearance and layout of the park. Exercise 1: Students practice what they learned in part 2 of the lesson. Exercise 2: Students use reasoning to extend what they know about parallelograms. Exercise 3: Students use reasoning and/or algebra to find unknown angle measures. Exercise 4: Students apply their learning to solve a multi-step real-world problem. Parallelogram A B © Houghton Mifflin Harcourt Publishing Company C D Opposite sides are congruent. Opposite angles are congruent. AB DC AD BC ∠ 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 . Chapter 6 249 Lesson 2 Notes The angles of a parallelogram also have an important property. It is stated in the following theorem, which you will prove as an exercise. Theorem T If a quadrilateral is a parallelogram, then opposite angles are congruent. PRACTICE 1. Prove the above theorem about opposite angles of a parallelogram. Given: ABCD is a parallelogram. Prove: ∠A ∠C and ∠B ∠D A (Hint: You only need to prove that ∠A ∠C. A similar argument can be used to prove that ∠B ∠D. Also, you may or may not need to use all the rows of the table in your proof.) D C Statements Reasons 1. ABCD is a parallelogram. 2. Draw DB. 1. Given 2. Through any two points there exists exactly one line. AB || DC; AD || BC 3. Definition of parallelogram ∠ADB ∠CBD; ∠ABD ∠CDB 4. Alternate Interior Angles Theorem DB DB 5. Reflexive Property of Congruence 6. ABD CDB 6. ASA Congruence Criterion 7. ∠A ∠C 7. CPCTC 3. 4. 5. © Houghton Mifflin Harcourt Publishing Company B ___ ___ ___ ___ ___ ___ ___ 2. Explain why consecutive angles of a parallelogram are supplementary. 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. 3. In the figure, JKLM is a parallelogram. Find the measure of each of the numbered angles. J m∠1 = 19°; m∠2 = 43°; m∠3 = 118°; K 1 3 m∠4 = 118°; m∠5 = 19° 4 N 62˚ 43˚ 2 5 M Chapter 6 L Lesson 2 249 4. A city planner is designing a park in the shape of a parallelogram. As shown in the figure, there will be two straight paths through which visitors may enter the park. The paths are bisectors of consecutive angles of the parallelogram, and the paths intersect at point P. A B P D C © Houghton Mifflin Harcourt Publishing Company a. Work directly on the parallelograms below and use a compass and straightedge to construct the bisectors of ∠A and ∠B. Then use a protractor to measure ∠APB in each case. A B A B P P D C D C Make a conjecture about ∠APB. ∠APB is a right angle. b. Write a paragraph proof to show that your conjecture is always true. (Hint: Suppose m∠BAP = x°, m∠ABP = y°, and m∠APB = z°. What do you know about x + y + z? What do you know about 2x + 2y?) By the Triangle Sum Theorem, x + y + z = 180. Also, m∠DAB = (2x)° and m∠ABC = (2y)°. By the Same-Side Interior Angles Postulate m∠DAB + m∠ABC = 180°. So 2x + 2y = 180 and x + y = 90. Substituting this in the first equation gives 90 + z = 180 and z = 90. When the city planner takes into account the dimensions of the park, she finds that point P lies on ___ DC , as shown. Explain why it must be the case that DC = 2AD. (Hint: Use congruent base angles to show that DAP and CPB are isosceles.) ___ A x˚ B y˚ x˚ y˚ z˚ D P C ∠DAP ∠BAP since AP is an angle bisector. Also, ∠DPA ∠BAP by the Alternate Interior Angles Theorem. Therefore, ∠DAP ∠DPA. This means ___ ___ ___ ___ ___ ___ DAP is isosceles, with AD DP . Similarly, BC PC . Also, BC AD as opposite sides of a parallelogram. So, DC = DP + PC = AD + BC = AD + © Houghton Mifflin Harcourt Publishing Company c. AD = 2AD. Chapter 6 Chapter 6 250 Lesson 2 250 Lesson 2 ADD I T I O NA L P R AC TI C E AND P RO BL E M S O LV I N G Assign these pages to help your students practice and apply important lesson concepts. For additional exercises, see the Student Edition. Answers Additional Practice 1. 108.8 cm 2. 91 cm 3. 217.6 cm 4. 123° 5. 123° 6. 57° 7. 117° 8. 63° 9. 71 10. 21 11. 10.5 12. 15 13. 30 14. (0, -3) 15. Possible answer: Statements Reasons 1. Given 2. m∠EDG = m∠EDH + m∠GDH, m∠FGD = m∠FGH + m∠DGH 2. Angle Add. Post. 3. m∠EDG + m∠FGD = 180° 3. cons. ∠s supp. © Houghton Mifflin Harcourt Publishing Company 1. DEFG is a parallelogram. 4. m∠EDH + m∠GDH + 4. Subst. (Steps m∠FGH + m∠DGH = 180° 2, 3) 5. m∠GDH + m∠DGH + m∠DHG = 180° 5. Triangle Sum Thm. 6. m∠GDH + m∠DGH + m∠DHG = m∠EDH + m∠GDH + m∠FGH + m∠DGH 6. Trans. Prop. of = 7. m∠DHG = m∠EDH + m∠FGH 7. Subtr. Prop. of = Problem Solving 1. m∠C = 135°; m∠D = 45° 2. 15 in. 3. 4.5 ft 4. 65° 5. B 6. H 7. D Chapter 6 251 Lesson 2 Name Class Notes 6-2 Date Additional Practice $JXUQH\LVDZKHHOHGFRWRUVWUHWFKHUXVHGLQKRVSLWDOV 0DQ\JXUQH\VDUHPDGHVRWKDWWKHEDVHZLOOIROGXSIRU HDV\VWRUDJHLQDQDPEXODQFH:KHQSDUWLDOO\IROGHGWKH EDVHIRUPVDSDUDOOHORJUDP,Q 678998FHQWLPHWHUV 8:FHQWLPHWHUVDQGP769)LQGHDFKPHDVXUH . 6: 76 BBBBBBBBBBBBBBBBBBBBBBBB P698 86 BBBBBBBBBBBBBBBBBBBBBBBBB P678 BBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBB P789 BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBB -./0LVDSDUDOOHORJUDP)LQGHDFKPHDVXUH P/ P. BBBBBBBBBBBBBBBBBBBBBBBB 0- BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBB 9:;<LVDSDUDOOHORJUDP)LQGHDFKPHDVXUH 9; ;= BBBBBBBBBBBBBBBBBBBBBBBB © Houghton Mifflin Harcourt Publishing Company =: BBBBBBBBBBBBBBBBBBBBBBBBB :< BBBBBBBBBBBBBBBBBBBBBBBB . BBBBBBBBBBBBBBBBBBBBBBBBB 7KUHHYHUWLFHVRI $%&'DUH%&DQG' )LQGWKHFRRUGLQDWHVRIYHUWH[$ BBBBBBBBBBBBBBBBBBBBB :ULWHDWZRFROXPQSURRI *LYHQ'()*LVDSDUDOOHORJUDP 3URYHP'+*P('+P)*+ Chapter 6 Lesson 2 251 Problem Solving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© Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company 7KHZDOOIUDPHVRQWKHVWDLUFDVHZDOOIRUPSDUDOOHORJUDPV $%&'DQG()*+ ) FP * FP + FP - FP $ & % ' Chapter 6 Chapter 6 252 Lesson 2 252 Lesson 2
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Pearson Geometry Common Core, 2011

Chapter 1

Tools of Geometry

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Chapter 2

Reasoning and Proof

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Chapter 3

Parallel and Perpendicular Lines

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Chapter 4

Congruent Triangles

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Chapter 5

Relationships Within Triangles

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Chapter 6

Polygons and Quadrilaterals

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Chapter 7

Similarity

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Chapter 8

Right Triangles and Trigonometry

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Transformations

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Area

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Surface Area and Volume

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Probability

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