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Home , Congruence (geometry)

Topic Congruence 19 and

1 How many lines of symmetry does the Transamerica Building in San Francisco have? You will find out in Lesson 19-2.

436

27290_436-437;1.indd 436 12/18/06 2:32:42 PM Vocabulary Choose the best term from the box. • symmetry • congruent • rotation • refl ection

1. The ? of a fi gure is the mirror image.

2. Two fi gures are ? if they have the same and same size.

3. A fi gure has ? if it can be folded along a so that both parts match exactly.

4. When a fi gure changes position by moving about a , it is a ? . 2 In this M.C. Escher drawing, what are some of the ways these horses Solid Figures have been moved? You will find out Name the solid fi gure for each object. in Lesson 19-1.

3 5. 6. 7. The Megaray is one of the world’s largest kites measuring 66 meters long and 38 meters across. What is the ’s ? You will find out in Lesson 19-3. Identify each shape.

8. 9. 10.

11. 12. 13.

14. 15. 16.

17. Writing to Explain Explain how the fi gures in Exercises 8–10 are alike and how they are diff erent.

Topic 19 437

27290_436-437;2.indd 2 9/13/07 1:13:56 PM Lesson 19-1 Congruent Figures When are figures congruent? Figures that are the same size and shape are congruent. MG 3.3 Identify congruent figures. You can use translations, reflections, and rotations to test if two figures are congruent.

Congruent Not Congruent

Guided Practice*

Do you know HOW? Do you UNDERSTAND? For 1 through 4, tell if the figures in each 5. If one of the house shapes above is 1 pair are congruent. __ rotated 4 turn, will the two shapes 1. 2. still be congruent? 6. Writing to Explain Can a and a square ever be congruent? 3. 4. Why or why not?

Independent Practice

For 7 through 15, tell if the figures in each pair are congruent. 7. 8. 9.

10. 11. 12.

13. 14. 15.

438 *For another example, see Set A on page 448.

27290_438-439;5.indd 438 12/19/06 9:22:46 AM Congruent figures Congruent figures Congruent figures can be related by can be related by can be related by a . a . a rotation.

Problem Solving

For 16 and 17, describe everything that is the same and everything that is different about each pair of figures. Then tell if the figures are congruent.

16. 17.

18. In the M.C. Escher drawing at the X B right, which horse(s) are a translation of the horse labeled X? C A Horse A C Horse A and C A B Horse B D Horse A, B, and C

Symmetry Drawing 78 By M.C. Escher 19. How many days are in 52 weeks? 20. Reasoning Use the diagram below. A 59 days C 365 days Frida wrote a message on paper and held it up to a mirror. What does the B 364 days D 366 days message say?

21. On a bus ride, Jasmine counted REFLECTION. A IS THIS 24 taxis and 12 bicycles. How many wheels did she count in all?

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Lesson 19-1 439

27290_438-439;7.indd 2 9/13/07 1:15:04 PM Lesson 19-2 Line Symmetry What is a line of symmetry? A figure is symmetric if it can be folded MG 3.4 Identify figures on a line to form two congruent halves that have bilateral and rotational symmetry. that fit on top of each other. The fold line is called a line of symmetry. This truck has one line of symmetry.

Guided Practice*

Do you know HOW? Do you UNDERSTAND? For 1 and 2, tell if each line is a line 5. Do some figures have no lines of symmetry. of symmetry?

1. 2. 6. How many lines of symmetry does the figure below have?

For 3 and 4, tell how many lines of symmetry each figure has.

3. 4. 7. Writing to Explain How many lines of symmetry does a bicycle tire have?

Independent Practice For 8 through 11, tell if each line is a line of symmetry. 8. 9. 10. 11.

For 12 through 15, tell how many lines of symmetry each figure has. 12. 13. 14. 15.

440 *For another example, see Set B on page 448 .

27290_440-441.indd 440 12/11/06 9:06:44 AM A figure can have A figure can have many A figure can have more than one line lines of symmetry. no lines of symmetry. of symmetry.

For 16 through 23, trace each figure and draw lines of symmetry if you can. 16. 17. 18. 19.

20. 21. 22. 23.

Problem Solving 24. How many lines of symmetry does 25. How many lines of symmetry does a scalene have? an have?

26. Reasoning Vanessa drew a figure 27. Draw a that does not and said that it had an infinite have a line of symmetry. number of lines of symmetry. What figure did she draw?

28. The Transamerica Building in San Francisco has one line of symmetry. Use the picture at the right to describe where the line of symmetry is.

29. Write 5 capital letters that have at least one line of symmetry.

30. How many lines of symmetry does a square have? A None C 4 lines Animated Glossary www.pearsonsuccessnet.com B 2 lines D 6 lines Lesson 19-2 441

27290_440-441;6.indd 2 9/13/07 1:16:10 PM Lesson 19-3 Rotational Symmetry set of What is rotational symmetry? When a figure can rotate onto itself in less than MG 3.4 Identify figures a full turn, the figure has rotational symmetry. that have bilateral and 1 rotational symmetry. If you rotate this figure __ turn, it has rotated 90°. Also MG 3.5. 4 This figure has rotational symmetry.

Guided Practice*

Do you know HOW? Do you UNDERSTAND? Does the figure have rotational 5. A figure that has been rotated a 1 symmetry? Write yes or no. __ ■ 4 turn has been rotated degrees. 1. 2. 6. A figure that has been rotated 180° I has been rotated ■ turn.

3. 4. 7. A figure that has been rotated a 3 __ turn has been rotated ■ degrees. L 4

Independent Practice

In 8 through 15, does the figure have rotational symmetry? Write yes or no. Give the least measure and turn that will rotate the figure onto itself. 8. 9. 10. 11.

12. H 13. 14. 15.

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442 *For another example, see Set C on page 449.

27290_442-443;19.indd 1 9/13/07 1:16:41 PM This figure has This figure does This figure has rotational symmetry. not have rotational rotational symmetry. It can be rotated 90°, symmetry. It must be It must be rotated 180°, or 270°, or a rotated 360°, or a full 180°, or a __ 1 turn, to __1 turn, a __1 turn, or a __3 2 4 2 4 turn, to land on itself. land on itself. turn, to land on itself.

Problem Solving

For 16 and 17, fill in and use the table to the right. 16. Valerie has 16 yards of fencing material Side A Side B to build a dog run. She wants to put Length Length Area the fence around a rectangular area. Complete the table to the right for 1 yd ■ 7 sq. yd the possibilities of the fencing. 2 yd ■ ■ ■ 5 yd ■ 17. Which area is the largest? What is 4 yd ■ ■ another name for this shape?

18. Measurement A backyard is in the 19. Which capital letter below has shape of an . rotational symmetry? One side measures 17 feet. What is A C the of the backyard? N T B D 17 feet Y E

20. Where would the parentheses have 21. The Megaray Kite has an area to go to make this equation true? of almost 1,500 square meters. Suppose the wind suddenly changes 18 Ϫ 2 ϩ 12 Ϫ 8 ϭ 20 and the kite moves 25 meters east. Has the shape or size of the kite changed? Explain.

Lesson 19-3 443

27290_442-443.indd 443 11/7/07 11:17:52 AM Lesson Problem Solving 19-4 Draw a Picture grid paper Lisa has been asked to draw a large arrow that is exactly the same shape MR 2.2 Apply strategies and results from simpler as the one shown on the grid at problems to more complex problems. the right. Also MR 2.0, MG 3.0. Make a large arrow that is exactly the same shape. Explain how you know it is the same shape.

Guided Practice*

Do you know HOW? Do you UNDERSTAND? For 1 and 2, make a large figure that is 3. Suppose that you drew the arrow exactly the same shape. Explain how you above so that it was pointing know it is the same shape. vertically. Would the shape of the 1. 2. arrow change?

4. Draw a picture of a shape. Then triple each side.

Independent Practice

Solve. 5. Draw a large figure. Then draw a smaller figure that is exactly the same shape.

6. If you were to cut out a hexagon to make a sign similar to the shape below, how • What do I know? would you draw it to make it twice the • What diagram can I use to help understand the problem? size? • Can I use , subtraction, multiplication, or division? • Is all of my work correct? • Did I answer the right question? • Is my answer reasonable?

444 *For another example, see Set D on page 449.

27290_444-445;4.indd Sec1:1 9/13/07 1:21:07 PM Plan Solve What do I know? I know the length Double the length of each side. of each side of the arrow. The arrow is 11 units long from left to right.

What am I asked To make an arrow to find? that is exactly the same shape The shapes are the same because the length of each side was doubled.

7. Stephen is listening to a book on 8. Six people are taking part in a charity tape. The book has 17 chapters and walk. Two people walked for 8 miles, each chapter is about 22 minutes three people walked for 6 miles, and long. How many minutes will it take one person walked for 10 miles. How to listen to the complete book? many miles did they walk in all?

9. Which can be rotated less than 10. Which of the following shapes has one complete turn and look exactly exactly four lines of symmetry? the same? A C A C

B D B D

11. Jackie and Kendall are part of their 12. Lawrence’s father said that he would school’s relay race team. Each put 12 dollars into Lawrence’s member of the team has to run for savings account for every 20 dollars one-half mile of a 3-mile race. Lawrence put into it. If after a year a Draw a picture to help you find his father had put 96 dollars into how many members are on the Lawrence’s account, how much did relay team. Lawrence put into his account? b How many other members are on the relay team besides Jackie and Kendall?

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Lesson 19-4 445

27290_444-445;4.indd Sec1:2 9/13/07 1:21:11 PM

1. Four of Mrs. Li’s students decorated 3. How many lines of symmetry does a bulletin board using shapes with an equilateral triangle have? (19-2) lines of symmetry. Whose shape has A 0 8 lines of symmetry? (19-2) B 1 WALL OF SYMMETRY C 2 Ralph Liza D 3

4. Cassidy made this shape for her school’s basketball team.

Patricia Dan Which shape below is the same shape? (19-4)

A

A Ralph B Liza C Patricia D Dan B

2. Milton built two congruent flower beds. Which could be the flower beds he built? (19-1)

AB CD C

A A and B B A and C C A and D D D B and C

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5. Which figure below has rotational 7. Which lists all the ways this figure symmetry? (19-3) could be rotated and land on itself? (19-3) A

B

A 90° only B 90° and 180° C C 90°, 180°, and 270° D This figure has no rotational symmetry. D 8. Which figure is congruent to the figure on 6. How many lines of symmetry does the right? (19-1) the logo below have? (19-2)

A

B

C

A 0 B 1 C 2 D D 4

Topic 19 Test Prep 447

27290_446-447.indd Sec2:447 12/11/06 10:55:00 AM

Set A, pages 438–439

Are the figures congruent? If so, tell if they are Remember that you can use related by a reflection, translation, or rotation. translations, reflections, and rotations to find if two figures are congruent. Are the figures in each pair congruent? 1. 2.

The are the same size and shape, 3. 4. so they are congruent. They are related by a reflection. The are not the same size, so they 5. 6. are not congruent. The stacked oval shapes are congruent. They are related by a rotation. The O and Q are different shapes. They are not congruent.

Set B, pages 440–441

How many lines of symmetry Remember that figures can have do the figures have? many lines of symmetry.

Fold the figure along the dashed Draw the lines of symmetry for line. The two halves are congruent each figure. and fit one on top of the other. It has one line of symmetry. 1. 2.

The figure may be folded five ways that result in congruent figures that 3 . 4. fit on top of each other. This figure has five lines of symmetry.

448

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Set C, pages 442-443

What kind of rotation does Remember, a figure has rotational the figure make? symmetry if it can rotate onto itself in less than a full turn. This figure can be rotated 1 1. 2. __ 90°, or a 4 turn, to land on itself.

This figure can be rotated 3. 4. 1 180°, or a __ turn, to land 2 on itself.

This figure can be rotated 1 __ 90°, 180°, or 270°, or a 4 1 3 turn, a __ turn, or a __ turn, 2 4 to land on itself.

Set D, pages 444-445

Make a letter “T” that is exactly Remember to measure the figure the same shape. Explain how before you draw it. you know it is the same shape. 1. Draw a letter “L” that is exactly the same shape. Explain how What do I know? The letter is 11 units in vertical you know it is the same shape. height and spans 9 units horizontally.

What am I To make a letter T that is exactly asked to find? the same shape.

Double the of the figure above.

Topic 19 Reteaching 449

27290_448-449;10.indd Sec2:2 9/13/07 1:22:43 PM