Chapter X6 Resource Masters. Course13

Transcription

1 hapter X6 Resource Masters ourse13

2 NAME This is an alphabetical list of new vocabular terms ou will learn in hapter 6. As ou stud the chapter, complete each term s definition or description. Remember to add the page number where ou found the term. Add this page to our math stud notebook to review vocabular at the end of the chapter. Vocabular Term acute triangle Reading to Learn Mathematics Vocabular uilder Found on Page Definition/Description/Eample Vocabular uilder adjacent angles alternate eterior angles alternate interior angles complementar angles corresponding angles equilateral triangle isosceles triangle line smmetr obtuse triangle Glencoe/McGraw-Hill vii Mathematics: Applications and oncepts, ourse 3

3 NAME Reading to Learn Mathematics Vocabular uilder (continued) Vocabular Term parallelogram Found on Page Definition/Description/Eample quadrilateral reflection rhombus right triangle rotation scalene triangle supplementar angles translation transversal trapezoid vertical angles Glencoe/McGraw-Hill viii Mathematics: Applications and oncepts, ourse 3

4 NAME Stud Guide and Intervention Line and Angle Relationships The relationship between pairs of angles can be used to find missing measures. Find the value of in the figure at the right. The two angles are supplementar, so their sum is Write an equation Subtract 35 from each side. 145 Simplif. Use the figure at the right. 35 a. Find m 3 if m and 7 are corresponding angles. Since corresponding angles are congruent, their measures are the same. m 3 m 7, so m b. Find m 4 if m and 5 are alternate interior angles. Since alternate interior angles are congruent, their measures are the same. m 4 m 5, so m Lesson 6 1 Find the value of in each figure For Eercises 4 7, use the figure at the right. 4. Find m 5 if m Find m 2 if m Find m 1 if m Find m 8 if m Find m 5 if m Glencoe/McGraw-Hill 307 Mathematics: Applications and oncepts, ourse 3

5 NAME Practice: Skills Line and Angle Relationships lassif each angle or angle pair using all names that appl acute right obtuse vertical complementar, supplementar adjacent Find the value of in each figure For Eercises 13 22, use the figure at the right. 13. Find m 5 if m Find m 2 if m Find m 3 if m Find m 8 if m Find m 5 if m Find m 3 if m Find m 4 if m Find m 1 if m Find m 6 if m Find m 3 if m Glencoe/McGraw-Hill 308 Mathematics: Applications and oncepts, ourse 3

6 NAME Practice: Word Problems Line and Angle Relationships 1. SIGN The support wire for a sign meets the wall and the overhang as shown below. If m 2 42, find m 1. Eplain our reasoning AIRPRTS The runwas at a local airport are laid out as shown below. Runwas A and are parallel, and runwa cuts across A and. If m 1 55, find m 2. Eplain our reasoning. A 1 2 m 1 48 ; 1 and 2 are complementar angles. m 2 55 ; 1 and 2 are corresponding angles. 3. RAILRADS East of the town of Rockport, the railroad tracks intersect Highwa 67 as shown below. If m 1 133, find m 2. Eplain our reasoning. 4. AMPING Jonna and Elizabeth found a level campsite and pitched their tent as shown below. If m 1 120, find m 2. Eplain our reasoning. Lesson X m 2 60 ; 1 and 2 are supplementar angles. m 2 47 ; 1 and 2 are supplementar angles. 5. ALPHAET The top and bottom segments of the letter Z are parallel as shown below. If m 1 43, find m 2. Eplain our reasoning. 6. FLRING Garret is designing a floor with diamond-shaped tiles as shown below. If m 1 125, find m 2. Eplain our reasoning. 1 2 m 2 43 ; 1 and 2 are alternate interior angles. 1 2 m ; 1 and 2 are vertical angles. Glencoe/McGraw-Hill 309 Mathematics: Applications and oncepts, ourse 3

7 Pre-Activit NAME Reading to Learn Mathematics Line and Angle Relationships omplete the Mini Lab at the top of page 256 in our tetbook. Write our answers below. 1. For each set of intersecting lines, identif the pairs of angles that have the same measure. 1 and 3, 2 and 4 2. What is true about the sum of the measures of the angles sharing a side? Their sum is 180. Reading the Lesson 3. Angles can be acute, right, obtuse, or straight. For each one of these possibilities draw an eample of the angle. acute angle Sample answer: right angle Sample answer: obtuse angle Sample answer: straight angle Sample answer: 4. How do ou know if two lines are perpendicular? Sample answer: The lines will intersect at right angles. 5. How can ou tell the difference between alternate interior and alternate eterior angles? Sample answer: Alternate interior angles lie between the two parallel lines, and alternate eterior angles lie on the outside of the parallel lines. Helping You Remember 6. Work with a partner. Have our partner draw two angles. Identif the tpes of angles our partner has drawn. See students work. Glencoe/McGraw-Hill 310 Mathematics: Applications and oncepts, ourse 3

8 NAME Enrichment Matchstick Puzzles The puzzles on this page involve a bo of wooden matches. Toothpicks or an other straight objects ma also be used. Use the square figure at the right to make each of the following figures. 1. Move 3 matches to make 2. Move 4 matches to make 3 identical squares. 3 identical squares. 3. Remove 2 matches to 4. Move 2 matches to make make 2 squares. 7 squares. (You ma cross the matches in this one onl.) Lesson X Use the heagonal figure at the right to make each of the following figures. 5. Move 4 matches to make 6. Move 4 matches to make 3 equilateral triangles. 4 identical diamonds. Use the square figure at the right to make each of the following figures. 7. Remove 4 matches to 8. Remove 6 matches to make 5 identical squares. make 5 identical squares. Glencoe/McGraw-Hill 311 Mathematics: Applications and oncepts, ourse 3

9 NAME Stud Guide and Intervention Triangles and Angles The sum of the measures of the angles of a triangle is 180. This can be used to find a missing angle measure in a triangle. Find the value of in A The sum of the measures is Simplif Subtract 116 from each side. 64 Simplif. A Triangles can be classified b the measures of their angles and b the lengths of their sides. lassif the triangle b its angles and b its sides. 21 E F Angle DEF has one obtuse angle. Sides DEF has two congruent sides. So, DEF is an obtuse isosceles triangle. D Find the value of in each triangle lassif each triangle b its angles and b its sides acute scalene right isosceles obtuse scalene Glencoe/McGraw-Hill 312 Mathematics: Applications and oncepts, ourse 3

10 Find the value of in each triangle NAME Practice: Skills Triangles and Angles lassif each triangle b its angles and b its sides acute equilateral obtuse isosceles right scalene Lesson acute isosceles acute scalene obtuse scalene obtuse isosceles right scalene acute isosceles Glencoe/McGraw-Hill 313 Mathematics: Applications and oncepts, ourse 3

11 NAME Practice: Word Problems Triangles and Angles MAPS For Eercises 1 and 2, use the figure that shows the towns of Lakeview, Peoria, and Alton. Lakeview 20 mi 20 mi Alton Peoria 1. The three towns form a triangle. lassif the triangle b its angles and b its sides. acute isosceles 2. Find the value of in the figure FITNESS The running path around the lake shown in the figure is triangular. lassif the triangle b its angles and b its sides. 4. FITNESS Refer to the triangular running track shown in Eercise 3. Find the value of ft lake 500 ft 372 ft right scalene 5. HIKING The trail shown in the figure is triangular. Find the value of in the figure. overlook 6. HIKING Refer to the triangular trail shown in Eercise 5. lassif the triangle b its angles and b its sides. obtuse isosceles 29 trail head waterfall Glencoe/McGraw-Hill 314 Mathematics: Applications and oncepts, ourse 3

12 Pre-Activit NAME Reading to Learn Mathematics Triangles and Angles omplete the Mini Lab at the top of page 262 in our tetbook. Write our answers below. 1. What do ou think is the sum of the measures of the three angles of an triangle? Eplain our reasoning. 180 ; The form a straight line. Reading the Lesson 2. How can ou determine whether a triangle is a right triangle? Sample answer: If the triangle has a 90 angle, then it is a right triangle. 3. If one angle of a triangle is either right or obtuse, what must be true of the other two angles? The must both be acute. 4. omplete the table. Tpe of Triangle Scalene Isosceles Equilateral Number of ongruent Sides no congruent sides at least two congruent sides three congruent sides Helping You Remember 5. omplete the diagram b drawing an eample of the triangle under each bo. Triangles Lesson 6 2 one obtuse angle one right angle three acute angles Sample answer: Glencoe/McGraw-Hill 315 Mathematics: Applications and oncepts, ourse 3

13 NAME Enrichment Three-Part Triangles If an equilateral triangle is divided into three equal parts, the parts can be colored to create puzzle pieces. If each of the three parts is colored with one of four different colors, a set of 24 different triangles results. Reflections are considered different, but rotations are not. 1. Three of the 24 possible triangles are shown above. olor these 24 triangles to show the complete set. The 24 triangles can be used to make man shapes. Here are two for ou to tr. In both problems, the border of the shape must be all the same color Sample answers are given. Glencoe/McGraw-Hill 316 Mathematics: Applications and oncepts, ourse 3

14 NAME Stud Guide and Intervention Special Right Triangles In a right triangle, the hpotenuse is alwas twice as long as the side opposite the 30 angle. Find each missing length. Step 1 Find c. c 2a Write the equation. c 2(10) or 20 Replace a with 10. Step 2 Find b. c 2 a 2 b 2 Pthagorean Theorem b 2 Replace c with 20 and a with b 2 Evaluate 20 2 and b Subtract 100 from each side. 300 b 2 Simplif. 300 b 2 Take the square root of each side b Use a calculator. 30 c b ft The length of c is 20 feet, and the length of b is about 17.3 feet. In a right triangle, the legs are alwas congruent. Find each missing length. Step 1 Find a. a and b are the same length, so a 17 centimeters. c a Step 2 You can find c using the method shown in Eample 1. The length of a is 17 centimeters, and the length of c is about 24.0 centimeters. 17 cm Find each missing length. Round to the nearest tenth if necessar. 1. b a c c b 3.5 in., c 4 in. a 1 mm, c 1.4 mm b 5 d, c 7.1 d c a 6 m b in. 1 mm cm a a 3 m, b 5.2 m b 13.9 ft, c 16 ft a 13 cm, c 18.4 cm b 8 ft b c 5 d c Lesson 6 3 Glencoe/McGraw-Hill 317 Mathematics: Applications and oncepts, ourse 3

15 NAME Find each missing length. Round to the nearest tenth if necessar. 1. b c a c 60 c b b 6.9 cm, c 8 cm a 2 ft, c 2.8 ft b 6 m, c 8.5 m c 60 a b 26 d 30 Practice: Skills Special Right Triangles 4 cm 30 2 ft b 14 mm a 13 d, b 24.2 mm, b 22.5 d c 28 mm a 7 mi, c 9.9 mi a 6 m 7 mi c b km 30 b c b a in. a 16 in., b 36.4 cm, b 10 km, b 27.7 in. c 42 cm c 14.1 km c cm b 12. a c 19 d a 19 d, b 22 m, c 26.9 d c 31.1 m c 22 m 38 ft 30 b 60 a a 19 ft, b 32.9 ft 13. The length of the hpotenuse of a right triangle is 5.4 feet. Find the lengths of the other sides. 2.7 ft, about 4.7 ft 14. The length of one of the legs in a right triangle is 3.7 meters. Find the lengths of the other sides. 3.7 m, about 5.2 m Glencoe/McGraw-Hill 318 Mathematics: Applications and oncepts, ourse 3

16 NAME Practice: Word Problems Special Right Triangles 1. SHADWS The shadow cast b a pole forms a right triangle, as shown below. What is the length of the shadow? 2. SHADWS Refer to the figure shown in Eercise 1. What is the height h of the pole? Round to the nearest tenth. about 15.6 ft 18 ft h 9 ft 3. MAPS The towns of akland and Summit are linked b a highwa and b a railroad, as shown below. What is the length of the section of highwa between akland and the intersection? akland 4. MAPS Refer to the figure in Eercise 3. What is the length d of the section of railroad linking the towns? Round to the nearest tenth. about 4.2 mi d 45 Summit 45 3 mi Intersection 3 mi 5. AESSIILITY A ramp is to be constructed to a platform 4 feet above the ground. The triangle formed b the ramp, the ground, and the platform is a right triangle. Find the length of the ramp. Platform 6. AESSIILITY Refer to the ramp described in Eercise 5. What is the distance d from the foot of the ramp to the platform? Round to the nearest tenth. about 6.9 ft Lesson ft 60 4 ft Ramp 30 d Glencoe/McGraw-Hill 319 Mathematics: Applications and oncepts, ourse 3

17 NAME Pre-Activit Reading to Learn Mathematics Special Right Triangles omplete the Mini Lab at the top of page 267 in our tetbook. Write our answers below. 1. What tpe of triangles have ou formed? right 2. What are the measures of the angles of the folded triangle? 90, 60, Measure and describe the relationship between the shortest and longest sides of this triangle. longest side is twice the length of shortest side 4. What are the measures of the angles of the triangle formed b folding the square? 90, 45, Measure and describe the relationship between the legs of this triangle. legs are the same length Reading the Lesson 6. omplete the following sentence. In a right triangle, the hpotenuse is alwas the length of the side opposite the 30 angle. twice 7. The two legs in a right triangle are alwas congruent. Wh? Sample answer: The two angle measures are the same, so the triangle is isosceles. Thus, the lengths of the legs are alwas the same. Helping You Remember 8. The sides of a triangle in a right triangle and right triangle each have a special relationship. How can ou remember which relationship goes with each triangle? Sample answer: In a right triangle, two angles have the same measure as well as the two legs. In a right triangle, the three angles are all different and so are the lengths of the sides, but the length of the hpotenuse is twice the length of the shorter leg. Glencoe/McGraw-Hill 320 Mathematics: Applications and oncepts, ourse 3

18 NAME Enrichment Tangrams The tangram puzzle, shown at the right, is made of seven pieces. There are five triangles, one square, and one parallelogram. The figure with the dotted lines shows one wa to make the tangram pieces. Start b folding a square into 16 equal parts reate a set of tangrams. Then use all seven pieces to make each shape shown. Record our solutions The square and the polgons in Eercises 1 4 are called conve because no outside angle measures less than 180. It is possible to make 13 different conve polgons with the tangram pieces: 1 triangle, 6 quadrilaterals, 2 pentagons, and 4 heagons. 5. Find the other 8 conve polgons. Record our solutions below. Lesson 6 3 There is more than one wa to make some of the shapes. Glencoe/McGraw-Hill 321 Mathematics: Applications and oncepts, ourse 3

19 NAME Stud Guide and Intervention lassifing Quadrilaterals The sum of the measures of the angles of a quadrilateral is 360. You can use this to find a missing angle measure in a quadrilateral. Find the value of in quadrilateral AD The sum of the measures is Simplif Subtract 260 from each side. 100 Simplif. A D The best description of a quadrilateral is the one that is the most specific. lassif each quadrilateral using the name that best describes it. a. The quadrilateral has both pairs of opposite sides parallel and congruent. It is a parallelogram. b. The quadrilateral has eactl one pair of parallel sides. It is a trapezoid. c. The quadrilateral is a parallelogram with four right angles. It is a rectangle. Find the value of in each quadrilateral lassif each quadrilateral using the name that best describes it square rhombus parallelogram Glencoe/McGraw-Hill 322 Mathematics: Applications and oncepts, ourse 3

20 Find the value of in each quadrilateral NAME Practice: Skills lassifing Quadrilaterals lassif each quadrilateral using the name that best describes it trapezoid rectangle square square parallelogram rhombus quadrilateral rectangle trapezoid Lesson 6 4 Glencoe/McGraw-Hill 323 Mathematics: Applications and oncepts, ourse 3

21 NAME Practice: Word Problems lassifing Quadrilaterals 1. AMPING The outline for a piece of canvas used to make a tent is shown below. What is the value of in the quadrilateral? AMPING Refer to the figure in Eercise 1. lassif the quadrilateral using the name that best describes it. trapezoid ART The figure shows part of the pattern from a piece of stained glass. What is the measure of? A ART Refer to the figure in Eercise 3. The sides of quadrilateral AD are all congruent. lassif the quadrilateral using the name that best describes it. rhombus 120 D 5. HME IMPRVEMENT The cross section of a wheelbarrow is shown below. What is the value of in the figure? 6. HME IMPRVEMENT Refer to the figure in Eercise 5. lassif the quadrilateral using the name that best describes it. quadrilateral Glencoe/McGraw-Hill 324 Mathematics: Applications and oncepts, ourse 3

22 NAME Pre-Activit Reading to Learn Mathematics lassifing Quadrilaterals omplete the Mini Lab at the top of page 272 in our tetbook. Write our answers below. 1. Name the shape of the figures formed when ou drew the diagonal. How man figures were formed? triangles; 2 2. You know that the sum of the angle measures of a triangle is 180. Use this fact to find the sum of the angle measures in a quadrilateral. Eplain our reasoning. 360 ; 2(180 ) Find the measure of each angle of our quadrilateral. ompare the sum of these measures to the sum ou found in Eercise 2. See students work. The are the same, 360. Reading the Lesson 4. Eplain what is done at each step in the following problem. Find the value of in quadrilateral AD. 75 D A m A m m m D 360 The sum of the measures is Replace m A with 85, m with 80, m with Simplif Subtract 240 from each side. 120 Simplif. 5. How can ou tell the difference between a square and a rhombus? Sample answer: A square will have four right angles in addition to four congruent sides. Helping You Remember 6. omplete the table below b placing an in each bo to show the correct information for each shape. opposite sides parallel 4 congruent sides eactl one pair opposite sides parallel 4 right angles opposite sides congruent Parallelogram Rhombus Square Rectangle Trapezoid Lesson 6 4 Glencoe/McGraw-Hill 325 Mathematics: Applications and oncepts, ourse 3

23 NAME Enrichment Diagonals and Altitudes In these activities, ou will use decimals to describe two different geometric relationships. 1. The diagonal of a square connects opposite corners. Measure one side and the diagonal of this square to the nearest tenth of a centimeter. 4.0 cm; 5.7 cm diagonal 2. n a separate sheet of paper, draw squares of various sizes. Use our squares to complete this table. Answers will var. Length of Side Length of Diagonal 3. Describe the relationship between the diagonal of a square and its side. The length of the diagonal is about 1.4 times the length of a side. 4. An equilateral triangle has three sides of the same length. The altitude of the triangle connects one corner with the middle of the opposite side at a 90 angle. Measure one side and the altitude of this equilateral triangle to the nearest tenth of a centimeter. 4.0 cm; 3.5 cm altitude 5. n a separate sheet of paper, draw equilateral triangles of various sizes. Use our figures to complete this table. Answers will var. Length of Side Length of Diagonal side 6. Describe the relationship between the altitude of an equilateral triangle and its side. The length of the altitude is about 0.87 times the length of one side. Glencoe/McGraw-Hill 326 Mathematics: Applications and oncepts, ourse 3

24 NAME Stud Guide and Intervention ongruent Polgons Two polgons are congruent if all pairs of corresponding angles are congruent and all pairs of corresponding sides are congruent. The letters identifing each polgon are written so that corresponding vertices appear in the same order. Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement. Angles The arcs indicate that A D, F, and E. Sides The side measures indicate that A D F, F E, and A E D. Since all pairs of corresponding sides and angles are congruent, the two triangles are congruent. ne congruence statement is A DFE. 2 cm A D E 2 cm F Lesson 6 5 In the figure, MN LJH. a. Find JL. M N corresponds to J L. So,M N J L. Since MN 2 centimeters, JL 2 centimeters. M H 30 3 cm 2 cm N 1.5 cm 60 b. Find m H. According to the congruence statement, and H are corresponding angles. So, H. Since m 60, m H 60. J L Determine whether the polgons shown are congruent. If so, name the corresponding parts and write a congruence statement. 1. es; 3 m R P 4 m G 6 m 6 m S In the figure, GFD TRE. Find each measure. 2. m R m M H 3 m G 45 ft P M, R H, S G, P R M H, R S H G, P S M G ; PRS MHG ft 100 F E 3. RT 20 ft 4. m E 30 D ft R T Glencoe/McGraw-Hill 327 Mathematics: Applications and oncepts, ourse 3

25 Determine whether the polgons shown are congruent. If so, name the corresponding parts and write a congruence statement. 1. R L 140 D M 22 m 38 R S m 40 N 100 W m m F D N M 29 m P 142 P 29 m E 38 V es; no es; V E, W, N S, L P, R, V E W, M R, N L S P, W R, V E R ; L M P R, N M S R ; V EWR NLM SPR 4. A 4 in F 24 m 24 m 3 in. 5 in m 5 in. A 20 m R 3 in. E 4 in. NAME D Practice: Skills ongruent Polgons 120 es; no es; A E, D, D R, E S,, T, U, A E D, D, D E R S, E S T, A E ; T U, D R U ; A ED quadrilateral DE quadrilateral RSTU D E R U S 80 T In the figure, HFI MLK. Find each measure. 7. m M ML 10 d 9. m K KM 7 d H 10 d 30 7 d F 6 d I K L M In the figure, quadrilateral AD quadrilateral EFGH. Find each measure. 11. m H EF 38 ft 13. m F HG 20 ft A 28 ft H G ft 20 ft D E F 25 ft Glencoe/McGraw-Hill 328 Mathematics: Applications and oncepts, ourse 3

26 NAME Practice: Word Problems ongruent Polgons AIRPLANES The diagram at the right is of an airplane as seen from above. The wings of the airplane form congruent quadrilaterals, so quadrilateral AD quadrilateral EFGH. Use this figure for Eercises 1 and Name an unlabeled wing part whose length is 3 meters. Eplain our answer. Sample answer: E F ; E F is the corresponding side to A in quadrilateral AD. D H 10 m 120 A E 3 m m G F Eplain how a qualit control person could find out if m D was correct? Sample answer: The qualit control person could check whether m HGF 110. Lesson WHALES The flukes of the eluga whale are shaped like triangles. Determine whether these triangles are congruent. If so, name the corresponding parts and write a congruence statement. (Hint: R Q is a side of each triangle.) R S P Q 4. PATTERNS Mand is making name tags in the shape of triangles. The all should be the same size. Eplain how she can use a pattern to make 25 name tags. How does she know the are all congruent? Sample answer: She can trace the triangle 25 times and cut out the triangles. The triangles are congruent because their corresponding sides and angles are congruent. P R S R, R Q R Q, Q P Q S, PRQ SRQ, P S, PQR SQR, PQR SQR 5. ALGERA Find the value of in the two congruent triangles. 8 cm 4 cm 14 cm 10 cm 2 6. NATURE Part of a spider s web is shown in the figure. Determine whether the two marked triangles are congruent. If so, name the corresponding parts and write a congruence statement. es; A D, A E, E, A D E, D E, A D ; A DE Glencoe/McGraw-Hill 329 Mathematics: Applications and oncepts, ourse 3

27 Pre-Activit NAME Reading to Learn Mathematics ongruent Polgons Read the introduction at the top of page 279 in our tetbook. Write our answers below. 1. How man different triangles are shown in the Winter Stars quilt? Eplain our reasoning and draw each triangle. 2; All triangles in this pattern have the same shape and size as one of the two triangles below. 2. op the quilt and label all matching triangles with the same number, starting with 1. Reading the Lesson 4. What does it mean for two angles to be congruent? The have the same measure. 5. What does it mean for two polgons to be congruent? The polgons corresponding sides and angles are congruent. 6. What does congruence mean in our own words? Sample answer: ongruent means the same size or measure. Helping You Remember 7. Make a simple drawing of the front of a house. Include at least two pairs of congruent polgons in our drawing. Then write a sentence describing the congruent polgons. See students work. Sample answer: The windows are congruent rectangles. Glencoe/McGraw-Hill 330 Mathematics: Applications and oncepts, ourse 3

28 NAME Enrichment rder of Rotational Smmetr The two eamples below show how to create designs with rotational smmetr. In both cases, point P is the center of the rotation. Notice that the order of rotational smmetr is the number of times the original figure maps onto itself in a complete rotation. The first step in making the design is to divide 360 b the order of rotation. Lesson 6 5 P P P Rotate 45 Repeat 6 times. Rotational Smmetr and trace rder 8 P P P Rotate 120 Repeat 1 time. Rotational Smmetr and trace rder 3 Write the order of rotational smmetr for each design Use a shape like the one shown to create a design with the given center and order of rotational smmetr. 6. rder 8 7. rder 3 8. rder 5 9. rder rder rder 6 Glencoe/McGraw-Hill 331 Mathematics: Applications and oncepts, ourse 3

29 NAME Determine whether the figure has line smmetr. If it does, trace the figure and draw all lines of smmetr. If not, write none. This figure has three lines of smmetr. Stud Guide and Intervention Smmetr A figure has line smmetr if it can be folded over a line so that one half of the figure matches the other half. This fold line is called the line of smmetr.some figures have more than one line of smmetr. A figure has rotational smmetr if it can be rotated or turned less than 360 about its center so that the figure looks eactl as it does in its original position. The degree measure of the angle through which the figure is rotated is called the angle of rotation. Determine whether the figure has rotational smmetr. Write es or no. If es, name its angles of rotation. Yes, this figure has rotational smmetr. It matches itself after being rotated 180. For Eercises 1 6, complete parts a and b for each figure. a. Determine whether the figure has line smmetr. If it does, draw all lines of smmetr. If not, write none. b. Determine whether the figure has rotational smmetr. Write es or no. If es, name its angles of rotation a. none a. four lines of smmetr b. no a. one line of smmetr b. es; 90, 180, 270 b. no a. none a. one line of smmetr b. no a. two lines of smmetr b. no b. es; 180 Glencoe/McGraw-Hill 332 Mathematics: Applications and oncepts, ourse 3

30 NAME Practice: Skills Smmetr For Eercises 1 12, complete parts a and b for each figure. a. Determine whether the figure has line smmetr. If it does, draw all lines of smmetr. If not, write none. b. Determine whether the figure has rotational smmetr. Write es or no. If es, name its angles of rotation a. one line of smmetr a. two lines of smmetr b. no b. es; 180 a. none; b. es; 120 ; 240 Lesson a. two lines of a. three lines of a. two lines of smmetr smmetr smmetr b. es; 180 b. es; 120 ; 240 b. es; a. one line of a. si lines of a. four lines of smmetr smmetr smmetr b. no b. es; 60 ; 120 ; 180 ; b. es; 90 ; 180 ; ; a. none b. es; 180 a. one line of a. two lines of smmetr smmetr b. no b. es; 180 Glencoe/McGraw-Hill 333 Mathematics: Applications and oncepts, ourse 3

31 NAME Practice: Word Problems Smmetr 1. FLAGS The flag of the ahamas is shown below. Determine whether the flag has line smmetr. If it does, draw all lines of smmetr. If not, write none. 2. FLAGS Refer to the flag in Eercise 1. Determine whether the flag has rotational smmetr. Write es or no. If es, name its angles of rotation. no 3. FLAGS The flag of Scotland is shown below. Determine whether the flag has line smmetr. If it does, draw all lines of smmetr. If not, write none. 4. FLAGS Refer to the flag in Eercise 3. Determine whether the flag has rotational smmetr. Write es or no. If es, name its angles of rotation. es; LGS Discuss all of the properties of smmetr that the logo below has. 6. FLWER F LIFE This design has been found on Native American pots, in caves, and on buildings worldwide. Eplain how to determine how man lines of smmetr it has. How man lines of smmetr are there? The logo has two lines of smmetr. It also has rotational smmetr of 180. Sample answer: Look for lines along which the design can be folded so that both sides have the same shape; 6. Glencoe/McGraw-Hill 334 Mathematics: Applications and oncepts, ourse 3

32 NAME Reading to Learn Mathematics Smmetr Pre-Activit omplete the Mini Lab at the top of page 286 in our tetbook. Write our answers below. 1. Draw a line down the center of our starfish outline. Then fold our paper across this line. What do ou notice about the two halves? The two halves match. 2. Are there other lines ou can draw on our outline that will produce the same result? If so, how man? es; 4 more 3. Place the transparenc over the outline on our tracing paper. Use our pencil point at the centers of the starfish to hold the transparenc in place. How man times can ou rotate the transparenc from its original position so that the two figures match? Do not count the original position. 4 Lesson Find the first angle of rotation b dividing 360 b the number of times the figures matched List the other angles of rotation b adding the first angle of rotation to the previous angle. Stop when ou reach , 144, 216, 288 Reading the Lesson 6. What is a line of smmetr? Sample answer: If a figure can be folded over a line so that one half of the figure matches the other half, the line is a line of smmetr. 7. What does it mean to rotate a figure? Sample answer: to turn the figure about its center 8. After a 360 rotation about its center, a figure looks eactl as it does in its original position. Does this mean that the figure has rotational smmetr? Eplain. No; ever figure has this propert. A figure has rotational smmetr if the angle of rotation is less than 360. Helping You Remember 9. Name two everda things that have rotational smmetr. Eplain how ou know these items have rotational smmetr. Sample answer: Pinwheel and snowflake; when ou rotate them a certain number of degrees less than 360, the appear to be in their original position. Glencoe/McGraw-Hill 335 Mathematics: Applications and oncepts, ourse 3

33 NAME Enrichment Matchstick Puzzles The puzzles on this page involve onl a bo of wooden matches. Toothpicks or an other straight objects ma also be used. 1. Move 4 matches to make 2. Remove 3 matches to make 3 equilateral triangles. 3 equilateral triangles. 3. Remove 5 matches to make 4. Move 6 matches to make a 5 congruent triangles. si-pointed star. 5. Move 6 matches to make a star with 6. Move 2 matches to make 12 congruent parallelograms. 6 equilateral triangles. Glencoe/McGraw-Hill 336 Mathematics: Applications and oncepts, ourse 3

34 NAME Stud Guide and Intervention Reflections When a figure is reflected across a line, ever point on the reflection is the same distance from the line of reflection as the corresponding point on the original figure. The image is congruent to the original figure, but the orientation is different from that of the original figure. Draw the image of quadrilateral AD after a reflection over the given line. A A D Step 1 ount the number of units between each verte and the line of reflection. D A 3 D A Step 2 To find the corresponding point for verte A, move along the line through verte A perpendicular to the line of reflection until ou are 3 units from the line on the opposite side. Draw a point and label it A. Repeat for each verte. Step 3 onnect the new vertices to form quadrilateral A D. Lesson 6 7 ' D D' A' ' Notice that if ou move along quadrilateral AD from A to to to D, ou are moving in the clockwise direction. However, if ou move along quadrilateral A D from A to to to D, ou are moving in the counterclockwise direction. A figure and its reflection have opposite orientations. Draw the image of the figure after a reflection over the given line S R T F I G H A D Glencoe/McGraw-Hill 337 Mathematics: Applications and oncepts, ourse 3

35 NAME Practice: Skills Reflections Draw the image of the figure after a reflection over the given line R K L A Q J S M A D Graph the figure with the given vertices. Then graph the image of the figure after a reflection over the given ais and write the coordinates of its vertices. 5. triangle A with vertices A(1, 4), 6. triangle DEF with vertices D( 1, 2), (4, 1), and (2, 5); -ais E( 3, 1), and F( 4, 5); -ais A (1, 4), (4, 1), and (2, 5) D (1, 2), E (3, 1), and F (4, 5) 7. trapezoid WXYZ with vertices W(2, 4), 8. rhombus QRST with vertices Q( 1, 5), X(2, 2), Y(4, 1), and Z(4, 3); -ais R( 4, 3), S( 1, 1), and T(2, 3); -ais W ( 2, 4), X ( 2, 2), Y ( 4, 1), and Z ( 4, 3) Q ( 1, 5), R ( 4, 3), S ( 1, 1), and T (2, 3) Glencoe/McGraw-Hill 338 Mathematics: Applications and oncepts, ourse 3

36 NAME Practice: Word Problems Reflections 1. ALPHAET The figure shows the letter V plotted on a coordinate sstem. Find the coordinates of points and D after the figure is reflected over the -ais. D 2. GREEK The figure shows the Greek letter gamma plotted on a coordinate sstem. Find the coordinates of points P and Q after the figure is reflected over the -ais. Then draw the reflected image. P Q (2, 0), D (0, 3) P ( 4, 4), Q ( 2, 3) 3. RAFTS andace is making a pattern for star-shaped ornaments. omplete the pattern shown so that the completed star has a vertical line of smmetr. 4. FLRING The Turners are replacing the flooring in their dining room. omplete the design shown so that the completed floor has a horizontal line of smmetr. Lesson FLAG Macedonia is a countr near Greece and Albania. The national flag of Macedonia has both vertical and horizontal smmetr. omplete the flag of Macedonia. 6. YTE Dasan is preparing a presentation on animal safet. Finish the drawing of a coote s footprint so that it has vertical smmetr. Glencoe/McGraw-Hill 339 Mathematics: Applications and oncepts, ourse 3

37 Pre-Activit NAME Reading to Learn Mathematics Reflections Read the introduction at the top of page 290 in our tetbook. Write our answers below. 1. ompare the shape and size of the bird to its image in the water. The size and shape of the bird and its image appear to be the same. 2. ompare the perpendicular distance from the water line to each of the points shown. What do ou observe? The distances from corresponding points on the bird and its image to the water line are equal. 3. The points A,, and appear counterclockwise on the bird. How are these points oriented on the bird s image? The points A,, and appear clockwise on the image of the bird. Reading the Lesson For Eercises 4 7, read the Ke oncept information on page 290 in our tetbook and refer to the graph at the right that shows quadrilaterals AD and A D and a line. 4. Find the distance of each verte from the given line. Are corresponding vertices, such as A and A, the same distance from the line? A: 1 unit, : 3 units, : 3 units, D: 1 unit, A : 1 unit, : 3 units, : 3 units, D : 1 unit; es 5. Are quadrilaterals AD and A D congruent? Eplain. Yes; corresponding sides have equal lengths and corresponding angles have equal measures. 6. Do the vertices A,,, and D appear clockwise or counterclockwise on quadrilateral AD? Do the vertices A,,, and D appear clockwise or counterclockwise on quadrilateral A D? Do the two figures have the same orientation? clockwise; clockwise; es ' ' D' A A' D 7. Is quadrilateral A D a reflection of quadrilateral AD over the given line? Wh or wh not? No; the figures do not have different orientations. Helping You Remember 8. Find two real-life eamples of objects that have line smmetr. Then draw a sketch of each so that the sketch can be reflected over a line to produce a drawing of the entire object. See students work. Glencoe/McGraw-Hill 340 Mathematics: Applications and oncepts, ourse 3

38 NAME Enrichment Your Move Tic-Tac-Toe is a game that is usuall plaed on a 3 3 figure similar to the one shown at the right. A variation of the 3 3 Tic-Tac-Toe game is plaed on a three-dimensional figure. In this game, an plaer completing a horizontal, vertical, or diagonal row or column on one level or more than one level wins the game. For eample, the left-most figure below shows two eamples of game-winning sequences, one with s winning horizontall and one with Xs winning verticall. hoose a partner and pla three-dimensional Tic-Tac-Toe. Use a pencil so the game can be erased and plaed again. Eample: 0 X Lesson 6 7 X X X Glencoe/McGraw-Hill 341 Mathematics: Applications and oncepts, ourse 3

39 NAME Stud Guide and Intervention Translations When a figured is translated, ever point is moved the same distance in the same direction. The translated figure is congruent to the original figure and has the same orientation. Draw the image of quadrilateral AD after a translation 2 units right and 3 units up. A A' ' D' ' 3 Step 1 To find the corresponding point for verte A, start at A and move 2 units to the right along the horizontal grid line and then move up 3 units along the vertical grid line. Draw a point and label it A. Repeat for each verte. A D D 2 ' Step 2 onnect the new vertices to form quadrilateral A D. A' ' A D' D Draw the image of the figure after the indicated translation units right and 4 units down 2. 3 units left and 2 units up S R G T F H 3. 2 units left and 3 units down 4. 2 units right and 1 unit up A D L K M N Glencoe/McGraw-Hill 342 Mathematics: Applications and oncepts, ourse 3

40 Draw the image of the figure after the indicated translation units left and 2. 4 units right and 3. 1 unit left and 4. 5 units right and 3 units up 1 unit up 2 units down 3 units down R NAME Graph the figure with the given vertices. Then graph the image of the figure after the indicated translation and write the coordinates of its vertices. 5. triangle A with 6. triangle XYZ with 7. triangle EFG with vertices A( 3, 1), vertices X(1, 2), Y(3, 5), vertices E(1, 4), F( 1, 1), ( 4, 4), and ( 1, 2) and Z(4, 1) translated and G(2, 1) translated translated 4 units right 5 units left and 3 units 3 units left and 1 unit and 1 unit up up down S T Practice: Skills Translations A D J M K L Q T R S A (1, 0), (0, 3), X ( 4, 1), Y ( 2, 2), E ( 2, 3), F ( 4, 0), and (3, 1) and Z ( 1, 4) and G ( 1, 2) 8. rhombus WXYZ with 9. rectangle QRST with 10. trapezoid DE with vertices W( 4, 3), X( 1, 1), vertices Q( 2, 4), vertices (2, 1), Y(2, 3), and Z( 1, 5) R( 2, 1), S( 4, 1), and (3, 3), D( 3, 3), and translated 2 units right T( 4, 4) translated E(0, 1) translated 1 unit and 5 units down 3 units right and 3 units left and 4 units up up Lesson X W ( 2, 2), X (1, 4), (1, 3), (2, 1), Y (4, 2), and Z (1, 0) Q (1, 1), R (1, 4), D ( 4, 1), and E ( 1, 3) S ( 1, 4) and T ( 1, 1) Glencoe/McGraw-Hill 343 Mathematics: Applications and oncepts, ourse 3

41 NAME Practice: Word Problems Translations 1. UILDINGS The figure shows an outline of the White House in Washington, D.., plotted on a coordinate sstem. Find the coordinates of points and D after the figure is translated 2 units right and 3 units up. 2. UILDINGS Refer to the figure in Eercise 1. Find the coordinates of points and D after the figure is translated 1 unit left and 4 units up. ( 2, 2), D (1, 0) (1, 1), D (4, 1) D 3. ALPHAET The figure shows a capital N plotted on a coordinate sstem. Find the coordinates of points F and G after the figure is translated 2 units right and 2 units down. F 4. ALPHAET Refer to the figure in Eercise 3. Find the coordinates of points F and G after the figure is translated 5 units right and 6 units down. F (1, 2), G (3, 5) G F ( 2, 2), G (0, 1) 5. QUILT The beginning of a quilt is shown below. Look for a pattern in the quilt. op and translate the quilt square to finish the quilt. 6. EAH Tlia is walking on the beach. op and translate her footprints to show her path in the sand. Glencoe/McGraw-Hill 344 Mathematics: Applications and oncepts, ourse 3

42 Pre-Activit NAME Reading to Learn Mathematics Translations Read the introduction at the top of page 296 in our tetbook. Write our answers below. 1. Describe the motion involved in moving the knight. left two squares and up one square 2. ompare the shape, size, and orientation of the knight in its original position to that of the knight in its new position. The are the same. Reading the Lesson For Eercises 3 6, read the Ke oncept information on page 296 in our tetbook and refer to the graph at the right that shows quadrilaterals AD and A D. 3. Determine how man units up and to the right each verte of A quadrilateral AD has moved to get to the corresponding verte of quadrilateral A D. Have all the vertices moved the same distance in the same direction? A to A : 2 units D up and 6 units to the right, to : 0 units up and 2 units to the right, to : 6 units up and 2 units to the right, D to D : 8 units up and 6 units to the right; no 4. Are quadrilaterals AD and A D congruent? Eplain. Yes; corresponding sides have equal lengths and corresponding angles have equal measures. 5. Do the vertices A,,, and D appear clockwise or counterclockwise on quadrilateral AD? Do the vertices A,,, and D appear clockwise or counterclockwise on quadrilateral A D? Do the two figures have the same orientation? clockwise; clockwise; es ' ' D' A' Lesson X Is quadrilateral A D a translation of quadrilateral AD? Wh or wh not? No; the points are not moved the same distance in the same direction. Helping You Remember 7. Translations and reflections are both tpes of transformations. ne wa to remember the difference between them is to note that a translation is a slide while a reflection is a flip. ome up with our own was to help ou distinguish between the different tpes of transformations. See students work. Glencoe/McGraw-Hill 345 Mathematics: Applications and oncepts, ourse 3

43 NAME Enrichment Reflections Reflections can create man interesting patterns and shapes. Reflect the pattern in Quadrant I over the vertical -ais into Quadrant II. Then reflect the pattern in Quadrants I and II over the horizontal -ais into Quadrants III and IV. Quadrants II I III IV Glencoe/McGraw-Hill 346 Mathematics: Applications and oncepts, ourse 3

44 A' A D NAME Graph trapezoid AD with vertices A(1, 3), (4, 4), (4, 0), and D(1, 1). Then graph the image of trapezoid AD after a rotation 90º counterclockwise about the origin and write the coordinates of its vertices. Stud Guide and Intervention Rotations When a figure is rotated about a point, ever point on the original figure has a corresponding point on the rotated image. A point and its corresponding point are the same distance from the center of rotation. The angles formed b connecting each point and its corresponding point to the center of rotation are all congruent. The rotated figure is congruent to the original figure and has the same orientation. Step 1 Graph trapezoid AD. Step 2 To find the corresponding point for verte A, draw a line segment between A and the origin. Then draw a second line segment starting at the origin that is the same length as the first segment and forms a 90º angle with the first segment. Draw a point at the end of the second segment and label it A. Step 3 ' Step 4 A ' ' A A' D A' D' D Repeat for verte. Repeat for vertices and D. Then connect the new vertices to form trapezoid A D. Graph the figure with the given vertices. Then graph the image of the figure after the indicated rotation about the origin and write the coordinates of its vertices. 1. triangle GHI with vertices G(1, 0), 2. polgon TUVW with vertices T(2, 4), H(3, 1), and I(2, 5); 90 counterclockwise U(3, 1), V( 1, 0), and W( 2, 3); 180 G (0, 1), T ( 2, 4), H ( 1, 3), U ( 3, 1), and I ( 5, 2) V (1, 0), and W (2, 3) Lesson 6 9 Glencoe/McGraw-Hill 347 Mathematics: Applications and oncepts, ourse 3

45 NAME Practice: Skills Rotations Graph the figure with the given vertices. Then graph the image of the figure after the indicated rotation about the origin and write the coordinates of its vertices. 1. triangle A with vertices A(1, 5), 2. triangle FGH with vertices F(4, 2), G(1, 1), (3, 4), and (4, 0); 90 counterclockwise and H(1, 5); 180 A ( 5, 1), ( 4, 3), and (0, 4) F ( 4, 2), G ( 1, 1), and H ( 1, 5) 3. triangle XYZ with vertices X(4, 4), 4. triangle LMN with vertices L( 1, 4), Y(0, 5), and Z(1, 2); 180 M( 4, 3), and N( 2, 1); 90 counterclockwise X ( 4, 4), Y (0, 5), and Z ( 1, 2) L (4, 1), M (3, 4), and N (1, 2) 5. square DEFG with vertices D( 4, 5), 6. quadrilateral TUVW with vertices E( 1, 5), F( 1, 2), and G( 4, 2); T( 2, 1),U( 1, 4), V( 4, 4), and W( 5, 0); 90 counterclockwise 180 D ( 5, 4), E ( 5, 1), T'(2, 1), U'(1, 4), V'(4, 4), and F ( 2, 1), and G ( 2, 4) W (5, 0) Glencoe/McGraw-Hill 348 Mathematics: Applications and oncepts, ourse 3

46 NAME Practice: Word Problems Rotations 1. ALPHAET Draw a figure on the grid below so that the figure together with its image after a 180 rotation will form a letter of the alphabet. Sample answer: 2. ALPHAET Draw a figure on the grid below so that the figure together with its images after 90, 180, and 270 counterclockwise rotations will form a letter of the alphabet. Sample answer: 3. QUILTS omplete the pattern for a quilt square b rotating the design 180 about the given point. What does the completed figure resemble? the numeral eight 4. QUILTS omplete the pattern for a quilt square b rotating the figure 90, 180, and 270 counterclockwise about the given point. 5. SNWFLAKE Mr. Ai is cutting paper snowflakes to decorate his classroom. omplete the snowflake below so that the completed figure has smmetr with 90, 180, and 270 as its angles of rotation. 6. LG The local swimming pool is having a contest, and all students are welcome to enter. The pool officials want a new logo that has rotational smmetr with 120 and 240 as its angles of rotation. The student whose logo is chosen will win a one-ear pass to the pool. In the space below, draw an entr for the contest. See students work. Lesson 6 9 Glencoe/McGraw-Hill 349 Mathematics: Applications and oncepts, ourse 3

47 Pre-Activit NAME Reading to Learn Mathematics Rotations omplete the Mini Lab at the top of page 300 in our tetbook. Write our answers below. 1. Measure the distances from points on the original figure and corresponding points on the image to the center of rotation. What do ou observe? These distances for corresponding points are equal. 2. Measure the angles formed b connecting the center of rotation to pairs of corresponding points. What do ou observe? The angles all have the same measure. Reading the Lesson For Eercises 3 6, read the Ke oncept information on page 300 of our tetbook and refer to the graph at the right that shows quadrilaterals AD and A D and a point. 3. Find the distance of each of the vertices A,, A, and from the given point. Are corresponding vertices, such as A and A, the same distance from the point? A: 3 units, : 1 unit, A : 1 unit, : 3 units; no A D ' A' ' D' 4. Are quadrilaterals AD and A D congruent? Eplain Yes; corresponding sides have equal lengths and corresponding angles have equal measures. 5. Do the vertices A,,, and D appear clockwise or counterclockwise on quadrilateral AD? Do the vertices A,,, and D appear clockwise or counterclockwise on quadrilateral A D? Do the two figures have the same orientation? clockwise; clockwise; es 6. Is quadrilateral A D a rotation of quadrilateral AD about the given point? Wh or wh not? No; corresponding vertices on the quadrilaterals are not the same distance from the point. Helping You Remember 7. omplete the table below to help ou remember the different tpes of transformations. Movement Turning about a point Flipping across a line Sliding in the plane Tpe of Transformation Rotation Reflection Translation Glencoe/McGraw-Hill 350 Mathematics: Applications and oncepts, ourse 3

48 NAME Enrichment Rotations Rotations occur when a figure or object (like the blade of a fan) moves about a pivot, or central point. Rotations can occur in clockwise or counterclockwise directions. Using the origin (0, 0) as the pivot point, rotate the pattern in Quadrant I 90 counterclockwise. After this rotation has been completed, rotate the pattern 90 counterclockwise into Quadrant III, then rotate the pattern 90 counterclockwise into Quadrant IV. Quadrants II I origin III IV Lesson 6 9 Glencoe/McGraw-Hill 351 Mathematics: Applications and oncepts, ourse 3