Geometric Ideas R 6-1 Lines, line segments, and rays are basic geometric ideas. They are sometimes described by the relationship they have to other lines, line segments, and rays. Draw Write Say Description E G F H EF GH Line EF is parallel to EF and GH are parallel line GH. lines. They are the same distance apart and will not cross each other. G N O P NP intersects GO Line NP intersects line GO. NP and GO are intersecting lines. They pass through the same point. V S T SU VT Line SU is perpendicular to line VT. SU and VT are perpendicular lines. They intersect and form square corners. U Use the diagram at the right. Name the following. 1. two parallel lines K E F H 2. a line segment D G 3. two perpendicular lines 4. two lines 5. a ray L A B C V I 6. two perpendicular rays 7. a point Use with Lesson 6-1. 73
Geometric Ideas P 6-1 Use the diagram at the right. Name the following. 1. three points P J Q R E I F 2. a ray N A 3. two intersecting lines but not perpendicular 4. two parallel lines 5. a line segment 6. two perpendicular lines 7. Reasoning Can a line segment have two midpoints? Explain. Test Prep 8. Which type of lines are shown by the figure? A. Intersecting B. Parallel C. Perpendicular D. Curved 9. Writing in Math Draw and label two perpendicular line segments KL and. MN A B Use with Lesson 6-1. 73
Measuring and Classifying Angles R 6-2 The chart below can help you to describe and identify an angle. Measure EFG. 90 E Remember to place the 0 mark on one 180 0 side of the angle. F G EFG measures 45. Draw an angle of 65. S W Place the center of the protractor on S. Line up SW with the 0 mark. Place a point at 65. Label it J. Draw SJ. Classifying Angles Acute measures between 0 and 90 Right measures exactly 90 Obtuse measures between 90 and 180 Straight measures exactly 180 Classify each angle as acute, right, obtuse, or straight. Then measure each angle. 1. 2. 3. Draw an angle with each measure. 4. 100 5. 170 6. 90 7. Reasoning ABC measures less than 180 but more than 90. Is ABC a right, an acute, an obtuse, or a straight angle? 74 Use with Lesson 6-2.
Measuring and Classifying Angles P 6-2 Classify each angle as acute, right, obtuse, or straight. Then measure each angle. (Hint: Draw longer sides if necessary.) 1. 2. Draw an angle with each measure. 3. 120 4. 180 5. Draw an acute angle. Label it with the letters A, B, and C. What is the measure of the angle? Test Prep 6. Which kind of angle is shown in the figure below? A. Acute B. Obtuse C. Right D. Straight 7. Writing in Math Explain how to use a protractor to measure an angle. 74 Use with Lesson 6-2.
Segments and Angles Related to Circles R 6-3 All the points on a circle are the same distance from the center. The center of C circle C is at point C. A radius goes from the center to any point on the circle. RS is a radius of circle R. R S (plural: radii) A chord has both endpoints on a circle. HD is a chord of circle O. O A diameter passes through the center of a circle. DR is a diameter of circle M. H D D A diameter of a circle is also the longest chord of that circle. A circle s diameter is twice as long as its radius. M R A central angle is an angle whose vertex is the center. ACB is a central angle. D C A B Use the terms at the top of this page to identify each figure in circle F. 1. point F R G 2. GFS 3. R S 4. U V 5. G F U F V S 6. Reasoning If a circle has a radius of 12 in., what is the circle s diameter? Use with Lesson 6-3. 75
Segments and Angles Related to Circles P 6-3 Identify each figure in circle D. E N 1. NDI 2. LE 3. DI L D I 4. AN A 5. point D 6. DA 7. Reasoning Is the radius of a circle half of the diameter? Explain. 8. Construct a circle that has a radius with an equal length to the given line segment. Test Prep 9. A line segment that connects two points on a circle passing through the center is called a A. Radius B. Diameter C. Chord D. Central angle 10. Writing in Math Brenda thinks that a chord can be many different lengths on a given circle. Is she right? Explain your thinking. Use with Lesson 6-3. 75
Polygons R 6-4 A polygon is a closed plane figure made up of line segments. Common polygons have names that tell the number of sides the polygon has. A regular polygon has sides of equal length and angles of equal measure. Triangle 3 sides Pentagon 5 sides Octagon 8 sides 60 3 in. 3 in. 60 60 3 in. Hexagon 6 sides Open Figure Quadrilateral 4 sides Each side is 3 in. long. Each angle is 60. Name each polygon. Then tell if it appears to be a regular polygon. 1. 2. 3. 4. 5. Reasoning Shakira sorted shapes into two different groups. Use geometric terms to describe how she sorted the shapes. Group A Group B 76 Use with Lesson 6-4.
Polygons P 6-4 Name each polygon. Then tell if it appears to be a regular polygon. 1. 2. 3. Name the polygon. Name the vertices. H T B Test Prep 4. Which polygon has eight sides? A. Quadrilateral B. Pentagon C. Hexagon D. Octagon 5. Writing in Math Draw two regular polygons and two that are irregular. Use geometric terms to describe one characteristic of each type. E 76 Use with Lesson 6-4.
Classifying Triangles R 6-5 You can classify triangles by the lengths of their sides and the sizes of their angles. acute all angles less than 90 60 8 cm 8 cm 60 60 8 cm equilateral all sides the same length This triangle is both equilateral and acute. Not all acute triangles are equilateral. right one right angle 16 in. 16 in. 24 in. isosceles two sides the same length This triangle is both isosceles and right. Not all right triangles are isosceles. obtuse one obtuse angle 7.5 ft 120 5 ft 4 ft scalene no sides the same length This triangle is both scalene and obtuse. Not all obtuse triangles are scalene. Remember that the sum of the measures of the angles of a triangle is 180. Classify each triangle by its sides and then by its angles. 1. 2. 3. 8 cm 8 cm 60 128 3 ft 3 ft 26 26 14 cm 60 60 3 ft 5 in. 4 in. 3 in. The measures of two angles of a triangle are given. Find the measure of the third angle. 4. 40, 100, 5. 14, 98, 6. 38, 38, Use with Lesson 6-5. 77
Classifying Triangles P 6-5 Classify each triangle by its sides and then by its angles. 1. 2. 12 ft 9.5 ft 4 m 60 4 m 8 ft 60 60 4 m The measures of two angles of a triangle are given. Find the measure of the third angle. 3. 47, 62, 4. 29, 90, 5. 75, 75, 6. 54, 36, 7. Judy bought a new tent for a camping trip. Look at the side of the tent with the opening to classify the triangle by its sides and its angles. Test Prep 8. Which describes a scalene triangle? A. 4 equal sides B. 3 equal sides C. 2 equal sides D. 0 equal sides 9. Writing in Math The lengths of two sides of a triangle are 15 in. each. The third side measures 10 in. What type of triangle is this? Explain your answer using geometric terms. Use with Lesson 6-5. 77
Classifying Quadrilaterals R 6-6 Quadrilateral Definition Example Parallelogram Rectangle Rhombus Square Trapezoid A quadrilateral with both pairs of opposite sides parallel and equal in length A parallelogram with four right angles A parallelogram with all sides the same length A rectangle with all sides the same length A quadrilateral with only one pair of parallel sides 5 in. 2 in. 2 in. 5 in. 5 ft 2 ft 2 ft 5 ft 4 in. 4 in. 4 in. 4 in. 1 ft 1 ft 1 ft 1 ft 2 in. 2 in. 3 in. 6 in. Remember that the sum of the measures of the angles of a quadrilateral is 360. Classify each quadrilateral. Be as specific as possible. 1. 6 ft 2. 4 in. 3. 3 ft 3 ft 4 in. 4 in. 3 ft 7 ft 9 ft 6 ft 4 in. The measures of three angles of a quadrilateral are given. Find the measure of the fourth angle. 6 ft 4. 65, 150, 89, 5. 100, 80, 100, 6. 82, 78, 90, 78 Use with Lesson 6-6.
Classifying Quadrilaterals P 6-6 Classify each quadrilateral. Be as specific as possible. 1. 2. 15 cm 8 cm 8 cm 15 cm 3. 4. 8 in. 17 cm 4 in. 4 in. 17 cm 17 cm 8 in. 17 cm The measures of three angles of a quadrilateral are given. Find the measure of the fourth angle. 5. 90, 145, 78, 6. 110, 54, 100, 7. Name the vertices of the square. F G Test Prep 8. Three of the angles of a quadrilateral measure 80, 100, and 55. Which is the measure of the fourth angle? A. 115 B. 120 C. 125 D. 130 9. Writing in Math Can a trapezoid have four obtuse angles? Explain. L A 78 Use with Lesson 6-6.
PROBLEM-SOLVING STRATEGY Solve a Simpler Problem R 6-7 Give Away During a grand opening, a mall provides prizes for the first 70 customers. The 1st customer receives a coupon book, the 2nd customer, a travel mug, and the 3rd customer, a T-shirt. If the prizes continue to be given away in the same pattern, which prize would the 70th customer receive? Read and Understand Step 1: What do you know? There are 3 different prizes given away in a pattern. Step 2: What are you trying to find? Which gift the 70th customer will receive Plan and Solve Step 3: What strategy will you use? Strategy: Solve a simpler problem I can divide 69 by 3. I know the 69th customer will receive the 3rd prize. The 70th customer will receive the coupon book. Look Back and Check Is your work correct? Yes, 69 can be divided by 3 evenly to find the 3rd prize. The 70th customer would be next for the 1st prize. Solve the simpler problem. Use the solution to help you solve the original problem. 1. Anna can choose from white or yellow paper. She can choose from thin or wide paintbrushes. She can choose from green, purple, or blue paint. How many different combinations of art supplies are possible? Use with Lesson 6-7. 79
PROBLEM-SOLVING STRATEGY Solve a Simpler Problem Solve the simpler problems. Use the solutions to help you solve the original problem. 1. Reggie is designing a triangular magazine rack with 5 shelves. The top shelf will hold 1 magazine. The second shelf will hold 3 magazines, and the third shelf will hold 5 magazines. This pattern continues to the bottom shelf. How many magazines will the magazine rack hold altogether? P 6-7 Simpler Problem What is the pattern? How many magazines will the fourth shelf hold? How many magazines will the bottom shelf hold? Solution: 2. At the deli, you receive 1 free sub after you buy 8 subs. How many free subs will you receive from the deli if you buy 24 subs? 3. The chef has 5 different kinds of pasta and 3 different flavors of sauce. How many different meals is she able to make? Use with Lesson 6-7. 79
PROBLEM-SOLVING SKILL Writing to Describe R 6-8 Use geometric terms to describe one characteristic of the lines in each group. Group 1 Group 2 E B Q C R T K A F D X U Y U J V You can brainstorm to help you describe the characteristics. Group 1 Group 2 no right angles not parallel all intersecting no perpendicular lines all right angles not parallel all intersecting all perpendicular lines When you write your description, use mathematical terms correctly and be brief. All of the lines in Group 1 are intersecting and not perpendicular. All of the lines in Group 2 are intersecting and are perpendicular. 1. Abdul sorted the polygons to the right into two groups. Use geometric terms to describe a characteristic in each group. Group A Group B 80 Use with Lesson 6-8.
PROBLEM-SOLVING SKILL Writing to Describe P 6-8 Solve each problem. You may use a brainstorming table to help plan your description. 1. Use geometric terms to describe three properties of a square. 2. Use mathematical terms to describe the numbers in each set. 83 3 27 35 9 1 28 6 42 86 72 8 Set C 3. Mariam created two patterns. Use mathematical terms to describe the numbers in each pattern. Set D Pattern 1: 12, 16, 20, 24, 28, 32 Pattern 2: 1, 2, 4, 8, 16, 32, 64 4. Use geometric terms to describe three properties of this trapezoid. 80 Use with Lesson 6-8.
Congruence and Similarity R 6-9 Similar polygons have the same angle measures, but the lengths are different. Think: same shape, different size 4.5 ft 53 7.5 ft 12 ft 37 14 ft 37 6 ft 53 9 ft Congruent polygons have the same angle measures and the lengths of the sides are the same. Think: same shape, same size 10 m 5 m 5 m 10 m 10 m 5 m 5 m 10 m Do the figures in each pair appear to be similar? If so, are they also congruent? 1. 2. 3. 4. Use with Lesson 6-9. 81
Congruence and Similarity P 6-9 Do the figures in each pair appear to be similar? If so, are they also congruent? 1. 2. Is each situation an example of congruent or similar figures? Explain your reasoning. 3. When baking bread, the baker uses two sizes of loaf pans. One loaf pan fits inside the other. 4. A friend gave you and five friends each a quarter. Test Prep 5. If Figure A is similar to Figure B, which is the measure of XYZ in Figure A? A. 90 B. 58 C. 45 D. 32 X Z Figure A 6. Writing in Math If two circles are congruent, are the diameters the same length? Explain. Y 58 B A 32 Figure B C Use with Lesson 6-9. 81
Transformations R 6-10 The size and shape of a figure do not change when it is slid, flipped, or turned. 1. A slide, or translation, moves a figure in a straight direction. 2. A flip, or reflection, of a figure gives its mirror image. 3. A turn, or rotation, moves a figure about a point. Turns can be measured in degrees. Here are some common turns. 1 90 or turn 4 1 180 or turn 2 3 270 or turn 360 or full turn 4 Tell whether the figures in each pair are related by a slide, a flip, or a turn. If a turn, tell what type of turn. 1. 2. 3. 4. 82 Use with Lesson 6-10.
Transformations P 6-10 Tell whether the figures in each pair are related by a flip, slide, or turn. If a turn, describe it. 1. 2. 3. On a compass, if you are standing at 0 and you turn to your right to make a 90 turn, what fraction of a turn is that? 4. If a figure is flipped over the dashed line and then rotated a 1 4 turn counterclockwise, which of the figures below shows the result? A. B. C. D. Test Prep 5. Which term describes the mirror image of a figure? A. Slide B. Flip C. Turn D. Pattern 6. Writing in Math Mark says that the figure above was flipped. Faith says that it made a 3 4 turn. Steven says that it made a 90 turn. Who is correct? 82 Use with Lesson 6-10.
Symmetry R 6-11 If a figure is symmetric, it can be split in half so that one side reflects onto the other side. The fold line is called the line of symmetry. This figure has two lines of symmetry. If you folded along one of the lines, the two sides would fit on top of each other. This figure has one line of symmetry. There is only one way to fold it exactly in half. This figure has no lines of symmetry. It cannot be folded in half so that one side reflects onto the other side. How many lines of symmetry does each figure have? You may trace the figure to check. 1. 2. 3. Part of a symmetric trademark is shown. Complete the drawing. 4. Use with Lesson 6-11. 83
Symmetry P 6-11 How many lines of symmetry does each figure have? You may fold your paper to check. 1. 2. 3. Reasoning How many lines of symmetry does an equilateral triangle have? Explain. Part of a symmetric trademark is shown. Complete each drawing. 4. 5. Test Prep 6. Which figure has more than two lines of symmetry? A. B. C. D. 7. Writing in Math Draw a shape or figure that has more than 2 lines of symmetry. Put the lines of symmetry into your drawing. Use with Lesson 6-11. 83
PROBLEM-SOLVING APPLICATIONS The Geometry of Life R 6-12 Juanita and her friend Kim decided to meet at the park. Use geometric terms to describe the location of their houses and the park in relation to the line segment. Juanita's house Kim's house Park J P K Juanita s house is located at point J on JK. Kim s house is located at point K on JK. The park is located at P, the midpoint of JK. Use the diagram above for question 1. 1. What is the shape of the chimney at Juanita s house? 2. Kenneth s mother ordered a pizza for his graduation party. The pizza has a diameter of 12 in. What is the measurement of the radius of the pizza? 3. If it is 3:00 P.M., what is the measure of the angle that can be formed between the hour hand and the minute hand? 4. Angelina saw the sign at the right. What is the name of this polygon? 84 Use with Lesson 6-12.
PROBLEM-SOLVING APPLICATION The Shape We Are In 1. Tarah hung an American flag from the front porch of her house. She placed it in a flag stand on one of the posts of the porch. Measure the angle of the flagpole from the post. P 6-12 2. Construct a circle. Use the measure of the radius below. B A 3. Karen and Willie took a walk on the streets near their home. The path of their walk is shown at the right. Classify the triangle formed by the path by its angles and its sides. 4. Part of a symmetric figure is shown. Complete the drawing. 84 Use with Lesson 6-12.