Study Guide and Intervention

Transcription

1 IO 1-1 tudy Guide and Intervention oints, Lines, and lanes ame oints, Lines, and lanes In geometry, a point is a location, a line contains points, and a plane is a flat surface that contains points and lines. If points are on the same line, they are collinear. If points on are the same plane, they are coplanar. xample a. a line containing point Use the figure to name each of the following. he line can be named as. lso, any two of the three points on the line can be used to name it.,, or b. a plane containing point he plane can be named as plane or can be named using three noncollinear points in the plane, such as plane, plane, and so on. Lesson 1-1 xercises efer to the figure. 1. ame a line that contains point. 2. What is another name for line m? m 3. ame a point not on. 4. ame the intersection of and. 5. ame a point not on line or line m. raw and label a plane for each relationship. 6. is in plane. 7. intersects at. X Y 8. oint X is collinear with points and. 9. oint Y is not collinear with points and. 10. Line contains points X and Y. Glencoe/cGraw-Hill 1 Glencoe Geometry

2 IO 1-1 tudy Guide and Intervention (continued) oints, Lines, and lanes oints, Lines, and lanes in pace pace is a boundless, three-dimensional set of all points. It contains lines and planes. xample a. How many planes appear in the figure? O here are three planes: plane, plane O, and plane. b. re points,, and coplanar? Yes. hey are contained in plane O. xercises efer to the figure. 1. ame a line that is not contained in plane. 2. ame a plane that contains point. 3. ame three collinear points. efer to the figure. 4. How many planes are shown in the figure? 5. re points,, G, and H coplanar? xplain. G J H I 6. ame a point coplanar with,, and. raw and label a figure for each relationship. 7. lanes and intersect in HJ. t s 8. Line r is in plane, line s is in plane, and lines r and s intersect at point J. H J r 9. Line t contains point H and line t does not lie in plane or plane. Glencoe/cGraw-Hill 2 Glencoe Geometry

3 1-1 IO kills ractice efer to the figure. oints, Lines, and lanes 1. ame a line that contains point. 2. ame a point contained in line n. p n G 3. What is another name for line p? 4. ame the plane containing lines n and p. Lesson 1-1 raw and label a figure for each relationship. 5. oint K lies on. 6. lane J contains line s. K J s 7. Y lies in plane and contains point, but does not contain point H. in plane U. Y H 8. Lines q and f intersect at point Z U q Z f efer to the figure. 9. How many planes are shown in the figure? 10. How many of the planes contain points and? W 11. ame four points that are coplanar. 12. re points,, and coplanar? xplain. Glencoe/cGraw-Hill 3 Glencoe Geometry

4 1-1 IO ractice efer to the figure. oints, Lines, and lanes 1. ame a line that contains points and. 2. ame a line that intersects the plane containing points,, and. j g h 3. ame the plane that contains and. raw and label a figure for each relationship. 4. K and G intersect at point in plane. K G 5. line contains L( 4, 4) and (2, 3). Line q is in the same coordinate plane but does not intersect L. Line q contains point. y q O x L efer to the figure. 6. How many planes are shown in the figure? 7. ame three collinear points. W X 8. re points,,, and W coplanar? xplain. VIULIZIO ame the geometric term(s) modeled by each object tip of pin 11. O strings 12. a car antenna 13. a library card Glencoe/cGraw-Hill 4 Glencoe Geometry

5 1-2 IO tudy Guide and Intervention Linear easure and recision easure Line egments part of a line between two endpoints is called a line segment. he lengths of and are written as and. When you measure a segment, the precision of the measurement is half of the smallest unit on the ruler. xample 1 xample 2 ind the length of. ind the length of. cm in. 1 2 he long marks are centimeters, and the shorter marks are millimeters. he length of is 3.4 centimeters. he measurement is accurate to within 0.5 millimeter, so is between 3.35 centimeters and 3.45 centimeters long. xercises he long marks are inches and the short marks are quarter inches. he length of is about inches. he measurement is accurate to within one half of a quarter inch, or 1 8 inch, so is between 1 5 inches and ind the length of each line segment or object. inches long. Lesson cm in in. 1 2 cm ind the precision for each measurement in mm cm 8. 2 ft mm yd 2 Glencoe/cGraw-Hill 7 Glencoe Geometry

6 1-2 IO tudy Guide and Intervention (continued) Linear easure and recision alculate easures On, to say that point is between points and means,, and are collinear and. On, 3 cm. We can say that the segments are congruent, or. lashes on the figure indicate which segments are congruent. xample 1 xample 2 ind. ind x and. 1.2 cm 1.9 cm alculate by adding and herefore, is 3.1 centimeters long. 2x 5 x 2x is between and. x 2x 2x 5 3x 2x 5 x 5 2x 5 2(5) 5 15 xercises ind the measurement of each segment. ssume that the art is not drawn to scale cm 2.5 cm in. 6 in. 3. X Z 3 4. W X 31 2 in. 4 in. X Y Z W 6 cm X Y ind x and if is between and. 5. 5x, 3x, and x, 5x 4, and x, 12, and x,, and 24. Use the figures to determine whether each pair of segments is congruent. 9. and 11 cm 10. X Y and Y Z X 5 cm 5 cm 3x 5 5x 1 11 cm Y 9x 2 Z Glencoe/cGraw-Hill 8 Glencoe Geometry

7 1-2 IO kills ractice Linear easure and recision ind the length of each line segment or object cm in. 1 2 ind the precision for each measurement feet centimeters inches 2 ind the measurement of each segment G H 1in in. 4.9 cm 5.2 cm 9.7 mm G H 15 mm Lesson 1-2 ind the value of the variable and YZ if Y is between X and Z. 9. XY 5p, YZ p, and XY XY 12, YZ 2g, and XZ XY 4m, YZ 3m, and XZ XY 2c 1, YZ 6c, and XZ 81 Use the figures to determine whether each pair of segments is congruent. 13., 14., 15. W X, W Z 2 m 9 ft 12 yd Y Z 3 m 3 m 10 yd 5 ft 5 ft 5 m 12 yd X W Glencoe/cGraw-Hill 9 Glencoe Geometry

8 1-2 IO ractice Linear easure and recision ind the length of each line segment or object in. 1 2 cm ind the precision for each measurement meters inches millimeters 4 ind the measurement of each segment W X 18.4 cm 4.7 cm in in. W X 89.6 cm Y 100 cm ind the value of the variable and KL if K is between J and L. 9. JK 6r, KL 3r, and JL JK 2s, KL s 2, and JL 5s 10 Use the figures to determine whether each pair of segments is congruent. 11. U, W 12., 13. G, 2 ft 12.7 in. 5x G H 2 ft 3 ft 6x U W 3 ft 12.9 in. 14. Y Jorge used the figure at the right to make a pattern for a mosaic he plans to inlay on a tabletop. ame all of the congruent segments in the figure. Glencoe/cGraw-Hill 10 Glencoe Geometry

9 1-3 IO tudy Guide and Intervention istance and idpoints istance etween wo oints istance on a umber Line istance in the oordinate lane a b b a or a b ythagorean heorem: a 2 b 2 c 2 istance ormula: d (x x 2 1 ) 2 (y 2 y 1 ) 2 ( 2, 1) O y (1, 3) x (1, 1) xample 1 xample 2 ind ( 4) ind the distance between ( 2, 1) and (1, 3). ythagorean heorem () 2 () 2 () 2 () 2 (3) 2 (4) 2 () istance ormula d (x x 2 1 ) 2 (y 2 y 1 ) 2 (1 2)) ( 2 (3 ( 1)) 2 (3) 2 (4) xercises Use the number line to find each measure G G 6. G G Lesson 1-3 Use the ythagorean heorem to find the distance between each pair of points. 9. (0, 0), (6, 8) 10. ( 2, 3), (3, 15) 11. (1, 2), (9, 13) 12. ( 12, 2), ( 9, 6) Use the istance ormula to find the distance between each pair of points. 13. (0, 0), (15, 20) 14. O( 12, 0), ( 8, 3) 15. (11, 12), (6, 2) 16. ( 2, 10), ( 4, 3) Glencoe/cGraw-Hill 13 Glencoe Geometry

10 1-3 IO tudy Guide and Intervention (continued) istance and idpoints idpoint of a egment idpoint on a If the coordinates of the endpoints of a segment are a and b, umber Line then the coordinate of the midpoint of the segment is a. b 2 If a segment has endpoints with coordinates (x idpoint on a 1, y 1 ) and (x 2, y 2 ), oordinate lane then the coordinates of the midpoint of the segment are x 1 x, 2 y 1 y xample 1 ind the coordinate of the midpoint of he coordinates of and are 3 and 1. If is the midpoint of, then the coordinate of is or xample 2 is the midpoint of for ( 2, 4) and (4, 1). ind the coordinates of. x 2 x 1, y y , or (1, 2.5) xercises Use the number line to find the coordinate of the midpoint of each segment G G G G ind the coordinates of the midpoint of a segment having the given endpoints. 9. (0, 0), (12, 8) 10. ( 12, 8), (6, 12) 11. (11, 2), ( 9, 13) 12. ( 2, 6), ( 9, 3) 13. (10, 22), (9, 10) 14. ( 11, 2), ( 19, 6) Glencoe/cGraw-Hill 14 Glencoe Geometry

11 1-3 IO kills ractice istance and idpoints Use the number line to find each measure. J K L 1. L 2. JL K 4. Use the ythagorean heorem to find the distance between each pair of points y y G O x O x 7. K(2, 3), (4, 4) 8. ( 3, 1), ( 2, 3) Use the istance ormula to find the distance between each pair of points. 9. Y(2, 0), (2, 6) 10. W( 2, 2), (5, 2) 11. ( 7, 3), (5, 2) 12. ( 3, 1), (2, 6) Use the number line to find the coordinate of the midpoint of each segment Lesson ind the coordinates of the midpoint of a segment having the given endpoints. 17. (3, 1), U(5, 3) 18. J( 4, 2), (5, 2) ind the coordinates of the missing endpoint given that is the midpoint of. 19. (2, 0), (5, 2) 20. (5, 4), (6, 3) 21. (3, 9), ( 1, 5) Glencoe/cGraw-Hill 15 Glencoe Geometry

12 1-3 IO ractice istance and idpoints Use the number line to find each measure. 1. VW 2. V V U V W Use the ythagorean heorem to find the distance between each pair of points. 5. y 6. Z y O x O x Use the istance ormula to find the distance between each pair of points. 7. L( 7, 0), Y(5, 9) 8. U(1, 3), (4, 6) Use the number line to find the coordinate of the midpoint of each segment ind the coordinates of the midpoint of a segment having the given endpoints. 13. K( 9, 3), H(5, 7) 14. W( 12, 7), ( 8, 4) ind the coordinates of the missing endpoint given that is the midpoint of. 15. (5, 8), (4, 3) 16. (2, 9), ( 1, 6) 17. ( 3, 8), (1, 2) 18. I he coordinates of the vertices of a quadrilateral are ( 1, 3), (3, 3), (5, 1), and U( 2, 1). ind the perimeter of the quadrilateral. ound to the nearest tenth. Glencoe/cGraw-Hill 16 Glencoe Geometry

13 1-4 IO tudy Guide and Intervention ngle easure easure ngles If two noncollinear rays have a common endpoint, they form an angle. he rays are the sides of the angle. he common endpoint is the vertex. he angle at the right can be named as,,, or 1. right angle is an angle whose measure is 90. n acute angle has measure less than 90. n obtuse angle has measure greater than 90 but less than 180. xample 1 xample easure each angle and classify it as right, acute, or obtuse. 1 a. ame all angles that have as a vertex. hree angles are 1, 2, and 3. or other angles, use three letters to name them:,, and. b. ame the sides of 1., a. Using a protractor, m , so is an acute angle. b. Using a protractor, m , so is an obtuse angle. c. Using a protractor, m 90. is a right angle. xercises efer to the figure. 1. ame the vertex of ame the sides of. 3. Write another name for Lesson 1-4 easure each angle in the figure and classify it as right, acute, or obtuse Glencoe/cGraw-Hill 19 Glencoe Geometry

14 1-4 IO tudy Guide and Intervention (continued) ngle easure ongruent ngles ngles that have the same measure are congruent angles. ray that divides an angle into two congruent angles is called an angle bisector. In the figure, is the angle bisector of. oint lies in the interior of and. xample efer to the figure above. If m 2x 14 and m x 34, find x and find m. ince bisects,, or m m. 2x 14 x 34 m (2x 14) (x 34) 2x 14 x x 34 x x x x 20 xercises bisects, and and are opposite rays. 1. If m 60 and m 4x 14, find the value of x. 2. If m 3x 13 and m 6x 2, find m. and are opposite rays, bisects, and bisects. 3. If m 6x 4 and m 7x 2, find m If m 1 4x 10 and m 2 5x, find m If m 2 6y 2 and m 1 8y 14, find m. 6. Is a right angle? xplain. Glencoe/cGraw-Hill 20 Glencoe Geometry

15 1-4 IO kills ractice ngle easure or xercises 1 12, use the figure at the right. ame the vertex of each angle W U V ame the sides of each angle V 8. 1 Write another name for each angle W easure each angle and classify it as right, acute, or obtuse O O O LG In the figure, and are opposite rays, bisects, and bisects. 17. If m 4x 16 and m 6x 4, find m. L Lesson If m 7x 8 and m 5x 10, find m. Glencoe/cGraw-Hill 21 Glencoe Geometry

16 1-4 IO ractice ngle easure or xercises 1 10, use the figure at the right. ame the vertex of each angle ame the sides of each angle O O 8. O Write another name for each angle easure each angle and classify it as right, acute, or obtuse. V W 11. UZW 12. YZW U X 13. ZW 14. UZ Z Y LG In the figure, and are opposite rays, bisects, and G bisects. 15. If m 4x 15 and m 6x 5, find m. 16. If m G 9x 3 and m G 13x 9, find m G. G 17. I IG he diagram shows a sign used to warn drivers of a school zone or crossing. easure and classify each numbered angle. 2 1 Glencoe/cGraw-Hill 22 Glencoe Geometry

17 1-5 IO tudy Guide and Intervention ngle elationships airs of ngles djacent angles are angles in the same plane that have a common vertex and a common side, but no common interior points. Vertical angles are two nonadjacent angles formed by two intersecting lines. pair of adjacent angles whose noncommon sides are opposite rays is called a linear pair. xample Identify each pair of angles as adjacent angles, vertical angles, and/or as a linear pair. a. c. and U have a common vertex and a common side, but no common interior points. hey are adjacent angles. U and 5 are adjacent angles whose noncommon sides are opposite rays. he angles form a linear pair. xercises b. d. 1 and 3 are nonadjacent angles formed by two intersecting lines. hey are vertical angles. 2 and 4 are also vertical angles G and are two angles whose measures have a sum of 90. hey are complementary. and G are two angles whose measures have a sum of 180. hey are supplementary. Identify each pair of angles as adjacent, vertical, and/or as a linear pair. U 1. 1 and and 6 V and and 2 or xercises 5 7, refer to the figure at the right. 5. Identify two obtuse vertical angles. V 6. Identify two acute adjacent angles. U 7. Identify an angle supplementary to U. 8. ind the measures of two complementary angles if the difference in their measures is 18. Lesson 1-5 Glencoe/cGraw-Hill 25 Glencoe Geometry

18 1-5 IO tudy Guide and Intervention (continued) ngle elationships erpendicular Lines Lines, rays, and segments that form four right angles are perpendicular. he right angle symbol indicates that the lines are perpendicular. In the figure at the right, is perpendicular to, or. xample ind x so that Z Z. If Z Z, then m Z 90. m Z m Z m Z um of parts whole (9x 5) (3x 1) 90 ubstitution 12x 6 90 implify. 12x 84 x 7 ivide each side by 12. ubtract 6 from each side. (9x 5) (3x 1) Z xercises 1. ind x and y so that. 2. ind m. 5x (9y 18) x 3. m 3x 10, m x, and. ind x. 4. If m 7y 3 and m 3y 3, find y so that. 5. ind x, m, and m. 3x (8x 2) 6. ind y, m, and m W. (4y 5) W (2y 5) V Glencoe/cGraw-Hill 26 Glencoe Geometry

19 1-5 IO kills ractice ngle elationships or xercises 1 6, use the figure at the right and a protractor. 1. ame two acute vertical angles. K 2. ame two obtuse vertical angles. H G J 3. ame a linear pair. 4. ame two acute adjacent angles. 5. ame an angle complementary to KH. 6. ame an angle supplementary to KG. 7. ind the measures of an angle and its complement if one angle measures 18 degrees more than the other. 8. he measure of the supplement of an angle is 36 less than the measure of the angle. ind the measures of the angles. LG or xercises 9 10, use the figure at the right. 9. If m 8x 18, find x so that. 10. If m 3y 10 and m y, find y so that is a right angle. etermine whether each statement can be assumed from the figure. xplain. 11. WZU is a right angle. W V 12. YZU and UZV are supplementary. X Y Z U 13. VZU is adjacent to YZX. Lesson 1-5 Glencoe/cGraw-Hill 27 Glencoe Geometry

20 1-5 IO ractice ngle elationships or xercises 1 4, use the figure at the right and a protractor. 1. ame two obtuse vertical angles. 2. ame a linear pair whose vertex is. 3. ame an angle not adjacent to but complementary to G. G H 4. ame an angle adjacent and supplementary to. 5. wo angles are complementary. he measure of one angle is 21 more than twice the measure of the other angle. ind the measures of the angles. 6. If a supplement of an angle has a measure 78 less than the measure of the angle, what are the measures of the angles? LG or xercises 7 8, use the figure at the right. 7. If m G 5x 10, find x so that. 8. If m G 16x 4 and m G 2x 13, find x so that G is a right angle. G etermine whether each statement can be assumed from the figure. xplain. 9. O and O are complementary. O 10. and is a linear pair. 11. and are vertical angles. 12. arren sketched a map of the cross streets nearest to his home for his friend iguel. escribe two different angle relationships between the streets. eacon Olive ain Glencoe/cGraw-Hill 28 Glencoe Geometry

21 IO 1-6 tudy Guide and Intervention olygons olygons polygon is a closed figure formed by a finite number of coplanar line segments. he sides that have a common endpoint must be noncollinear and each side intersects exactly two other sides at their endpoints. polygon is named according to its number of sides. regular polygon has congruent sides and congruent angles. polygon can be concave or convex. xample ame each polygon by its number of sides. hen classify it as concave or convex and regular or irregular. a. b. I H L Lesson 1-6 c. G he polygon has 4 sides, so it is a quadrilateral. It is concave because part of or lies in the interior of the figure. ecause it is concave, it cannot have all its angles congruent and so it is irregular. d. J K he figure is not closed, so it is not a polygon. he polygon has 5 sides, so it is a pentagon. It is convex. ll sides are congruent and all angles are congruent, so it is a regular pentagon. he figure has 8 congruent sides and 8 congruent angles. It is convex and is a regular octagon. xercises ame each polygon by its number of sides. hen classify it as concave or convex and regular or irregular Glencoe/cGraw-Hill 31 Glencoe Geometry

22 1-6 IO tudy Guide and Intervention (continued) olygons erimeter he perimeter of a polygon is the sum of the lengths of all the sides of the polygon. here are special formulas for the perimeter of a square or a rectangle. xample Write an expression or formula for the perimeter of each polygon. ind the perimeter. a. 3 in. a 4 in. b c 5 in. a b c in. b. 5 cm s 5 cm s s 5 cm s 5 cm 4s 4(5) 20 cm c. 2 ft w 3 ft 2 2w 2(3) 2(2) 10 ft w xercises ind the perimeter of each figure cm 3 cm 5.5 ft 3.5 cm square yd 27 yd 12 yd 14 yd 24 yd 1 cm ind the length of each side of the polygon for the given perimeter x x x 2 x x rectangle 2x Glencoe/cGraw-Hill 32 Glencoe Geometry

23 IO 1-6 kills ractice olygons ame each polygon by its number of sides and then classify it as convex or concave and regular or irregular Lesson ind the perimeter of each figure yd yd 20 yd 40 yd 4 m 6 m 2 m 3 m 5 m 10 in. 2 in. 2 in. 2 in. 2 in. 2 in. 10 in. 2 in. OOI GOY ind the perimeter of each polygon. 10. triangle with vertices (3, 5), (3, 1), and (0, 1) 11. quadrilateral with vertices ( 3, 2), (1, 2), (1, 4), and ( 3, 4) 12. quadrilateral LO with vertices L( 1, 4), (3, 4), (2, 1), and O( 2, 1) LG ind the length of each side of the polygon for the given perimeter millimeters kilometers feet 4w 1 w Glencoe/cGraw-Hill 33 Glencoe Geometry

24 1-6 IO ractice olygons ame each polygon by its number of sides and then classify it as convex or concave and regular or irregular ind the perimeter of each figure mm mm 21 mi 33 mi 10 mm 18 mm 32 mi 2 cm 6 cm 14 cm 4 cm 6 cm 6 cm 14 cm 4 cm OOI GOY ind the perimeter of each polygon. 7. quadrilateral O with vertices O( 3, 2), (1, 5), (6, 4), and (5, 2) 8. pentagon UVW with vertices (0, 0), (3, 2), U(2, 5), V( 2, 5), and W( 3, 2) LG ind the length of each side of the polygon for the given perimeter inches centimeters feet 6n 8 n 3x 5 2x 3 x 9 2x 2 5x 4 WIG or xercises 12 13, use the following information. Jasmine plans to sew fringe around the scarf shown in the diagram. 12. How many inches of fringe does she need to purchase? 16 in. 4 in. 4 in. 16 in. 13. If Jasmine doubles the width of the scarf, how many inches of fringe will she need? Glencoe/cGraw-Hill 34 Glencoe Geometry