Applications. 40 Shapes and Designs. 1. Tell whether each diagram shows an angle formed by a wedge, two sides meeting at a common point, or a turn.
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1 Applications. Tell whether each diagram shows an angle formed by a wedge, two sides meeting at a common point, or a turn. a. b. c. 2. Give the degree measure of each turn. a. One right-angle turn b. Four right-angle turns c. Five right-angle turns d. One half of a right-angle turn e. One ninth of a right-angle turn f. One fourth of a right-angle turn 40 Shapes and Designs
2 3. At the start of each hour, the minute hand of a clock points straight up at the 2. In parts (a) (f), determine the angle through which the minute hand turns as the given amount of time passes. Notice that only the minute hand is illustrated on the clock. Make a sketch to illustrate each situation. The curved arrow is pointing clockwise here because that is the direction that the minute hand turns. a. 5 minutes b. 30 minutes c. 20 minutes d. one hour e. 5 minutes f. one and a half hours 4. Without using an angle ruler, decide whether the measure of each angle is closest to 308,608,908, 208, 508, 808, 2708, or 3608.Be prepared to explain your reasoning. a. b. c. d. e. f. g. h. Investigation 2 Polygons and Angles 4
3 5. You have learned that a 908 angle is called a right angle. An angle with measure less than 908 is an acute angle. An angle with measure greater than 908 and less than 808 is an obtuse angle. An angle with measure exactly 808 is a straight angle. Decide whether each angle in Exercise 4 is right, acute, obtuse, straight, or none of these. For Exercises 6 9, find the measure of the angle labeled x, without measuring x 30 x x 27 x 35 For Exercises 0 3, a worker bee has located flowers with nectar and is preparing to do her dance. The dots represent the hive, the sun, and flowers. Estimate the measure of each angle. Use an angle ruler to check your estimate. 0.. Sun Sun Hive Hive Flowers Flowers Hive Sun Hive Sun Flowers Flowers For: Bee Dance Activity Visit: PHSchool.com Web Code: amd Shapes and Designs
4 4. Draw an angle for each measure. Include a curved arrow indicating the turn. a. 458 b. 258 c. 808 d Without measuring, decide whether the angles in each pair have the same measure. If they do not, tell which angle has the greater measure. Then find the measure of the angles with an angle ruler to check your work. a. 2 b. 2 c. 2 Investigation 2 Polygons and Angles 43
5 6. Estimate the measure of each angle, then check your answers with an angle ruler. a. b. c. d. e. 7. For each polygon below, measure the angles with an angle ruler. a. b. 8. You have read about how worker bees communicate the location of flowers. Suppose the angle a worker bee indicates is off by 8. How will this affect the other bees ability to locate the flowers? Explain. 44 Shapes and Designs
6 9. A bee leaves the hive and wants to fly to a rose but instead ends up at a daisy. How many degrees did the bee travel off course? Estimate your answer. Then check your answer with an angle ruler. Rose Hive Daisy 20. Little Bee left point A for a flower patch. Big Bee left point B for the same flower patch. However, both bees were 58 off course. Little Bee landed on the patch and Big Bee did not. Explain why Big Bee did not hit the patch and Little Bee did, if they were both off course by 58. For: Help with Exercise 20 Web Code: ame-3220 Big Bee A Little Bee 5º 5º B 2. Lines L and L 2 are parallel lines cut by a transversal. The measure of one of the angles is given. Based on what you discovered in Problem 2.5, find the measures of the other angles. a b 20 d e f g h L 2 L Investigation 2 Polygons and Angles 45
7 22. In parts (a) (c), lines L and L 2 are intersected by a transversal. The measures of some of the angles formed are given. In each part, tell whether you think the lines are parallel. Explain. a. b. 06 L 64 L 06 L 2 60 L 2 c. 35 L 25 L a. Draw any two intersecting lines, L and L 2. Measure the four angles formed around the point of intersection. b. What patterns do you observe among the angle measures? c. Draw two more pairs of intersecting lines and measure the angles formed. Do you observe the same patterns as in part (b)? 24. Multiple Choice Use the angle measures to determine which of the following shapes is a parallelogram. The shapes may not be drawn to scale. A. B C D Shapes and Designs
8 25. How did you know which shape was a parallelogram in Exercise 24? 26. Liang said that an equilateral triangle must have angles totaling 3608 because he can cut it into two right triangles, as in the diagram. Is Liang s statement correct? Explain Connections Is the statement below true or false? Justify your answer. The region inside a polygon can be tiled by triangles. 28. The number 360 was chosen for the number of degrees in a full turn. The number may have been chosen because it has many factors. a. List all the factors of 360. b. What is the prime factorization of 360? 29. A right angle can be thought of as a quarter of a complete rotation. 3 a. How many degrees is of a quarter of a rotation? b. How many degrees is two times a quarter of a rotation? 3 c. How many degrees is 2 times a quarter of a rotation? Replace the j with a number that will make the sentence true. 2 j = 3. = 40 j 32. = 33. = 20 j j For: Multiple-Choice Skills Practice Web Code: ama-3254 Investigation 2 Polygons and Angles 47
9 34. A full turn is If a bee turns around 808, like the one at the right, she has made a half turn. a. What fraction of a turn is 908? b. What fraction of a turn is 2708? c. How many turns is 7208? d. How many degrees is the fraction of a turn? 35. The minute hand on a watch makes a complete rotation every hour. The hand makes half of a full rotation in 30 minutes. a. In how many minutes does the hand make of a rotation? 6 b. In how many minutes does the hand make of half a rotation? 6 c. What fraction of an hour is of half a 6 rotation? d. How many degrees has the minute hand moved through in of half a rotation? The circular region is divided into four equal wedges formed by angles with vertices at the center of the circle. Such angles are called central angles of the circle. Each central angle shown measures In parts (a) (c), sketch a circular region divided into the given number of equal wedges. Then find the measure of the central angles. a. 8 equal edges b. 6 equal wedges c. 3 equal wedges d. Find another way to divide the circular region into equal wedges so that the central angles have whole number degree measures. Give the number of wedges and the measure of the central angles. What strategy did you use? 48 Shapes and Designs
10 37. A ruler is used to measure the length of line segments. An angle ruler is used to measure the size of (or turn in) angles. a. What is the unit of measurement for each kind of ruler? b. Write a few sentences comparing the method for measuring angles to the method for measuring line segments. 38. Skateboarders use angle measures to describe their turns. Explain what a skateboarder would mean by each statement. a. I did a 720. b. I did a 540. c. I did a In the figure below, the blue segments represent half of a polygon. The red vertical line is the line of symmetry for the complete polygon. y A(4, 0) (0, 0) B(, 5) (0, 5) x a. Copy the figure onto a sheet of grid paper. Then draw the missing half of the polygon. b. On the new half of the figure, what are the coordinates of the point that corresponds to point A? What are the coordinates of the point that corresponds to point B? c. Describe some properties of the polygon. Investigation 2 Polygons and Angles 49
11 40. Multiple Choice Which choice is a 808 rotation of the figure below? F. G. H. J. Extensions 4. Design a new polar coordinate grid for playing Four in a Row. Play your game with a friend or a member of your family. Explain the ideas that led to your new design. Compare playing on your new grid to playing on the grids given in Problem The midpoint of a line segment is the point that divides it into two segments of equal length. Trace the parallelogram below onto a sheet of paper. Connect the midpoints of two opposite sides. Describe the two quadrilaterals that are formed. Are they parallelograms? Explain. 50 Shapes and Designs
12 43. a. In the equilateral triangle below, the midpoints of two of the sides have been marked and then connected by a line segment. How does the length of this segment compare to the length of the third side of the triangle? Does the segment appear to be parallel to the third side of the triangle? b. Draw an isosceles triangle. Locate the midpoints of two of the sides. Then draw a line segment connecting the midpoints. Compare the segment with the third side of the triangle. Do the observations you made in part (a) also apply in this case? Now connect the midpoints of a different pair of sides. Do the same observations hold? c. Draw a scalene triangle. Locate the midpoints of two of the sides. Then draw a line segment connecting the midpoints. Compare the segment with the third side of the triangle. Do the observations you made in part (a) also apply in this case? Now connect the midpoints of a different pair of sides. Do the same observations hold? 44. Two lines are perpendicular if they intersect to form right angles. Tell whether the statement below is true or false. Justify your answer. If a transversal is perpendicular to one line in a pair of parallel lines, then it must also be perpendicular to the other line. Perpendicular Lines Investigation 2 Polygons and Angles 5
13 Astronomers use two types of angles to locate objects in the sky. The altitudinal (al tuh too di nuhl) angle is the angle from the horizon to the object. The horizon has an altitudinal angle of 08. The point directly overhead, called the zenith, has an altitudinal angle of 908. Zenith (90 ) (45 ) (45 ) (0 ) Horizon (0 ) Horizon The azimuthal (az uh myoothr uhl) angle is the angle of rotation from north to the object. To find the azimuthal angle of an object, face north and then rotate clockwise until you are facing the object. The angle through which you turn is the azimuthal angle. To find the azimuthal angle between two objects, face one of the objects and then turn until you are facing the other. The angle through which you turn is the azimuthal angle between the objects. NW (35 ) N (0 ) NE (45 ) W (270 ) E (90 ) SW (225 ) S (80 ) SE (35 ) For: Information about astronomy Web Code: ame Shapes and Designs
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