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Unit9: Angles and Lines

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Angle, Line, segmengt, sexagesimal system, bisector, angles/hours, minutes, seconds

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Unit9: Angles and Lines

1. 1. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Angles and Lines Matem´ticas 1o E.S.O. a Alberto Pardo Milan´s e -
2. 2. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises 1 Lines 2 Perpendicular bisector of a segment and bisector of an angle 3 Sexagesimal system and angles 4 Sexagesimal system and time 5 Adding and Subtracting in the sexagesimal system 6 ExercisesAlberto Pardo Milan´s e Angles and Lines
3. 3. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises LinesAlberto Pardo Milan´s e Angles and Lines
4. 4. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Lines Plane Geometry Plane Geometry deals with ﬁgures in a plane (circles, lines, polygons. . . ) A line is a set of points extending in two opposite directions without end, It’s a straight one-dimensional ﬁgure with no thickness and extending inﬁnitely in both directions. A line is sometimes called a straight line. Points are said to be collinear points if they lie on a single straight line. A plane is a ﬂat surface that has no thickness and extends without ending in ALL directions. Points are said to be coplanar points if they lie on a common plane. A line segment is a ﬁnite portion of an inﬁnite line. A line segment always has a beginning and an end.Alberto Pardo Milan´s e Angles and Lines
5. 5. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Lines Plane Geometry Plane Geometry deals with ﬁgures in a plane (circles, lines, polygons. . . ) A line is a set of points extending in two opposite directions without end, It’s a straight one-dimensional ﬁgure with no thickness and extending inﬁnitely in both directions. A line is sometimes called a straight line. Points are said to be collinear points if they lie on a single straight line. A plane is a ﬂat surface that has no thickness and extends without ending in ALL directions. Points are said to be coplanar points if they lie on a common plane. A line segment is a ﬁnite portion of an inﬁnite line. A line segment always has a beginning and an end.Alberto Pardo Milan´s e Angles and Lines
6. 6. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Lines Perpendicular bisector of a segment and bisector of an angle Lines that intersect in a point are called intersecting lines. Lines that do not intersect are called parallel lines (in the plane). Two lines that lie on top of one another are called coincident lines. Two lines or line segments which are perpendicular are said to be orthogonal (two lines are said to be perpendicular if they meet at a 90o angle). Given two intersecting lines, the point of intersection is called the vertex and the amount of rotation about the vertex required to bring one line into correspondence with the other is called the angle between them.Alberto Pardo Milan´s e Angles and Lines
7. 7. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Lines One full rotation corresponds to 360 degrees. Half a full rotation is called a straight angle, and a quarter of a full rotation is called a right angle. An angle less than a right angle is called an acute angle, an angle greater than a right angle (but less than a straight angle) is called an obtuse angle, and an angle greater than a straight angle (but less than a full angle) is called a reﬂex angle.Alberto Pardo Milan´s e Angles and Lines
8. 8. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Perpendicular bisector of a segment and bisector of an angleAlberto Pardo Milan´s e Angles and Lines
9. 9. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Perpendicular bisector of a segment and bisector of an angle Bisector of a line segment A perpendicular bisector of a line segment AB is a line perpendicular to AB and passing through the midpoint M of AB.Alberto Pardo Milan´s e Angles and Lines
10. 10. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Perpendicular bisector of a segment and bisector of an angle Bisector of an angle The bisector of an angle is the line that divides the angle into two equal parts.Alberto Pardo Milan´s e Angles and Lines
11. 11. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Sexagesimal system and anglesAlberto Pardo Milan´s e Angles and Lines
12. 12. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Sexagesimal system and angles The size of an angle There are several ways to measure the size of an angle, for example using units of degrees: In a complete circle there are three hundred and sixty degrees. You can express angles to any precision using decimal numbers. Example: 30 degrees and a half is 30.5 degrees. There is another way to state the size of an angle, subdividing a degree. The degree is divided into sixty parts called minutes. These minutes are further divided into sixty parts called seconds.Alberto Pardo Milan´s e Angles and Lines
13. 13. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Sexagesimal system and angles The size of an angle There are several ways to measure the size of an angle, for example using units of degrees: In a complete circle there are three hundred and sixty degrees. You can express angles to any precision using decimal numbers. Example: 30 degrees and a half is 30.5 degrees. There is another way to state the size of an angle, subdividing a degree. The degree is divided into sixty parts called minutes. These minutes are further divided into sixty parts called seconds.Alberto Pardo Milan´s e Angles and Lines
14. 14. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Sexagesimal system and angles Angles: Degrees, minutes and seconds In a full circle there are 360 degrees, each degree is split up into 60 parts called minutes (each part being 1/60 of a degree), each minute is split up into 60 parts called seconds (each part being 1/60 of a minute). Example: 40 degrees, 20 minutes and 50 seconds is usually written this way: 40o 20 50 . Example: 30.5 degrees is 30o and 30 , because 0.5 degrees are 0.5 · 60 = 30 minutes.Alberto Pardo Milan´s e Angles and Lines
15. 15. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Sexagesimal system and angles Angles: Degrees, minutes and seconds In a full circle there are 360 degrees, each degree is split up into 60 parts called minutes (each part being 1/60 of a degree), each minute is split up into 60 parts called seconds (each part being 1/60 of a minute). Example: 40 degrees, 20 minutes and 50 seconds is usually written this way: 40o 20 50 . Example: 30.5 degrees is 30o and 30 , because 0.5 degrees are 0.5 · 60 = 30 minutes.Alberto Pardo Milan´s e Angles and Lines
16. 16. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Sexagesimal system and timeAlberto Pardo Milan´s e Angles and Lines
17. 17. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Sexagesimal system and time Time: Hours, minutes and seconds To measure time we can use hours: In a day there are twenty-four hours. You can express time to any precision using decimal numbers. Example: 3 hours and a quarter is 3.25 hours. There is another way to state the amount of time, subdividing an hour. The hour is divided into sixty parts called minutes. These minutes are further divided into sixty parts called seconds. In a day there are 24 hours, each hour is split up into 60 parts called minutes (each part being 1/60 of an hour), each minute is split up into 60 parts called seconds (each part being 1/60 of a minute). Example: 14 hours, 56 minutes and 12 seconds is usually written this way: 14 hr 56 min 12 sec. Example: 3.25 hours is 3 hr and 15 min, because 0.25 hours are 0.25 · 60 = 15 minutes.Alberto Pardo Milan´s e Angles and Lines
18. 18. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Sexagesimal system and time Time: Hours, minutes and seconds To measure time we can use hours: In a day there are twenty-four hours. You can express time to any precision using decimal numbers. Example: 3 hours and a quarter is 3.25 hours. There is another way to state the amount of time, subdividing an hour. The hour is divided into sixty parts called minutes. These minutes are further divided into sixty parts called seconds. In a day there are 24 hours, each hour is split up into 60 parts called minutes (each part being 1/60 of an hour), each minute is split up into 60 parts called seconds (each part being 1/60 of a minute). Example: 14 hours, 56 minutes and 12 seconds is usually written this way: 14 hr 56 min 12 sec. Example: 3.25 hours is 3 hr and 15 min, because 0.25 hours are 0.25 · 60 = 15 minutes.Alberto Pardo Milan´s e Angles and Lines
19. 19. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Sexagesimal system and time Time: Hours, minutes and seconds To measure time we can use hours: In a day there are twenty-four hours. You can express time to any precision using decimal numbers. Example: 3 hours and a quarter is 3.25 hours. There is another way to state the amount of time, subdividing an hour. The hour is divided into sixty parts called minutes. These minutes are further divided into sixty parts called seconds. In a day there are 24 hours, each hour is split up into 60 parts called minutes (each part being 1/60 of an hour), each minute is split up into 60 parts called seconds (each part being 1/60 of a minute). Example: 14 hours, 56 minutes and 12 seconds is usually written this way: 14 hr 56 min 12 sec. Example: 3.25 hours is 3 hr and 15 min, because 0.25 hours are 0.25 · 60 = 15 minutes.Alberto Pardo Milan´s e Angles and Lines
20. 20. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Adding and Subtracting in the sexagesimal systemAlberto Pardo Milan´s e Angles and Lines
21. 21. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Adding and Subtracting in the sexagesimal system Add in the sexagesimal system You must add or subtract the units separately. But you may need to do some adjusting if the hours end up 24 or more, the minutes end up 60 or more, the seconds end up 60 or more,or less than zero. Adding Add the hours. If the hours are 24 or more, subtract 24 from the hours and add 1 to the days. Add the minutes. If the minutes are 60 or more, subtract 60 from the hours and add 1 to the hours. And so on. Example: 40o 20 50 + 10o 33 6 = 50o 55 56 . Example: 14 hr 20 min 50 sec + 15 hr 43 min 26 sec =29 hr 63 min 76 sec =1 day 6 hr 4 min 16 sec.Alberto Pardo Milan´s e Angles and Lines
22. 22. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Adding and Subtracting in the sexagesimal system Add in the sexagesimal system You must add or subtract the units separately. But you may need to do some adjusting if the hours end up 24 or more, the minutes end up 60 or more, the seconds end up 60 or more,or less than zero. Adding Add the hours. If the hours are 24 or more, subtract 24 from the hours and add 1 to the days. Add the minutes. If the minutes are 60 or more, subtract 60 from the hours and add 1 to the hours. And so on. Example: 40o 20 50 + 10o 33 6 = 50o 55 56 . Example: 14 hr 20 min 50 sec + 15 hr 43 min 26 sec =29 hr 63 min 76 sec =1 day 6 hr 4 min 16 sec.Alberto Pardo Milan´s e Angles and Lines
23. 23. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Adding and Subtracting in the sexagesimal system Add in the sexagesimal system You must add or subtract the units separately. But you may need to do some adjusting if the hours end up 24 or more, the minutes end up 60 or more, the seconds end up 60 or more,or less than zero. Adding Add the hours. If the hours are 24 or more, subtract 24 from the hours and add 1 to the days. Add the minutes. If the minutes are 60 or more, subtract 60 from the hours and add 1 to the hours. And so on. Example: 40o 20 50 + 10o 33 6 = 50o 55 56 . Example: 14 hr 20 min 50 sec + 15 hr 43 min 26 sec =29 hr 63 min 76 sec =1 day 6 hr 4 min 16 sec.Alberto Pardo Milan´s e Angles and Lines
24. 24. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Adding and Subtracting in the sexagesimal system Add in the sexagesimal system You must add or subtract the units separately. But you may need to do some adjusting if the hours end up 24 or more, the minutes end up 60 or more, the seconds end up 60 or more,or less than zero. Adding Add the hours. If the hours are 24 or more, subtract 24 from the hours and add 1 to the days. Add the minutes. If the minutes are 60 or more, subtract 60 from the hours and add 1 to the hours. And so on. Example: 40o 20 50 + 10o 33 6 = 50o 55 56 . Example: 14 hr 20 min 50 sec + 15 hr 43 min 26 sec =29 hr 63 min 76 sec =1 day 6 hr 4 min 16 sec.Alberto Pardo Milan´s e Angles and Lines
25. 25. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Adding and Subtracting in the sexagesimal system Subtract in the sexagesimal system Subtracting Subtract the hours. If the hours are negative, add 24 to the hours and subtract 1 from days. Subtract the minutes. If the minutes are negative, add 60 to the minutes and subtract 1 from hours. And so on. Example: 23 hr 40 min 51 sec − 15 hr 33 min 21 sec = 8 hr 7 min 30 sec. Example: 45o 33 12 −11o 42 20 =44o 92 72 −11o 42 20 =33o 50 52 .Alberto Pardo Milan´s e Angles and Lines
26. 26. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Adding and Subtracting in the sexagesimal system Subtract in the sexagesimal system Subtracting Subtract the hours. If the hours are negative, add 24 to the hours and subtract 1 from days. Subtract the minutes. If the minutes are negative, add 60 to the minutes and subtract 1 from hours. And so on. Example: 23 hr 40 min 51 sec − 15 hr 33 min 21 sec = 8 hr 7 min 30 sec. Example: 45o 33 12 −11o 42 20 =44o 92 72 −11o 42 20 =33o 50 52 .Alberto Pardo Milan´s e Angles and Lines
27. 27. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Adding and Subtracting in the sexagesimal system Subtract in the sexagesimal system Subtracting Subtract the hours. If the hours are negative, add 24 to the hours and subtract 1 from days. Subtract the minutes. If the minutes are negative, add 60 to the minutes and subtract 1 from hours. And so on. Example: 23 hr 40 min 51 sec − 15 hr 33 min 21 sec = 8 hr 7 min 30 sec. Example: 45o 33 12 −11o 42 20 =44o 92 72 −11o 42 20 =33o 50 52 .Alberto Pardo Milan´s e Angles and Lines
28. 28. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Adding and Subtracting in the sexagesimal system Subtract in the sexagesimal system Subtracting Subtract the hours. If the hours are negative, add 24 to the hours and subtract 1 from days. Subtract the minutes. If the minutes are negative, add 60 to the minutes and subtract 1 from hours. And so on. Example: 23 hr 40 min 51 sec − 15 hr 33 min 21 sec = 8 hr 7 min 30 sec. Example: 45o 33 12 −11o 42 20 =44o 92 72 −11o 42 20 =33o 50 52 .Alberto Pardo Milan´s e Angles and Lines
29. 29. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Adding and Subtracting in the sexagesimal system Subtract in the sexagesimal system Subtracting Subtract the hours. If the hours are negative, add 24 to the hours and subtract 1 from days. Subtract the minutes. If the minutes are negative, add 60 to the minutes and subtract 1 from hours. And so on. Example: 23 hr 40 min 51 sec − 15 hr 33 min 21 sec = 8 hr 7 min 30 sec. Example: 45o 33 12 −11o 42 20 =44o 92 72 −11o 42 20 =33o 50 52 .Alberto Pardo Milan´s e Angles and Lines
30. 30. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises ExercisesAlberto Pardo Milan´s e Angles and Lines
31. 31. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 1 Find ﬁve objects in your classroom that have right angles:Alberto Pardo Milan´s e Angles and Lines
32. 32. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 1 Find ﬁve objects in your classroom that have right angles:Alberto Pardo Milan´s e Angles and Lines
33. 33. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 2 Using the set square, draw an angle of a)45o b)30o c)105o d)15oAlberto Pardo Milan´s e Angles and Lines
34. 34. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 2 45o 30o 105o 15oAlberto Pardo Milan´s e Angles and Lines
35. 35. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 3 For each angle below, state whether the angle is acute, obtuse or reﬂex:Alberto Pardo Milan´s e Angles and Lines
36. 36. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 3 For each angle below, state whether the angle is acute, obtuse or reﬂex:Alberto Pardo Milan´s e Angles and Lines
37. 37. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 4 Measure the following angles:Alberto Pardo Milan´s e Angles and Lines
38. 38. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 4 Measure the following angles:Alberto Pardo Milan´s e Angles and Lines
39. 39. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 6 Write down the following amounts: 1 43o 34 25 : 2 5o 17 30 : 3 3 hr 14 min 18 sec: 4 15 hr 56 min 24 sec:Alberto Pardo Milan´s e Angles and Lines
40. 40. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 6 Write down the following amounts: 1 43o 34 25 : forty-three degrees, thirty-four minutes and twenty-ﬁve seconds. 2 5o 17 30 : ﬁve degrees, seventeen minutes and thirty seconds. 3 3 hr 14 min 18 sec: three hours, fourteen minutes and eighteen seconds. 4 15 hr 56 min 24 sec: ﬁfteen hours, ﬁfty-six minutes and twenty-four seconds.Alberto Pardo Milan´s e Angles and Lines
41. 41. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 6 Write down the following amounts: 1 43o 34 25 : forty-three degrees, thirty-four minutes and twenty-ﬁve seconds. 2 5o 17 30 : ﬁve degrees, seventeen minutes and thirty seconds. 3 3 hr 14 min 18 sec: three hours, fourteen minutes and eighteen seconds. 4 15 hr 56 min 24 sec: ﬁfteen hours, ﬁfty-six minutes and twenty-four seconds.Alberto Pardo Milan´s e Angles and Lines
42. 42. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 6 Write down the following amounts: 1 43o 34 25 : forty-three degrees, thirty-four minutes and twenty-ﬁve seconds. 2 5o 17 30 : ﬁve degrees, seventeen minutes and thirty seconds. 3 3 hr 14 min 18 sec: three hours, fourteen minutes and eighteen seconds. 4 15 hr 56 min 24 sec: ﬁfteen hours, ﬁfty-six minutes and twenty-four seconds.Alberto Pardo Milan´s e Angles and Lines
43. 43. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 6 Write down the following amounts: 1 43o 34 25 : forty-three degrees, thirty-four minutes and twenty-ﬁve seconds. 2 5o 17 30 : ﬁve degrees, seventeen minutes and thirty seconds. 3 3 hr 14 min 18 sec: three hours, fourteen minutes and eighteen seconds. 4 15 hr 56 min 24 sec: ﬁfteen hours, ﬁfty-six minutes and twenty-four seconds.Alberto Pardo Milan´s e Angles and Lines
44. 44. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 6 Write down the following amounts: 1 43o 34 25 : forty-three degrees, thirty-four minutes and twenty-ﬁve seconds. 2 5o 17 30 : ﬁve degrees, seventeen minutes and thirty seconds. 3 3 hr 14 min 18 sec: three hours, fourteen minutes and eighteen seconds. 4 15 hr 56 min 24 sec: ﬁfteen hours, ﬁfty-six minutes and twenty-four seconds.Alberto Pardo Milan´s e Angles and Lines
45. 45. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 7 Find: 1 34o 14 15 + 11o 13 15 = 2 55o 44 52 + 24o 53 34 = 3 3 hr 34 min 17 sec + 4 hr 11 min 34 sec = 4 18 hr 34 min 37 sec + 12 hr 26 min 34 sec =Alberto Pardo Milan´s e Angles and Lines
46. 46. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 7 Find: 1 34o 14 15 + 11o 13 15 =45o 27 30 2 55o 44 52 + 24o 53 34 =80o 38 26 3 3 hr 34 min 17 sec + 4 hr 11 min 34 sec =7 hr 45 min 51 sec 4 18 hr 34 min 37 sec + 12 hr 26 min 34 sec =1 day 7 hr 1 min 11 secAlberto Pardo Milan´s e Angles and Lines
47. 47. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 7 Find: 1 34o 14 15 + 11o 13 15 =45o 27 30 2 55o 44 52 + 24o 53 34 =80o 38 26 3 3 hr 34 min 17 sec + 4 hr 11 min 34 sec =7 hr 45 min 51 sec 4 18 hr 34 min 37 sec + 12 hr 26 min 34 sec =1 day 7 hr 1 min 11 secAlberto Pardo Milan´s e Angles and Lines
48. 48. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 7 Find: 1 34o 14 15 + 11o 13 15 =45o 27 30 2 55o 44 52 + 24o 53 34 =80o 38 26 3 3 hr 34 min 17 sec + 4 hr 11 min 34 sec =7 hr 45 min 51 sec 4 18 hr 34 min 37 sec + 12 hr 26 min 34 sec =1 day 7 hr 1 min 11 secAlberto Pardo Milan´s e Angles and Lines
49. 49. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 7 Find: 1 34o 14 15 + 11o 13 15 =45o 27 30 2 55o 44 52 + 24o 53 34 =80o 38 26 3 3 hr 34 min 17 sec + 4 hr 11 min 34 sec =7 hr 45 min 51 sec 4 18 hr 34 min 37 sec + 12 hr 26 min 34 sec =1 day 7 hr 1 min 11 secAlberto Pardo Milan´s e Angles and Lines
50. 50. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 7 Find: 1 34o 14 15 + 11o 13 15 =45o 27 30 2 55o 44 52 + 24o 53 34 =80o 38 26 3 3 hr 34 min 17 sec + 4 hr 11 min 34 sec =7 hr 45 min 51 sec 4 18 hr 34 min 37 sec + 12 hr 26 min 34 sec =1 day 7 hr 1 min 11 secAlberto Pardo Milan´s e Angles and Lines
51. 51. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 8 Find: 1 45o 21 45 − 33o 12 15 = 2 53o 24 12 − 32o 43 44 = 3 6 hr 24 min 15 sec − 4 hr 1 min 5 sec = 4 8 hr 24 min 13 sec − 5 hr 25 min 36 sec =Alberto Pardo Milan´s e Angles and Lines
52. 52. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 8 Find: 1 45o 21 45 − 33o 12 15 =12o 9 30 2 53o 24 12 − 32o 43 44 =20o 40 26 3 6 hr 24 min 15 sec − 4 hr 1 min 5 sec =2 hr 23 min 10sec 4 8 hr 24 min 13 sec − 5 hr 25 min 36 sec =2 hr 58 min 37secAlberto Pardo Milan´s e Angles and Lines
53. 53. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 8 Find: 1 45o 21 45 − 33o 12 15 =12o 9 30 2 53o 24 12 − 32o 43 44 =20o 40 26 3 6 hr 24 min 15 sec − 4 hr 1 min 5 sec =2 hr 23 min 10sec 4 8 hr 24 min 13 sec − 5 hr 25 min 36 sec =2 hr 58 min 37secAlberto Pardo Milan´s e Angles and Lines
54. 54. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 8 Find: 1 45o 21 45 − 33o 12 15 =12o 9 30 2 53o 24 12 − 32o 43 44 =20o 40 26 3 6 hr 24 min 15 sec − 4 hr 1 min 5 sec =2 hr 23 min 10sec 4 8 hr 24 min 13 sec − 5 hr 25 min 36 sec =2 hr 58 min 37secAlberto Pardo Milan´s e Angles and Lines
55. 55. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 8 Find: 1 45o 21 45 − 33o 12 15 =12o 9 30 2 53o 24 12 − 32o 43 44 =20o 40 26 3 6 hr 24 min 15 sec − 4 hr 1 min 5 sec =2 hr 23 min 10sec 4 8 hr 24 min 13 sec − 5 hr 25 min 36 sec =2 hr 58 min 37secAlberto Pardo Milan´s e Angles and Lines
56. 56. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 8 Find: 1 45o 21 45 − 33o 12 15 =12o 9 30 2 53o 24 12 − 32o 43 44 =20o 40 26 3 6 hr 24 min 15 sec − 4 hr 1 min 5 sec =2 hr 23 min 10sec 4 8 hr 24 min 13 sec − 5 hr 25 min 36 sec =2 hr 58 min 37secAlberto Pardo Milan´s e Angles and Lines
57. 57. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 9 Luis is a marathon runner. He trains two days: on the ﬁrst day he runs the marathon in 2 hr 48 min 35 sec; and on the second day he runs it in 2 hr 45 min 30 sec. How long does Luis run over the two days in total?Alberto Pardo Milan´s e Angles and Lines
58. 58. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 9 Luis is a marathon runner. He trains two days: on the ﬁrst day he runs the marathon in 2 hr 48 min 35 sec; and on the second day he runs it in 2 hr 45 min 30 sec. How long does Luis run over the two days in total? Data: First day: 2 hr 48 min 35 sec Second day: 2 hr 45 min 30 sec 2 hr 48 min 35 sec + 2 hr 45 min 30 sec 5 hr 34 min 5 sec Answer: Luis runs 5 hr 34 min 5 sec in total.Alberto Pardo Milan´s e Angles and Lines
59. 59. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 9 Luis is a marathon runner. He trains two days: on the ﬁrst day he runs the marathon in 2 hr 48 min 35 sec; and on the second day he runs it in 2 hr 45 min 30 sec. How long does Luis run over the two days in total? Data: First day: 2 hr 48 min 35 sec Second day: 2 hr 45 min 30 sec 2 hr 48 min 35 sec + 2 hr 45 min 30 sec 5 hr 34 min 5 sec Answer: Luis runs 5 hr 34 min 5 sec in total.Alberto Pardo Milan´s e Angles and Lines
60. 60. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 9 Luis is a marathon runner. He trains two days: on the ﬁrst day he runs the marathon in 2 hr 48 min 35 sec; and on the second day he runs it in 2 hr 45 min 30 sec. How long does Luis run over the two days in total? Data: First day: 2 hr 48 min 35 sec Second day: 2 hr 45 min 30 sec 2 hr 48 min 35 sec + 2 hr 45 min 30 sec 5 hr 34 min 5 sec Answer: Luis runs 5 hr 34 min 5 sec in total.Alberto Pardo Milan´s e Angles and Lines
61. 61. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 9 Luis is a marathon runner. He trains two days: on the ﬁrst day he runs the marathon in 2 hr 48 min 35 sec; and on the second day he runs it in 2 hr 45 min 30 sec. How long does Luis run over the two days in total? Data: First day: 2 hr 48 min 35 sec Second day: 2 hr 45 min 30 sec 2 hr 48 min 35 sec + 2 hr 45 min 30 sec 5 hr 34 min 5 sec Answer: Luis runs 5 hr 34 min 5 sec in total.Alberto Pardo Milan´s e Angles and Lines
62. 62. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 10 Ana spent 7 hr 20 min 28 sec on painting a room in the weekend. She worked the same amount of time each day. How much time did she work each day?Alberto Pardo Milan´s e Angles and Lines
63. 63. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 10 Ana spent 7 hr 20 min 28 sec on painting a room in the weekend. She worked the same amount of time each day. How much time did she work each day? Data: She Ana spent 7 hr 20 min 28 sec 1 hr 40 min 14 sec + 1 hr 40 min 14 sec Answer: She worked 3 hr 20 min 28 sec 1 hr 40 min 14 sec a day.Alberto Pardo Milan´s e Angles and Lines
64. 64. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 10 Ana spent 7 hr 20 min 28 sec on painting a room in the weekend. She worked the same amount of time each day. How much time did she work each day? Data: She Ana spent 7 hr 20 min 28 sec 1 hr 40 min 14 sec + 1 hr 40 min 14 sec Answer: She worked 3 hr 20 min 28 sec 1 hr 40 min 14 sec a day.Alberto Pardo Milan´s e Angles and Lines
65. 65. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 10 Ana spent 7 hr 20 min 28 sec on painting a room in the weekend. She worked the same amount of time each day. How much time did she work each day? Data: She Ana spent 7 hr 20 min 28 sec 1 hr 40 min 14 sec + 1 hr 40 min 14 sec Answer: She worked 3 hr 20 min 28 sec 1 hr 40 min 14 sec a day.Alberto Pardo Milan´s e Angles and Lines
66. 66. Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises Exercises Exercise 10 Ana spent 7 hr 20 min 28 sec on painting a room in the weekend. She worked the same amount of time each day. How much time did she work each day? Data: She Ana spent 7 hr 20 min 28 sec 1 hr 40 min 14 sec + 1 hr 40 min 14 sec Answer: She worked 3 hr 20 min 28 sec 1 hr 40 min 14 sec a day.Alberto Pardo Milan´s e Angles and Lines