1. The document describes steps for performing operations with angles using a compass, including copying an angle, drawing a bisector, adding angles, and subtracting angles. 2. To copy an angle, arcs are drawn from the vertices and the distance between points on the original angle is transferred to the new angle. 3. To draw a bisector, arcs are drawn from a vertex and the intersection of two extended arcs forms the bisector. 4. Adding angles involves copying one angle and then transferring the distance of another angle outward, while subtracting involves copying one angle inside another.

1. OPERATIONS WITH ANGLES

2. Exercise 1
Given the angle A, copy it onto the ray from V´

3. We have the angle A, that
we want to copy onto the
ray from V´

4. STEPS:
1.Centering at V, we draw an arc (no matter
the length you take with the compass).
You get the points 1 and 2

5. STEPS:
1.Centering at V, we draw an arc (no matter
the length you take with the compass).
You get the points 1 and 2

6. STEPS:
1.Centering at V, we draw an arc (no matter
the length you take with the compass).
You get the points 1 and 2

7. STEPS:
2.With the same length in the compass, we
draw the same arc from V´ getting the point 1

8. STEPS:
2.With the same length in the compass, we
draw the same arc from V´ getting the point 1

9. STEPS:
2.With the same length in the compass, we
draw the same arc from V´ getting the point 1

10. STEPS:
3.Now we take the distance 1-2 from the given arc and
we transport it to the new one, from the point 1

11. STEPS:
3.Now we take the distance 1-2 from the given arc and
we transport it to the new one, from the point 1

12. STEPS:
3.Now we take the distance 1-2 from the given arc and
we transport it to the new one, from the point 1

13. STEPS:
3.Now we take the distance 1-2 from the given arc and
we transport it to the new one, from the point 1

14. STEPS:
4.We get the point 2. Joining the vertex V´ and the
point 2 we obtain the angle copied.

15. STEPS:
4.We get the point 2. Joining the vertex V´ and the
point 2 we obtain the angle copied.

16. Exercise 2
Draw the bisector of the given angle A

17. We have the angle A and
we want to draw the
bisector of this angle

18. STEPS:
1.Centering at the vertex, we draw an arc
(no matter the length you take with the
compass). You get the points 1 and 2

19. STEPS:
1.Centering at the vertex, we draw an arc
(no matter the length you take with the
compass). You get the points 1 and 2

20. STEPS:
1.Centering at the vertex, we draw an arc
(no matter the length you take with the
compass). You get the points 1 and 2

21. STEPS:
2.Centering at each point and opening the
compass further than the middle, draw
two arcs with the same length

22. STEPS:
2.Centering at each point and opening the
compass further than the middle, draw
two arcs with the same length

23. STEPS:
2.Centering at each point and opening the
compass further than the middle, draw
two arcs with the same length

24. STEPS:
2.Centering at each point and opening the
compass further than the middle, draw
two arcs with the same length

25. STEPS:
3.Where the two arcs intersect , join it with
the vertex and you obtain the bisector of
the angle

26. Exercise 3
Given the following angles A, B and C, carry out
the requested operations

27. We have to add the angle
A and the angle B

28. STEPS:
1.We have to copy the angle A
onto the ray as we saw
before. To do so, we are
going to draw the same arc
in all of them

29. STEPS:
1.We have to copy the angle A
onto the ray as we saw
before. To do so, we are
going to draw the same arc
in all of them

30. STEPS:
2.In the first angle, we get the
points 1 and 2

31. STEPS:
3.In the second angle, we get
the points 3 and 4

32. STEPS:
3.In the second angle, we get
the points 3 and 4

33. STEPS:
3.In the second angle, we get
the points 3 and 4

34. STEPS:
4.In the third angle, we get
the points 5 and 6

35. STEPS:
4.In the third angle, we get
the points 5 and 6

36. STEPS:
4.In the third angle, we get
the points 5 and 6

37. STEPS:
5.Now, we copy the first angle
onto the ray. To do it, we
draw the same arc and we
obtain the point 1. Then, on
this arc we transport the
distance 1-2 of the original
angle

38. STEPS:
5.Now, we copy the first angle
onto the ray. To do it, we
draw the same arc and we
obtain the point 1. Then, on
this arc we transport the
distance 1-2 of the original
angle

39. STEPS:
5.Now, we copy the first angle
onto the ray. To do it, we
draw the same arc and we
obtain the point 1. Then, on
this arc we transport the
distance 1-2 of the original
angle

40. STEPS:
5.Now, we copy the first angle
onto the ray. To do it, we
draw the same arc and we
obtain the point 1. Then, on
this arc we transport the
distance 1-2 of the original
angle

41. STEPS:
5.Now, we copy the first angle
onto the ray. To do it, we
draw the same arc and we
obtain the point 1. Then, on
this arc we transport the
distance 1-2 of the original
angle

42. STEPS:
5.Now, we copy the first angle
onto the ray. To do it, we
draw the same arc and we
obtain the point 1. Then, on
this arc we transport the
distance 1-2 of the original
angle

43. STEPS:
5.Now, we copy the first angle
onto the ray. To do it, we
draw the same arc and we
obtain the point 1. Then, on
this arc we transport the
distance 1-2 of the original
angle

44. STEPS:
7.We join the point 2 with the
vertex V´and we have the
angle A copied

45. STEPS:
7.Now, we have to copy the
angle B outside the angle A.
To do so, we have to take
the distance 3-4 from the
angle B and transport it
onto the same arc

46. STEPS:
7.Now, we have to copy the
angle B outside the angle A.
To do so, we have to take
the distance 3-4 from the
angle B and transport it
onto the same arc

47. STEPS:
7.Now, we have to copy the
angle B outside the angle A.
To do so, we have to take
the distance 3-4 from the
angle B and transport it
onto the same arc

48. STEPS:
7.Now, we have to copy the
angle B outside the angle A.
To do so, we have to take
the distance 3-4 from the
angle B and transport it
onto the same arc

49. STEPS:
7.Now, we have to copy the
angle B outside the angle A.
To do so, we have to take
the distance 3-4 from the
angle B and transport it
onto the same arc

50. STEPS:
8.The result is the space
marked in light pink color

51. We have to substract the
angle A and the angle B

52. STEPS:
1.We have to copy first the
angle B onto the ray and
then, copy the angle A
inside the angle B

53. STEPS:
2.We have to transport the
distance 3-4 of the angle B
onto the ray

54. STEPS:
2.We have to transport the
distance 3-4 of the angle B
onto the ray

55. STEPS:
2.We have to transport the
distance 3-4 of the angle B
onto the ray

56. STEPS:
2.We have to transport the
distance 3-4 of the angle B
onto the ray

57. STEPS:
2.We have to transport the
distance 3-4 of the angle B
onto the ray

58. STEPS:
3.Join the vertex 4
with the vertex V´
and you get the
angle B copied

59. STEPS:
4.Now, we have to
copy the angle A
inside the angle B.
To do so, we have
to transport the
distance 1-2 of the
angle A inside the
angle B

60. STEPS:
4.Now, we have to
copy the angle A
inside the angle B.
To do so, we have
to transport the
distance 1-2 of the
angle A inside the
angle B

61. STEPS:
4.Now, we have to
copy the angle A
inside the angle B.
To do so, we have
to transport the
distance 1-2 of the
angle A inside the
angle B

62. STEPS:
4.Now, we have to
copy the angle A
inside the angle B.
To do so, we have
to transport the
distance 1-2 of the
angle A inside the
angle B

63. STEPS:
4.Now, we have to
copy the angle A
inside the angle B.
To do so, we have
to transport the
distance 1-2 of the
angle A inside the
angle B

64. STEPS:
4.Now, we have to
copy the angle A
inside the angle B.
To do so, we have
to transport the
distance 1-2 of the
angle A inside the
angle B

65. STEPS:
5.The result of the
operation is the
space marked in
light pink color

66. We have to do two
operations. First we have
to add the angles B and C
and then, substract with
the result, the angle A

67. STEPS:
1.We have to copy first the
angle B. So, we have to
transport the distance 3-4
onto the ray

68. STEPS:
1.We have to copy first the
angle B. So, we have to
transport the distance 3-4
onto the ray

69. STEPS:
1.We have to copy first the
angle B. So, we have to
transport the distance 3-4
onto the ray

70. STEPS:
1.We have to copy first the
angle B. So, we have to
transport the distance 3-4
onto the ray

71. STEPS:
2.Once we get the
angle B we have
to add it the
angle C. So, we
have to transport
the distance 5-6
outward the
angle B

72. STEPS:
2.Once we get the
angle B we have
to add it the
angle C. So, we
have to transport
the distance 5-6
outward the
angle B

73. STEPS:
2.Once we get the
angle B we have
to add it the
angle C. So, we
have to transport
the distance 5-6
outward the
angle B

74. STEPS:
2.Once we get the
angle B we have
to add it the
angle C. So, we
have to transport
the distance 5-6
outward the
angle B

75. STEPS:
2.Once we get the
addittion of the
angles B and C,
we have to
substract the
angle A to the
result.

76. STEPS:
3.So, we have to
transport the
distance 1-2 of the
angle A inward the
result of the
previous operation

77. STEPS:
3.So, we have to
transport the
distance 1-2 of the
angle A inward the
result of the
previous operation

78. STEPS:
3.So, we have to
transport the
distance 1-2 of the
angle A inward the
result of the
previous operation

79. STEPS:
3.So, we have to
transport the
distance 1-2 of the
angle A inward the
result of the
previous operation

80. STEPS:
3.So, we have to
transport the
distance 1-2 of the
angle A inward the
result of the
previous operation

81. STEPS:
3.So, we have to
transport the
distance 1-2 of the
angle A inward the
result of the
previous operation

82. STEPS:
4.The result is the
space marked in
light pink color