This document discusses directed angles and their measurement. It defines an angle as the union of two rays with a common endpoint, called the vertex. A directed angle specifies an order to the rays, written as an ordered pair with the initial side first. The standard position for a directed angle places the vertex at the origin and the initial side on the positive x-axis. Angles are measured in degrees, with positive values for counterclockwise rotation and negative for clockwise. The document provides examples of directed angle notation, standard position, and positive and negative angle measures. It includes exercises for students to practice identifying directed angles, their measures and positions.

1. REDSEA STEM HIGH SCHOOL
MA1.01 DIRECTED ANGLE
2019-2020
MR George Alfy Shaker
01119135313
George.Alfy@stemredsea.moe.edu.eg

3. Angle is :
• Angle is union of two rays with
a common start point.
• The common point is called
«Vertex» of the angle and the
two rays are called "sides“ of
the angle.

4. i.e.: In the figure opposite:
• The Vertex of the angle is B
and the two sides of the angle
are BA & BC .
• BA ∪ B C = (ABC)
and is written as  ABC or
 CBA or  B .

5. Degree Measure System
 degree measure depends on dividing the circle
into 360 equal arcs in length, then:
1- The central angle subtends one of these arcs, its
measure equals one degree (1o)
2- Each degree is subdivided into 60 equal
divisions , each division is called a minute and is
denoted by (1')
3- Each minute is subdivided into 60 equal
divisions, each division is called " a second" and is
denoted by (1'')
i.e.: 1c = 60' and 1' = 60''

6. Directed Angle
 We shall now put a further
emphasis on the order of the
two rays forming the angle,
then it is written in the form
of an ordered pair (OA , OB)
 OA is the initial side and OB is the terminal
side of the angle of vertex O as in figure (1) .

7.  If the initial side is OB and
the terminal side is OA , then
it is written as ( OB , OA ) in
figure (2).

8.  The directed angle
 AOB = ( OA , OB )
 The directed angle
 BOA = ( OB , OA )

9. Question
 Is ( OA , OB ) = ( OB , OA ) ?
 Is  AOB =  BOA ?
 Is the directed angle  AOB = the directed
angle  BOA ?

10. Standard position of the
directed angle
An directed angle is in the standard position if it
satisfies the next two conditions :
1) Its vertex is the origin of rectangular coordinate
system.
2) Its initial side lies on the positive direction of
the x-axis.

11. Is the directed angle  AOB in the standard
position? Explain your answer.

12. Exercise (1) :
Which one of the following ordered pairs
expresses a directed angle in its standard
position?

13. Exercise (2) :
Which of the following directed angles is in the
standard position?

14. Positive and negative measures
of a directed angle
In figure (1):
The directed angle,
resulting from an
anticlockwise rotation
has a positive measure.

15. In figure (2):
The directed angle,
resulting from a
clockwise rotation has
a negative measure.

16. Exercise (3) :
Find the measure of the directed angle in each of
the following figure:

17. Is

18. Angle's position in the
orthogonal coordinate plane:
The orthogonal coordinate
plane is divided into four
quadrants.

20. 



21. Exercise (4) :

24. Exercise (4) :

25. Solution :

26. Exercise (5) :

27. Exercise (6) :

29. Exercise (7) :

30. Equivalent angles

33. Exercise (8) :
Find a positive and a negative measure of an
angle co-terminal with each of the following
angles:

35. Exercise (9) :

37. Homework

38. Is

39. Is

40. Is

41. Is

43. Is

44. Is