1 directed angle
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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 directed angle
- 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:
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- 18. Angle's position in the orthogonal coordinate plane: The orthogonal coordinate plane is divided into four quadrants.
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- 21. Exercise (4) :
- 24. Exercise (4) :
- 25. Solution :
- 26. Exercise (5) :
- 27. Exercise (6) :
- 29. Exercise (7) :
- 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
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