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Angles

Angles. Side 1. Vertex. Side 2. Two rays with a common endpoint. Model Item Notation. An angle is Two rays with a common end point. The parts are the sides ( rays ) , the vertex ( common point), interior space, and exterior space.

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Angles

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  1. Angles Side 1 Vertex Side 2 Two rays with a common endpoint.

  2. Model Item Notation An angle is Two rays with a common end point. The parts are the sides ( rays ) , the vertex ( common point), interior space, and exterior space.

  3. Angles are classified by rotation of the rays.

  4. Zero degrees 90 degrees

  5. Straight Angle: 180 degrees.

  6. Obtuse angles: between 90 and 180 degrees. Acute angles are < 90 degrees

  7. Types of Angles Acute angles are less than 90 degrees Right angles are equal to 90 degrees. [ Looks like letter L ] Obtuse angles are greater than 90 degrees but less than 180 degrees. Straight angles look like lines and are equal to 180 degrees.

  8. Types of Angles Angles are differentiated by the quantify of rotation of the rays as if they were hands of a clock. No rotation is zero degrees and totally straight is 180 degrees. 90 degrees 45 degrees

  9. Types of Angles Smallest Zero Acute Small Right Middle Large Obtuse Straight Largest

  10. Measuring AnglesThe Protractor

  11. Measuring AnglesThe Protractor The smaller number is for the acute angles and the larger number is for the obtuse angles. Notice, the numbers add up to 180.

  12. 500

  13. 1400 400

  14. 250 600 600 350

  15. 570 350 530 350

  16. Adjacent Angles 1 2 Same vertex, Common ray, and no common interior

  17. Non-Adjacent Angles 2 1 Not the same endpoint.

  18. Non-Adjacent Angles B T A G Overlapping Interiors is not allowed.

  19. 4 3 5 6 2 7 1 9 8

  20. How Many Angles ? 2 + 1 = 3

  21. How Many Angles ? 3 + 2 + 1 = 6

  22. How Many Angles ? 3 + 2 + 1 = 10 4 +

  23. Did you see the pattern? 2 + 1 = 3 3 + 2 + 1 = 6 4 + 3 + 2 + 1 = 10 Total angles = sum of countdown of the smallest angle totals.

  24. 500 Vertex Position One ray must be horizontal. Reading a protractor

  25. Protractor Postulate For on a given plane, choose any point O between A and B. Consider and and all the rays that can be drawn from O on one side of . B A O

  26. Protractor Postulate These rays can be paired with the real numbers from 0 to 180 in such a way that: is paired with 0 and with 180. 0 180 B A O

  27. Protractor Postulate These rays can be paired with the real numbers from 0 to 180 in such a way that: If is paired with x and is paired with y, then P X Q Y 0 180 B A O

  28. Protractor Postulate These rays can be paired with the real numbers from 0 to 180 in such a way that: If is paired with x and is paired with y, then = 50 P Example 100 Q 150 0 180 B A O

  29. Example 2 70 120 C T A Top Scale 500 500 Bottom Scale

  30. Angle Addition Postulate A B O C If point B lies in the interior of then And

  31. Angle Addition Postulate B A O C If is a straight angle and B is any point not on then

  32. Note: The angle addition postulate is just like the segment addition postulate. When the two angles form a straight line then they are called linear pairs. Euclid referred to this concept as … “The sum of the parts equals the whole.”

  33. Angle Addition Applications A B 310 220 O C 530

  34. Example 2 A 5x +13 B 4x +1 220 O C Find the values of the angles. 4x +1 +22 = 5x +13 4x +23 = 5x +13 10 = x Substitute back into expressions.

  35. Summary Angles are 2 rays with a common end point. There are 4 types of angles: Acute – less than 900 Right = 900 Obtuse – between 900and 1800 Straight = 1800

  36. Summary 2 Angles can be indicated by numbers, the vertex, or by 3 letter of which the middle letter is the vertex. Angles are measured with a protractor. The Protractor Postulate establishes measuring angles with a protractor. The Angle Addition Postulate establishes the sum of two adjacent angles is indeed the sum of the two angles.

  37. C’est fini. Good day and good luck.

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