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GBK Geometry

GBK Geometry. Jordan Johnson. Today’s plan. Greeting Lesson: Navigation & the Protractor Postulate Homework / Questions Clean-up. Points of Sail. Bearing / Azimuth.

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GBK Geometry

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  1. GBK Geometry Jordan Johnson

  2. Today’s plan • Greeting • Lesson: Navigation & the Protractor Postulate • Homework / Questions • Clean-up

  3. Points of Sail

  4. Bearing / Azimuth “The US Army defines the azimuth between Point A and Point B as the angle, measured in the clockwise direction, between the north reference ray and Ray AB. For example, if the bearing between Point A and Point B is E 45° S, the azimuth between Point A and Point B is 135°.”[2][3]

  5. Measurement of Angles • Definitions: • A degree is 1/360 of the distance rotated in a circle. • A rotation is the set of positions that are occupied by a ray as it turns 360°. • A half-rotation is the set of positions for 180°.

  6. Angle Measure:The Protractor Postulate • The Protractor Postulate: • It is possible to number the rays in a half-rotation with coordinates from 0 to 180, such that positive number differences measure angles. • Note: This is the angle version of the Ruler Postulate. • This means if rays OA and OB have coordinates a° and b°, AOB = |a – b|. A B O

  7. Betweenness of Rays • Definition: • A ray OB is between rays OA and OC (in the same half-rotation) iff its coordinate is between theirs. • As with points (A-B-C), “OB is between OA and OC” is abbreviated “OA-OB-OC”. A B O C

  8. Angle Addition:The Betweenness of Rays Theorem • If OA-OB-OC, then AOB + BOC = AOC. A B O C

  9. Fields of Vision

  10. Methods of bisection • Compass and straightedge • Ruler (for segments) or protractor (for angles) • Paper folding

  11. Definitions • A point is the midpoint of a line segment iff it divides the segment into two equal segments. • A line bisects an angle iff it divides the angle into two equal angles. • Two things are congruent iff they would coincide exactly when superimposed: C A D B

  12. Definitions, cont. • A corollary is a theorem that is easy (usually trivially easy) to prove as a consequence of another theorem or a postulate.

  13. Two Basic Corollaries • Corollary of the Ruler Postulate: • A line segment has exactly one midpoint. • Corollary of the Protractor Postulate: • An angle has exactly one ray that bisects it.

  14. Homework • Asg #19: Read Ch. 3, Lesson 3 (pp. 93-97) & do: • Exercises #1-7, 16-20, 34-43, 54 • Bonus: Set III. • Due Wednesday, 10/17. • Test corrections are due Thursday (periods 1 & 2) or Friday (period 7). • Asg #20: Read Ch. 3, Lesson 4 (pp. 100-103) & do: • Set I, 1-6 and 10-14. • Set II, 31-36, 41-47. • Bonus: Set III. • Due Monday, 10/22.

  15. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and back tables). • See you tomorrow!

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