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1-6: Exploring Angles

1-6: Exploring Angles. Expectations: You will be able to use the definitions of rays and angles. You will be able to solve problems involving congruent angles. Ray.

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1-6: Exploring Angles

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  1. 1-6: Exploring Angles Expectations: You will be able to use the definitions of rays and angles. You will be able to solve problems involving congruent angles. 1.6: Exploring Angles

  2. Ray • Defn: A ray with endpoint D containing E and F, denoted DE or DF, is the union of DE and all points F, such that E is between D and F. F E D 1.6: Exploring Angles

  3. What do you mean? • AB : The distance between A and B. • AB: The segment with endpoints A and B. • AB: The ray with endpoint A, containing B • AB: The line through A and B. 1.6: Exploring Angles

  4. Opposite Rays • Opposite Rays: AB and AC are opposite rays iff A, B and C are _____________ and A is between B and C. C A B 1.6: Exploring Angles

  5. Angles • Defn: An angle is the union of two _____ with the same endpoint, but no other points in common. The common endpoint is the __________ of the angle and all other points on the rays make up the _______ of the angle. 1.6: Exploring Angles

  6. Angle Y is the vertex and the sides are YX and YZ. X Y Z 1.6: Exploring Angles

  7. _____ is the symbol for an angle. 1.6: Exploring Angles

  8. 3 ways to name an angle • 1. If there is only one angle, we may name the angle by its vertex. A 1.6: Exploring Angles

  9. 3 ways to name an angle • 2. We can place the vertex between a point from each side of the angle. Z X Y 1.6: Exploring Angles

  10. 3 ways to name an angle • 3. If there are 2 or more angles, we can use numbers. 2 1 3 4 1.6: Exploring Angles

  11. Angles and the Plane • Angles split the plane into 3 parts: interior exterior 1.6: Exploring Angles

  12. Measure of an Angle • Defn: The measure of ∠ABC, written _______, is a real number between 0 and 180. • Technically, an angle ________ have measure of 0 or 180. 1.6: Exploring Angles

  13. Protractor Postulate • Given AB and any real number r, s.t. 0<r<180, there are exactly 2 rays AC such that m∠CAB = r (one on each side of AB). 1.6: Exploring Angles

  14. C1 45° 45° C2 Given AB and let r = 45 A B 1.6: Exploring Angles

  15. Angle Addition Postulate(Somewhat obvious, but really important!) • If R is in the interior of ∠ABC, then m∠ABR + m∠RBC = __________. • If m∠ABR+ m∠RBC=m∠ABC, then R is in the interior of _________. 1.6: Exploring Angles

  16. gsp angle addition 1.6: Exploring Angles

  17. A R B C 1.6: Exploring Angles

  18. Determine the value of a if m∠WXZ = 59°. Y W Z 38° a X 1.6: Exploring Angles

  19. Determine the value of x if m∠WXZ = 102°. Y W Z (3x + 9)° (7x – 7)° X 1.6: Exploring Angles

  20. Acute, Right, Obtuse Angles • Defn: Acute, Right, Obtuse Angles: Let a be the measure of ∠A. Then angle A is: • a. Acute iff ________. • b. Right iff ________. • c. Obtuse iff ________. • d. Straight Angle iff _______. • e. Zero Angle if ________. 1.6: Exploring Angles

  21. Congruent Angles • Defn: ∠A is congruent to ∠B, written ∠A ≅∠B, iff ___________________. 1.6: Exploring Angles

  22. Angle Bisector • Remember to bisect means to cut exactly in half, so… • Defn: BD is the bisector of ∠ABC iff D is in the interior of ∠ABC and _____________. 1.6: Exploring Angles

  23. Solve for x and the measure of all three angles if XY bisects ∠WXZ. Y Z Z 15x-37 8x+12 X 1.6: Exploring Angles

  24. Which of the following is true of a ray KL? It extends infinitely in both directions. Its endpoint is L. It has length. It contains exactly 2 points, K and L. It is a portion of a line beginning with point K. 1.6: Exploring Angles

  25. The notation FG represents: The length of a line The length of a segment The length of a ray Two points A plane 1.6: Exploring Angles

  26. Assignment • Pages 50 – 51, • 17 – 25 (odds), 33 and 35 1.6: Exploring Angles

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