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Deductive Reasoning

“The proof is in the pudding.”. “Indubitably.”. Le pompt de pompt le solve de crime!". Je solve le crime. Pompt de pompt pompt.". Deductive Reasoning. 2-4 Special Pairs of Angles. 2-4 Special Pairs of Angles. Complementary angles are pairs of angles whose measure sums up to 90 0. 60. 30.

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Deductive Reasoning

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  1. “The proof is in the pudding.” “Indubitably.” Le pompt de pompt le solve de crime!" Je solve le crime. Pompt de pompt pompt." Deductive Reasoning 2-4 Special Pairs of Angles

  2. 2-4 Special Pairs of Angles Complementary angles are pairs of angles whose measure sums up to 900. 60 30 50 40 Note that when adjacent pairs are complementary, they form the letter L.

  3. Complementary angles can be seen visually as well as being computed. Note a dyslexic L. 45 45 35 55 35 + 55 = 90

  4. Complementary angles are like a marriage. Each angle is the complement of the other. Complementary Angles Find the complement of X. X Note that 50 is the complement of 40 And that 40 is the complement of 50. They are complements of each other just as husband and wives are spouses of each other. 80 10 50 40 20 70 45 45 40 50 30 60 Do you think you understand ? 55 35

  5. Each angle is the complement of the other. Complementary Angles Find the complement of X. 1 89 75 15 Tough Question ? Makes you really think! 100 None A 90 – 100 = -10 B Angles are never negative. Therefore, no solution or none. Note, that not all angles have complements. 90 - C 90- D

  6. Each angle is the complement of the other. Complementary Angles Find the complement of X. 90 – (90- C) 1 89 90 – 90 + C 75 15 100 None + C A 90 -A B 90 - B Stuck ? 90 - C C 90- D D Just do what you did for all the others.

  7. Now let’s examine another special pair of angles. Supplementary angles are pairs of angles whose measure sums up to 1800. 120 110 70 60 linear pairs Note that when the supplementary pairs are adjacent, they form a straight line. Such situations are called linear pairs.

  8. It is easy to recognize supplementary angles when they are linear pairs – form a straight line. When supplementary angles are adjacent to each other, They form a straight line. Note that this always looks like a tree branch.

  9. When the pair of angles form a straight line, it is not necessary to add up the angles. We can see the sum is 1800. The angles are supplementary. 120 60 linear pairs

  10. 130 50 The protractor gives the measure of both angle simultaneously. The smaller number matches the acute angle and The larger number matches the obtuse angle.

  11. When the angles are not adjacent, then the sums must be calculate to determine if the angles are supplementary Which angles are supplementary?

  12. Supplementary angles are like a marriage. Each angle is the supplement of the other. Find the supplement of X. Supplementary Angles Note that 40 and 140 are supplements or each other. 10 170 40 140 80 100 140 40 160 20 90 90 179 1

  13. Supplementary angles are like a marriage. Each angle is the supplement of the other. Find the supplement of X. Supplementary Angles Notice, the all angles have supplements while they do not always have complements. 10 170 40 140 80 100 Restated Unlike complements, supplements always have a companion pair. 140 40 160 20 A 180 - A B 180 - B

  14. Students sometimes confuse complementary with supplementary. An easy way to remember which is which it to think of the following: C comes before S And 90 comes before 180.

  15. When two lines, rays, or segments cross each other, the X generates 2 pairs of opposite angles called Vertical Angles. Draw 2 intersecting lines on tracing paper or patty paper. Next, fold the paper at the vertex so that one segment fold on top of the next segment. What do you observe?

  16. The opposite angles, the vertical angles, are congruent. NO ! Is this a proof? But is does suggests the following theorem. Vertical Angles are congruent. Now we will prove the theorem.

  17. Prove: Statements Reasons Given Substitution Reflexive Property Subtr. Prop. Of Equality

  18. Where is treasure located on the map? X marks the spot.

  19. “Vertical angles are congruent” is the most important and easiest theorem in geometry. Every time you see an X, the theorem can be applied. Vertical angles will come up in problems over and over again. Recognize the X and you will be rewarded with information.

  20. Vertical Angle Problems VA Complete and justify why. b a c 140 140 Vertical angles are congruent.

  21. Vertical Angle Problems Complete and justify why. b a c a + 140 = 180 140 a = 40 140 Vertical angles are congruent. 40 Angle Add. Postulate 40 Vertical angles are congruent.

  22. Vertical Angle Problems Complete and justify why. b a c VA a + 140 = 180 140 a = 40 140 Vertical angles are congruent. 40 Angle Add. Postulate 40 Vertical angles are congruent.

  23. Find x. X = 4 Sample Problems B T 8x + 6 2x + 30 8x + 6 38 A 8(4) + 6 K D 32 + 6 = 38 8x + 6 = 2x + 30 142 8x = 2x + 24 38 + y = 180 6x = 24 y = 142 x = 4

  24. Let’s try some sample problems Find the complement and supplement of each angle. 90 – 5 = 85 Complement ____________________ Supplement _____________________ 180 – 5 = 175 90 - 30 = 60 Complement ____________________ Supplement _____________________ 180 – 30 = 150 90 – 100 = none Complement ____________________ Supplement _____________________ 180 – 100 = 80 90 – k Complement ____________________ Supplement _____________________ 180 – k

  25. Name 2 right angles. Name a pair of adjacent complementary angles. Name a pair of non-adjacent complementary angles. Name a supplement of Name another pair of supplementary angles.

  26. Name a pair of non-adjacent complementary angles. Name a supplement of Name another pair of supplementary angles.

  27. Name 2 congruent supplementary angles. B C A E D Name 2 supplementary angles that are not congruent. Name 2 complementary angles. Name a straight angle.

  28. Name 2 congruent supplementary angles. B C A E D Name 2 supplementary angles that are not congruent.

  29. Name 2 congruent supplementary angles. B C A E D Name 2 supplementary angles that are not congruent. Name 2 complementary angles. Name a straight angle.

  30. B C A D 50 40 0 E F

  31. B C A D 50 40 0 E F

  32. B C A D 50 40 0 E F

  33. B C A D 0 E F

  34. B C A D 50 40 0 E F 50 40

  35. B C 90 A D 50 40 0 E F 50 50 + x + 40 = 180 40 x + 90 = 180 x = 90 90

  36. B C 90 A D 50 40 0 E F 50 50 + x + 40 = 180 40 x + 90 = 180 x = 90 90 90

  37. 140 B C 90 A D 50 40 40 50 0 90 E F

  38. 140 B C 130 90 A D 50 40 40 50 0 90 E F

  39. 140 B C 130 90 A D 50 40 40 50 130 0 90 E F

  40. 140 B C 130 90 A D 50 40 40 50 130 0 90 E F 90

  41. Summary There are three special pairs of angles. Complementary Angles 60 30 Pairs of angle whose sum = 900 . Supplementary Angles Pairs of angle whose sum = 1800 . Vertical Angles

  42. Each pair is associated with shapes. Complementary Angles 60 The letter L. 30 Supplementary Angles A tree branch. Vertical Angles X marks the spot.

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

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