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Midpoints and Congruence (Geometry 2_3)

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Students consider the definition of midpoint and the congruence of two line segments.

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Midpoints and Congruence (Geometry 2_3)

  1. 1. Congruent Segments You will learn to identify congruent segments and find the midpoints of segments. What You'll Learn
  2. 2. Congruent Segments In geometry, two segments with the same length are called ________ _________ You will learn to identify congruent segments and find the midpoints of segments. What You'll Learn
  3. 3. Congruent Segments In geometry, two segments with the same length are called ________ _________ congruent segments You will learn to identify congruent segments and find the midpoints of segments. What You'll Learn
  4. 4. Congruent Segments In geometry, two segments with the same length are called ________ _________ congruent segments You will learn to identify congruent segments and find the midpoints of segments. What You'll Learn Two segments are congruent if and only if ________________________ Definition of Congruent Segments
  5. 5. Congruent Segments In geometry, two segments with the same length are called ________ _________ congruent segments You will learn to identify congruent segments and find the midpoints of segments. What You'll Learn they have the same length Two segments are congruent if and only if ________________________ Definition of Congruent Segments
  6. 6. Congruent Segments B A C R S P Q
  7. 7. Congruent Segments B A C R S P Q
  8. 8. Congruent Segments B A C R S P Q
  9. 9. Congruent Segments Use the number line to determine if the statement is True or False. Explain you reasoning. x 11 -5 -3 -1 1 3 5 7 9 -6 -4 0 4 8 2 10 6 -2 R S T Y
  10. 10. Congruent Segments Use the number line to determine if the statement is True or False. Explain you reasoning. x 11 -5 -3 -1 1 3 5 7 9 -6 -4 0 4 8 2 10 6 -2 R S T Y Because RS = 4 and TY = 5,
  11. 11. Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. These statements are called ________.
  12. 12. Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. These statements are called ________. theorems
  13. 13. Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. These statements are called ________. theorems Theorems are statements that can be justified by using logical reasoning.
  14. 14. Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. These statements are called ________. theorems Theorems are statements that can be justified by using logical reasoning. Congruence of segments is reflexive. 2 – 1
  15. 15. Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. These statements are called ________. theorems Theorems are statements that can be justified by using logical reasoning. Congruence of segments is reflexive. 2 – 1
  16. 16. Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. These statements are called ________. theorems Theorems are statements that can be justified by using logical reasoning. Congruence of segments is reflexive. 2 – 1 Congruence of segments is symmetric. 2 – 2
  17. 17. Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. These statements are called ________. theorems Theorems are statements that can be justified by using logical reasoning. Congruence of segments is reflexive. 2 – 1 Congruence of segments is symmetric. 2 – 2
  18. 18. Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. These statements are called ________. theorems Theorems are statements that can be justified by using logical reasoning. Congruence of segments is reflexive. 2 – 1 Congruence of segments is symmetric. 2 – 2 Congruence of segments is transitive. 2 – 3
  19. 19. Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. These statements are called ________. theorems Theorems are statements that can be justified by using logical reasoning. Congruence of segments is reflexive. 2 – 1 Congruence of segments is symmetric. 2 – 2 Congruence of segments is transitive. 2 – 3
  20. 20. Congruent Segments There is a unique point on every segment called the _______.
  21. 21. Congruent Segments There is a unique point on every segment called the _______. midpoint
  22. 22. Congruent Segments There is a unique point on every segment called the _______. midpoint What do you notice about SM and MT ? On the number line below, M is the midpoint of . x 11 -5 -3 -1 1 3 5 7 9 -6 -4 0 4 8 2 10 6 -2 S M T
  23. 23. Congruent Segments There is a unique point on every segment called the _______. midpoint What do you notice about SM and MT ? On the number line below, M is the midpoint of . x 11 -5 -3 -1 1 3 5 7 9 -6 -4 0 4 8 2 10 6 -2 S M T
  24. 24. Congruent Segments There is a unique point on every segment called the _______. midpoint What do you notice about SM and MT ? SM = MT On the number line below, M is the midpoint of . x 11 -5 -3 -1 1 3 5 7 9 -6 -4 0 4 8 2 10 6 -2 S M T
  25. 25. Congruent Segments A point M is the midpoint of a segment if and only if M is between S and T and SM = MT Definition of Midpoint
  26. 26. Congruent Segments SM = MT A point M is the midpoint of a segment if and only if M is between S and T and SM = MT Definition of Midpoint M S T
  27. 27. Congruent Segments SM = MT The midpoint of a segment separates the segment into two segments of _____ _____. A point M is the midpoint of a segment if and only if M is between S and T and SM = MT Definition of Midpoint M S T
  28. 28. Congruent Segments SM = MT The midpoint of a segment separates the segment into two segments of _____ _____. equal length A point M is the midpoint of a segment if and only if M is between S and T and SM = MT Definition of Midpoint M S T
  29. 29. Congruent Segments SM = MT The midpoint of a segment separates the segment into two segments of _____ _____. equal length So, by the definition of congruent segments, the two segments are _________. A point M is the midpoint of a segment if and only if M is between S and T and SM = MT Definition of Midpoint M S T
  30. 30. Congruent Segments SM = MT The midpoint of a segment separates the segment into two segments of _____ _____. equal length So, by the definition of congruent segments, the two segments are _________. congruent A point M is the midpoint of a segment if and only if M is between S and T and SM = MT Definition of Midpoint M S T
  31. 31. Congruent Segments Find the value of x. In the figure, B is the midpoint of . 5x - 6 2x C B A
  32. 32. Congruent Segments Find the value of x. Since B is the midpoint: AB = BC In the figure, B is the midpoint of . 5x - 6 2x C B A
  33. 33. Congruent Segments Find the value of x. Since B is the midpoint: AB = BC Write the equation involving x : 5x – 6 = 2x In the figure, B is the midpoint of . 5x - 6 2x C B A
  34. 34. Congruent Segments Find the value of x. Since B is the midpoint: AB = BC Write the equation involving x : 5x – 6 = 2x Solve for x : 5x – 2x – 6 = 2x – 2x In the figure, B is the midpoint of . 5x - 6 2x C B A
  35. 35. Congruent Segments Find the value of x. Since B is the midpoint: AB = BC Write the equation involving x : 5x – 6 = 2x Solve for x : 5x – 2x – 6 = 2x – 2x 3x – 6 + 6 = 0 + 6 3x = 6 In the figure, B is the midpoint of . 5x - 6 2x C B A
  36. 36. Congruent Segments Find the value of x. Since B is the midpoint: AB = BC Write the equation involving x : 5x – 6 = 2x Solve for x : 5x – 2x – 6 = 2x – 2x 3x – 6 + 6 = 0 + 6 3x = 6 In the figure, B is the midpoint of . 5x - 6 2x C B A x = 2
  37. 37. Congruent Segments Find the value of x. Since B is the midpoint: AB = BC Write the equation involving x : 5x – 6 = 2x Solve for x : 5x – 2x – 6 = 2x – 2x 3x – 6 + 6 = 0 + 6 3x = 6 AB = 5x – 6 = 5(2) – 6 = 10 – 6 = 4 Check! In the figure, B is the midpoint of . 5x - 6 2x C B A x = 2
  38. 38. Congruent Segments Find the value of x. Since B is the midpoint: AB = BC Write the equation involving x : 5x – 6 = 2x Solve for x : 5x – 2x – 6 = 2x – 2x 3x – 6 + 6 = 0 + 6 3x = 6 AB = 5x – 6 = 5(2) – 6 = 10 – 6 = 4 BC = 2x = 2(2) = 4 Check! In the figure, B is the midpoint of . 5x - 6 2x C B A x = 2
  39. 39. Congruent Segments To bisect something means to separate it into ___ congruent parts.
  40. 40. Congruent Segments To bisect something means to separate it into ___ congruent parts. two
  41. 41. Congruent Segments To bisect something means to separate it into ___ congruent parts. two The ________ of a segment bisects the segment because it separates the segment into two congruent segments.
  42. 42. Congruent Segments To bisect something means to separate it into ___ congruent parts. two The ________ of a segment bisects the segment because it separates the segment into two congruent segments. midpoint
  43. 43. Congruent Segments To bisect something means to separate it into ___ congruent parts. two The ________ of a segment bisects the segment because it separates the segment into two congruent segments. A point, line, ray, or plane can also bisect a segment. midpoint B A E G C D
  44. 44. Congruent Segments To bisect something means to separate it into ___ congruent parts. two The ________ of a segment bisects the segment because it separates the segment into two congruent segments. A point, line, ray, or plane can also bisect a segment. midpoint B A E G C D
  45. 45. Congruent Segments To bisect something means to separate it into ___ congruent parts. two The ________ of a segment bisects the segment because it separates the segment into two congruent segments. A point, line, ray, or plane can also bisect a segment. midpoint B A E G C D
  46. 46. Congruent Segments To bisect something means to separate it into ___ congruent parts. two The ________ of a segment bisects the segment because it separates the segment into two congruent segments. A point, line, ray, or plane can also bisect a segment. midpoint B A E G C D
  47. 47. Congruent Segments To bisect something means to separate it into ___ congruent parts. two The ________ of a segment bisects the segment because it separates the segment into two congruent segments. A point, line, ray, or plane can also bisect a segment. midpoint B A E G C D
  48. 48. Congruent Segments End of Lesson

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