1. Lesson 1-5 Segments and Their Measures 1

2. Objectives Measure segments. Add segment lengths.

3. Key Vocabulary Coordinate Distance Length Between Congruent Congruent Segments

4. Postulates Postulate 5: Segment Addition Postulate

5. The Number Line (a ruler) To every point on a line there corresponds a unique real number. To every real number there corresponds a unique point on the line. Definition: Coordinate – The line number at corresponds to a point on a line.

6. Points On A Line The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. In the diagram, x 1 and x 2 are coordinates. The small numbers are subscripts. The coordinates are read as “x sub 1” and “x sub 2.” AB x1x1 x2x2 Names of points Coordinates of points AB x1x1 x2x2 AB

7. Distance Definition: Distance – is a numerical description of how far apart objects are. It is the absolute value of the difference between the coordinates of A and B. The distance is always positive. AB is also called the length of segment AB. AB x1x1 x2x2 Names of points Coordinates of points

8. The Ruler Points on a line can be paired with the real numbers in such a way that: Any two chosen points can be paired with coordinates on a ruler. The distance between any two points on a number line is the absolute value of the difference of the real numbers corresponding to the points. Formula: Take the absolute value of the difference of the two coordinates a and b: │a – b │

9. The Ruler The length AB can be found by |x 2 -x 1 |. **The symbol for the length of AB is AB.** AB x1x1x1x1 x2x2x2x2 AB = |x 2 – x 1 |

10. Example: Find AB.AB Point A is at 2.5 and B is at 5. So, AB = |5 - 2.5| = 2.5

11. Note: It doesn’t matter how you place the ruler. For example, if the ruler in The Example is placed so that A is aligned with 1, then B aligns with 3.5. The difference in the coordinates is the same.AB

12. Ruler: Example PK =| 3 - -2 | = 5 Remember: Distance is always positive Find the distance between P and K. Note: The coordinates are the numbers on the ruler or number line! The capital letters are the names of the points. Therefore, the coordinates of points P and K are 3 and -2 respectively. Substituting the coordinates in the formula │a – b │

13. Measure the total length of the shark’s tooth to the nearest inch. Then measure the length of the exposed part. 1.Align the zero mark of the ruler with one end of the shark’s tooth. SOLUTION Use a ruler to measure in inches. 2.Find the length of the shark’s tooth, AC. 4 1 2 4 1 2 – 0 = AC = 3.Find the length of the exposed part, BC. 8 7 4 1 2 8 3 1 – = BC = 8 1 Example 1

14. Example 2 A. Find the length of AB using the ruler. The ruler is marked in millimeters. Point B is closer to the 42 mm mark. Answer: AB is 42 millimeters long.

15. Example 2 B. Find the length of AB using the ruler. Each centimeter is divided into fourths. Point B is closer to the 4.5 cm mark. Answer: AB is 4.5 centimeters long.

16. Your Turn: A.2 mm B.1.8 mm C.18 mm D.20 mm

17. Your Turn A.1 cm B.2 cm C.2.5 cm D.3 cm B.

18. Example 3 Each inch is divided into sixteenths. Point E is closer to the 3-inch mark. A.

19. Example 3 B.

20. Your Turn: A. B. C. D.

21. Your Turn: A. B. C. D. B.

22. Betweenness Definition: X is between A and B if AX + XB = AB. AX + XB = ABAX + XB > AB Betweenness refers to collinear points only. X is between A and BX is not between A and B Because the 3 points are not collinear.

23. Is Alex between Ty and Josh? In order for a point to be between 2 others, all 3 points MUST BE collinear!! TyAlexJosh Yes! No, but why not? How about now?

24. Between When three points lie on a line, you can say that one of them is between the other two. This concept applies to collinear points only. A B C Point B is between points A and C.

25. Postulate 5: Segment Addition Postulate If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. AC ABC ABBC

26. Segment Addition If C is between A and B, then AC + CB = AB. Example: If AC = x, CB = 2x and AB = 12, then, find x, AC and CB. AC + CB = AB x + 2x = 12 3x = 12 x = 4 2x x 12 x = 4 AC = 4 CB = 8 Step 1: Draw a figure Step 2: Label fig. with given info. Step 3: Write an equation Step 4: Solve and find all the answers

27. Example 4 Find XZ. Assume that the figure is not drawn to scale. XZ is the measure of XZ. Point Y is between X and Z. XZ can be found by adding XY and YZ. ___ Add.

28. Your Turn: A.16.8 mm B.57.4 mm C.67.2 mm D.84 mm Find BD. Assume that the figure is not drawn to scale. B C D 16.8 mm 50.4 mm

29. Example 5 Find LM. Assume that the figure is not drawn to scale. Point M is between L and N. LM + MN= LNBetweenness of points LM + 2.6= 4Substitution LM + 2.6 – 2.6= 4 – 2.6Subtract 2.6 from each side. LM= 1.4Simplify.

30. Your Turn: Find TU. Assume that the figure is not drawn to scale. T U V 3 in A. B. C. D. in.

31. Use the map to find the distance from Athens to Albany. ANSWER The distance from Athens to Albany is 170 miles. SOLUTION Because the three cities lie on a line, you can use the Segment Addition Postulate. AM = 80 miles Use map. MB = 90 miles Use map. AB = AM + MB Segment Addition Postulate = 80 + 90 Substitute. = 170 Add. Example 6

32. Use the diagram to find EF. 16 – 10 = 10 + EF – 10 Subtract 10 from each side. 6 = EF Simplify. 16 = 10 + EF Substitute 16 for DF and 10 for DE. SOLUTION DF = DE + EF Segment Addition Postulate Example 7

33. Find the length. ANSWER 20 ANSWER 8 1.Find AC. 2.Find ST. Your Turn:

34. Example 8 ALGEBRA Find the value of x and ST if T is between S and U, ST = 7x, SU = 45, and TU = 5x – 3. ST + TU= SUBetweenness of points 7x + 5x – 3= 45Substitute known values. 7x + 5x – 3 + 3= 45 + 3Add 3 to each side. 12x= 48Simplify. Draw a figure to represent this situation.

35. Example 8 x= 4Simplify. ST= 7xGiven = 7(4)x = 4 = 28Multiply. Answer: x = 4, ST = 28 Now find ST.

36. Your Turn: A.n = 3; WX = 8 B.n = 3; WX = 9 C.n = 9; WX = 27 D.n = 9; WX = 44 ALGEBRA Find the value of n and WX if W is between X and Y, WX = 6n – 10, XY = 17, and WY = 3n.

37. Definition of congruent Congruent means same, different from equal. Congruent means having the same measure. ≅

38. Congruent Segments Definition: If numbers are equal the objects are congruent. AB: the segment AB ( an object ) AB: the distance from A to B ( a number ) Congruent segments can be marked with dashes. Correct notation: Incorrect notation: Segments with equal lengths. (congruent symbol: )

39. How to mark congruent segments in a figure A B C D E

40. How to mark congruent segments in figures A B C DEDE FGFG

41. ANSWER DE and FG have the same length. So, DE  FG. Are the segments shown in the coordinate plane congruent? For a vertical segment, subtract the y-coordinates. FG = –3 –1 = –4 = 4 SOLUTION For a horizontal segment, subtract the x-coordinates. DE = 1 – (–3) = 4 = 4 Example 9

42. 1.A(–2, 3), B(3, 3), C(–3, 4), D(–3, –1) Plot the points in a coordinate plane. Then decide whether AB and CD are congruent. 2.A(0, 5), B(0, –1), C(5, 0), D(–1, 0) ANSWER yes; ANSWER yes; Your Turn:

43. Joke Time How do crazy people go through the forest? They take the psycho path. What do you call a boomerang that doesn't work? A stick.

44. Assignment Section 1.5, pg. 31 – 33: #1 – 45 odd