1. Cover Slide Use Segments and Congruence

2. Between Segment Relationships and Rays Between L J K N K is between J and L.N is not between J and L. (Refer to the figure above) In order for K to be between J and L : all three points must be collinear K must lie on segment JL

3. Segment Definitions Segment consists of two points called endpoints and all the points between the endpoints C B Example: Named: by its two endpoints under the segment symbol The segment above can be named: or BCCB

4. Ray Definitions Ray consists of a segment that is extended indefinitely in one direction N Example: Named: by two points under the ray symbol Ways to name the ray above with endpoint N: or NBNC B C Ray NB consists of NB and all other points C such that B is between N and C. the endpoint must be named first

5. Opposite Rays Definitions Opposite Rays two collinear rays with a common endpoint L Example: BT Because L is between B and T, LB and LT are opposite rays. Postulates (or Axioms) statements assumed to be true accepted without proof

6. Length Definitions Length ( or Measure ) the distance between two points C B BC means segment BC BC means the measure of segment BC (without the segment symbol over the letters) The length or measure of a segment means the distance between its two endpoints.

7. Summary of Denotations C B means segment BC means the measure of segment BC (without the segment symbol over the letters) BC means line BC means ray BC BC

8. Ruler Post/ Distance on a Number Line Distance on a number line 0123456 -6-6 -5-5 -4-4 -3-3 -2-2 -1 BAND Distance (or the length of a segment) must be a positive number. The Ruler Postulate 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 that point. The distance between two points A ad B, written as AB, is the absolute value of the difference of their coordinates AB. Postulate 1 - Ruler Postulate

9. Distance Number Line Prob. 1 & 2 Computing Distance on a Number Line 0123456 -6-6 -5-5 -4-4 -3-3 -2-2 -1 BAND 1. Find the distance between points N and B. 2. Find the distance between points A and D. NB =  - 5 - 3  =  - 8  NB = 8 NB =  3 – ( - 5 )  =  8  NB = 8 or AD =  - 2 - 5  =  - 7  AD = 7 AD =  5 – ( - 2 )  =  7  AD = 7 or

10. Distance Number Line Prob. 3 & 4 Distance on a Number Line 0123456 -6-6 -5-5 -4-4 -3-3 -2-2 -1 BAND 3. Find the distance between points A and B. 4. Find the distance between points A and N. AB =  - 5 - ( - 2 )  =  - 3  AB = 3 AB =  - 2 - ( - 5 )  =  3  AB = 3 or AN =  - 2 - 3  =  - 5  AN = 5 AN =  3 – ( - 2 )  =  5  AN = 5 or

11. Segment Add Postulate If L is between B and T, then BL + LT = BT. If BL + LT = BT, then L is between B and T. Postulate 2 - Segment Addition Postulate Measures of Segments A B C AB = 16, Example: BC = 7 Find AC AB + BC = AC 16 + 7 = AC 23 = AC LBT

12. Apply Segm + Post prob. 1 & 2 A B C Applying the Segment Addition Postulate 1. AB = 12, AC = 41 2. AB = 3x – 2, BC = 5x + 6 AB + BC = AC 12 + x = 41 BC = 29 AB + BC = AC 3x – 2 + 5x + 6 = 36 AB = 10 Find BC Find AB B is between A and C AC = 36 8x + 4 = 36 8x = 32 x = 4 AB = 3x – 2 AB = 12 – 2 x = 29

13. Apply Segm + Post prob. 3 & 4 A B C Applying the Segment Addition Postulate 3. BC = 3(AB), AC = 72 4. AB = 22 - x, BC = 5x - 4 AB + BC = AC x + 3 x = 72 BC = 54 AB + BC = AC 22 - x + 5x - 4 = 6x + 4 AC = 46 Find BC Find AC B is between A and C AC = 6x + 4 4x + 18 = 6x + 4 14 = 2x 7 = x AC = 6x + 4 AC = 42 + 4 4x = 72 AB = x BC = 3x x = 18 BC = 3x BC = 3(18)

14. Congruent objects Definitions Congruent (Objects) objects that have the same size and shape Example: the congruent symbol is:  The triangles above appear to be congruent. More specific definitions for congruence will be used in the discussion of future geometric figures.

15. Congruent segments Definitions Congruent Segments (  segments  segments that are equal in length Example: 2 in A B C D AB and CD have equal lengths AB =  CD AB   CD Therefore, AB is congruent to CD The definition of  segments indicates the two statements are equivalent. They will be used interchangeably.

16. Congruent segments cont. Definitions Congruent Segments (  segments  segments that are equal in length Example: 2 in A B C D AB and CD have equal lengths AB =  CD AB   CD Therefore, AB is congruent to CD The small red “hash marks” indicate  segments.