1. (1.5) Division of Segments and Angles!!! By: Lauren Coggins, Kanak Chattopadhyay, and Morgan Muller What?!

2. Definitions and Their Converses O Definitions are ALWAYS Reversible. O Theorems are NOT ALWAYS Reversible. (Their converses are not always true.)

3. Midpoints Definition: If a point is the midpoint of a segment, then it divides the segment into two  segments. Ex. Given: C is the midpoint of AB Conclusion: AC  CB A C B 1.) C is the midpoint 1.) G Of AB 2.) AC  CB 2.) If a point is the midpoint of a segment, then it ÷s the segment into 2  segments

4. Midpoints Converse: If a point divides a segment into two  segment, then it is the midpoint of the segment. Ex. Given: PI  IE Conclusion: I is the midpoint of PE P I E 1.) PI  IE 1.) G 2.) I is the midpoint of PE 2.) If a point ÷s the segment into 2  segments, then it is the midpoint of the segment

5. Sample Problems O If M is the midpoint of FE, what conclusions can we draw? O Conclusions: O -AM  MB (If a point is the midpoint of a segment, then it divides the segment into 2 congruent segments.) O Point M bisects AB A M B

6. Trisection Points Definition: If two points trisect a segment, then they divide the segment into three congruent segments. Ex. Given: A and K are trisection points of CE. Conclusion: CA  AK  KE. CAKECAKE

7. S R 1.) A and E are 1.) G trisection points. 2.) CA  AK  KE 2.) If 2 points ÷ a segment into 2  segments, then they trisect the segment

8. Trisection Points Converse: If two points divide a segment into three congruent segments, then they trisect the segment. Ex. Given: CA  AN  NE. Conclusion: A and N trisect CE. CA N E 1.) CA  AN  NE 1.) G 2.) A and N trisect CE 2.) If 2 points ÷ a segment into 3  segments, then it is the midpoint of the segment

9. Angle Bisector Definition: If a ray bisects an angle, then it divides the angle into 2  angles. Ex. Given: IN bisects MIT Prove: MIN  TIN I T M N 1.) IN bisects MIT 1.) G 2.) MIN  TIN 2.) If a ray bisects an angle, then it ÷s the angle into 2  angles

10. Angle Bisector Converse: If a ray ÷s an  into 2  s, then it bisects the  Ex. Given: MIN  TIN Conclusion: IN bisects MIT 1.) MIN  TIN 1.) G 2.) IN bisects 2.) If a ray ÷s an  into 2  s, MIT then it bisects the  I T M N

11. Sample Problems O If OB is the bisector of AOC, then AOB is congruent to COB. (If a ray bisects an angle then it divides the angle into 2 congruent angles). O A B C

12. Angle Trisectors… Definition: Two rays that divide an angle into three congruent angles trisect the angle. The two dividing rays are called trisectors of the angle. Definition in “if then” Form: If 2 rays trisect an angle, then they divide the angle into three congruent angles. A B T H S Converse: If 2 rays divide the angle into 3 congruent angles, then they trisect the angle. For Example… If BAT  TAH  HAS, then AT and AH trisect BAS. Converse: If AT and AH trisect BAS, then BAT  TAH  HAS.

13. P I E N S Sample Problems Given: PS  SE Conclusion: S is the midpoint of PE Reason: If a point ÷s a segment into 2  segments, then it is the midpoint of the segment.

14. Sample Problems M I L K S Given: Points L and K are trisectors of IS Conclusion: IL  LK  KS Reason: If 2 points trisect a segment, then they divide the segment into 3  segments.

15. Sample Problems S N A P Given: ASN  PSA Conclusion: SA bisects PSN Reason: If a ray divides an angle into 2  angles, then it bisects the angle.

16. QUIZ TIME!!!!!!! Bisector Problems… Find CAR if AR bisects CAE and CAE equals 1.) 80 40 2.) 74 18 37 9 3.) 54 22 27 11 4.) 30 ½ 15 15 5.) 26 38 13 19 C A R E

17. QUIZ TIME!!!!!!! Given: LK bisects TI and RE TR = 6x; IE = 8x TL = 9; RK = 5 Perimeter of TREI = 84 Find: IE T I E R L K Answer = x = 4 IE = 32 units

18. QUIZ TIME!!!!!!! S O L I D 5x 7x 3x OD and OI divide straight angle SOL into three angles whose measures are in the ratio 5:7:2. Find mDOI. Answer: x = 12 mDOI = 84

19. Works Cited Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. New York: McDougal, Little & Company, 1991. Print. Oh! I remember!