# A Generalization of Rolle's Theorem with Application to by Dieudonne J.

By Dieudonne J.

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68 FUNDAMENTALS OF COLLEGE GEOMETRY ELEMENTARY If 3. If If 4. If If land m intersect at right angles, they are perpendicular. he drives, he should not drink. he drinks, he should not drive. the natural number is not even, it is odd. the natural number is not odd, it is even. The equivalence truth table. The step. of contrapositive statements numbers under each column Ifx T T F F ~q) - 18. is shown by the following indicates the order of each (~q~ 19. 21. T F T F T T T T F T F T T F T T F F T T 3 1 4 2 3 2 22.

12. Complements ofthe same angle are congruent. Given: Lx and L(J are complementary angles. Ly and L(J are complementary angles. Prove: Lx == Ly. LL Given: Lx is the complement Conclusion: Lx == Ly. 8. Proof Complements REASONS 1. Lx and L(J are complementary Ly and L(J are complementary 2. mLx+ mL(J Ai. Ai. = 90. mLy + mL(J = 90. 3. mLx+mL8 = mLy+mL8. 4. mLx = mLy. c-= L Y. 1. Given. 2. Ifa=c,b=c,thena=b. 4. Subtractive property f). Lx == Ly ~ mLx = of equaliti mLy'. It is important that each statement in the proof be substantiated by areas.

14. Diamonds are expensive. 15. Those who study will pass this course. 16. The sides of an equilateral triangle are congruent to each other. 17. The person who steals will surely be caught. 18. To be successful, one must work. 19. The worke-r will be a success. 20. You must be satisfied or your money will be refunded. 21. With your looks, I'd be a movie star. 8. Modus ponens. An implication by itself is of little value. However, if we know "p implies q" and that p is also true, we must accept q as true.