Oliver Shi Postulates/Theorems- 1. Through any two points there is exactly one line.
2. Through any three nonlinear points there is exactly one plane.
3. If two points lie in a plane, then the line containing those points lies in the plane. 4. If two unique lines intersect, then they intersect at exactly one point.
5. If two unique planes intersect, then they intersect at exactly one line.
6. Law of Detachment: If a->b is a true statement and a is true, then b is true.
7. Law of Syllogism: If a->b and b->a are true statements, then a->b is a true statement. 8. Law of Contrapositive: A conditional statement and its contrapositive have the same truth value.
9. Addition Property of Equality: If a = b, then a + c = b +c
10. Subtraction Property of Equality: If a = b, then a - c = b - c
11. Multiplication Property of Equality: If a = b. then ac = bc
12. Division Property of Equality: If a = b, c ≠ 0, then a/c = b/c
13. Reflexive Property of Equality: If a = a, then a = a
14. Symmetric Property of Equality If a = b, then b = a
15. Transitive Property of Equality If a = b and b = c, then a = c
16. Substitution Property of Equality If a = b, then b can be substituted for a in any equation. 17. Distributive Property: a(b + c) = ab + ac, and a(b - c) = ab - ac
18. Ruler Postulate: The points on a line can be put into a one-to-one correspondence with the real numbers
19. Segment Addition Postulate: If B is between A and C then AB + BC = AC
20. Protractor Postulate: If point X is on ray AB, then all rays that can be drawn from X can be put into a one-to-one correspondence with the real numbers from 0 to 180 21. Angle Addition Postulate: If X is in the interior of ∠ABC, then m∠ABX+ m∠BCX = m∠ABC. 22. Linear Pair Theorem: If two angles form a 180 degree angle or a linear pair, then they are supplementary. 23. Congruent Supplementary Theorem: If two angles are supplementary to the same angle, then they are congruent. 24. Right Angle Congruence Theorem: All right angles are congruent. 25. Congruent Complements Theorem: If two angles are complementary to the same angle, then they are congruent. 26. Common Segments Theorem: If A, B, C, and D are all collinear, the points are arranged on a segment such that A and D are the endpoints, and B is between A and C, and C is between B and D. If line segment AB is congruent to line segment CD, then line segment AC is congruent to line segment BD. 27. Vertical Angles Theorem: Vertical angles are always congruent 28. If two congruent angles are supplementary, then they are both a right angle.
I chose theses postulates and theorems, because they are very rudimentary yet important. Without these, solving geometric equations would be nearly impossible. These postulates are also extremely practical, so practical that I use them nearly every day in and out of school. All of these theorems are a necessity to geometry and can be easily memorized. These are the reasons I chose these postulates.