Chapter 1 Summary
Definitions
Some of these are technically "undefined," but so be it...
- point, line, segment, ray, triangle
- angle
- two rays with a common endpoint
- Kinds of angles: acute, right, obtuse, straight
- collinear points
- points that all lie on the same line
- noncollinear points
- points that are not collinear, i.e. not all lie on the same line (some might, but not all)
- betweenness of points
- (an undefined term)
- triangle inequality
- Any three points are either collinear or determine a triangle. In a triangle, the sum of the lengths of any two sides is greater than the length of the third (p. 19).
- Midpoint
- The point that bisects a segment
- bisect
- To bisect an angle or segment is to divide it into two congruent angles or segments. A point, line, segment or ray can bisect a segment; a line, segment or ray can bisect an angle (p. 28,29).
- trisect
- To divide a segment or angle into three congruent parts (p.29)
- postulate
- A statement accepted without proof.
- definition
- A definition states them meaning of a term. A definition can be used in place of the term without changing meaning, and a term can be used in place of its definition without changing meaning (reversibility) (p. 40).
- theorem
- A statement that is proven (p.40).
- conditional statement (or implication)
- An if...then... statement. We also write "if p then q" as "p ⇒ q"
- converse
- The converse of "p ⇒ q" is "q ⇒ p". The converse of a true statement is not always true.
- inverse
- The inverse of "p ⇒ q" is "~p ⇒ ~q". (Meaning "not p ⇒ not q") The inverse of a true statement is not always true.
- contrapositive
- The contrapositive of "p ⇒ q" is "~q ⇒ ~p". (Meaning "not q ⇒ not p") Unlike the converse and the inverse, the contrapositive of a true statement is always true.
- equivalence
- If p⇒q and q⇒p then we say "p if and only if q" or "p iff q" or p⇔q. In this case, p and q may be used interchangeably.