Unit 2

1. Unit 2 Reasoning & Proof

2. Vocabulary • Each word needs a page in your log Definition/Explanation: Ways to Name: Vocabulary Word Relationship: Drawing/Example:

3. Point • Basic undefined term in geometry • Location represented by a dot • The geometric figure formed at the intersection of two distinct lines. • Named with italicized capital letter: D, M, P

4. m Line • Basic undefined term in geometry • A line is the straight path connecting two points and extending beyond the points in both directions. • Made up of points with no thickness or width • Named by two points on the line or small italicized letters • means line AB or BA • m A B

5. Line Segment • All points between two given points (including the given points themselves). • Measurable part of the line between two endpoints including all points in between • Named by endpoints of segment • means Segment CD or Segment DC • C and D are the endpoints of the segment C D

6. Plane • A flat surface with no depth extending in all directions. • Any three noncollinear points lie on one and only one plane. • So do any two distinct intersecting lines. • A plane is a two-dimensional figure. • Named by three non-collinear points or capital script letter • ADL, LAD, LDA, DAL, DLA, ALD or P A L D P

7. Collinear • Points that lie on the same line • Complementary Angles • Two acute angles that add up to 90° • Also adjacent form a right angle. • Coplanar • Points that lie in the same plane • Supplementary Angles • Two angles that add up to 180°

8. Ray • A part of a line starting at a particular point and extending infinitely in one direction. • Named by end point and one other letter • or E F

9. L Angle • Two rays sharing a common endpoint. • Intersection of two noncollinear rays at common endpoint. • Rays are called sides and common endpoint is called a vertex • Typically measured in degrees or radians • Named by 3 letters--vertex in center position • KLM or MLK M K

10. Congruent • Exactly equal in size, length, measure and shape. • For any set of congruent geometric figures, corresponding sides, angles, faces, etc. are congruent (CPCTC). • Congruent segments, sides, and angles are often marked

11. Parallel Lines • Two distinct coplanar lines that do not intersect. • Parallel lines have the same slope. • Named by B D A C

12. Perpendicular Lines • At a 90° angle. • Perpendicular lines have slopes that are negative reciprocals • Named by G E F H

13. Adjacent Angles • Two angles in a plane which share a common vertex and a common side but do not overlap and have no common interior points.

14. Vertical Angles • Nonadjacent angles opposite one another at the intersection of two lines. • Vertical angles are congruent. • Angle 1 and 3 are congruent vertical angles. • Angle 2 and 4 are congruent vertical angles. 2 3 1 4

15. Linear Pair • A pair of adjacent angles formed by intersecting lines. • Non-common sides are opposite rays • Linear pairs of angles are supplementary. • Angles 1 and 2, 2 and 3, 3 and 4, 1 and 4 are linear pairs. 2 3 1 4

16. Theorem • An assertion that can be proved true using the rules of logic. • Is proven from axioms, definitions, undefined terms, postulates, or other theorems already known to be true. • A major result that has been proved to be true

17. Axiom • A statement accepted as true without proof. • So simple and direct that it is unquestionably true. • Postulate • Statement that describes a fundamental relationship between the basic terms of geometry • Accepted as true without proof • Corollary • Statement that can b easily proven • Undefined Terms • Readily understood words that are not formally explained by more basic words and concepts • Point, line, plane

18. Proof • Five Key Elements • Given • Draw Diagrams • Prove • Statement • Reasons • Step-by-step explanation that uses definitions, axioms, postulates, and previously proven theorems to draw a conclusion about a geometric statement. • Logical argument in which each statement is supported by a statement that is accepted as true.

19. Two-Column Proofs • Formal Proof • Statements & reasons organized into two columns

20. Algebraic Proofs • Group of algebraic steps used to solve problems (deductive argument) • Uses Properties of Equality for Real Numbers • Reflexive • Symmetric • Transitive • Addition & Subtraction • Multiplication & Division • Substitution • Distibutive

21. Flow Proofs • Organizes a series of statements in logical order, starting with the given statement • Statement written in box with reason written below box • Arrows indicate how statements are related

22. Indirect Proof • Uses indirect reasoning • Assume conclusion is false • Show that assumption leads to contradiction • Since assumption false, conclusion must be true • Also called proof by contradiction

23. Coordinate Proof • Uses figures in the coordinate plane and algebra to prove geometric concepts • Placing Figures • Use the origin as a vertex or center of the figure • Place at least one leg on an axis • Keep figure in 1st quadrant if possible • Use coordinates to make computations as simple as possible.

24. Paragraph Proof • Informal Proof • Paragraph written to explain why a conjecture for a given statement is true.

25. Theorems and Postulates • Midpoint Theorem • If M is the midpoint of , then . • Segment Addition Postulate • If B is between A and C, then AB+BC=AC. • If AB+BC=AC, then B is between A and C. • Angle Addition Postulate • If R is in the interior of , then . • If , then R is in the interior of .

26. Angles formed by Parallel lines • Transversals • Corresponding • Alternate Interior • Alternate Exterior • Consecutive

27. Reasoning • Inductive Reasoning • Uses specific examples to arrive at a general conclusion • Lacks logical certainty • Deductive Reasoning • Uses facts, rules, definitions, or properties to reach logical conclusions • Conjecture • Educated guess

28. If-Then Statements • A compound statement in the form “if A, then B”, where A and B are statements • Statement • Any sentence that is true or false, but not both • Compound Statement • A statement formed by joining two or more statements

29. If-Then Statements • Hypothesis • Statement that follows if in a conditional • Conclusion • Statement that follows then in a conditional • Counterexample • Used to show that a statement is not always true • Negation • Adds not to statement ()

30. If-Then Statements • Conditional Statement () • Statement that can be written in if-then form • Converse () • Exchanging the hypothesis and conclusion • Inverse () • Negating the hypothesis and conclusion • Contrapositive () • Exchange & negate the hypothesis & conclusion

31. If-Then Statements • Related Conditionals • Converses, Inverses, and conditionals that are based on a given conditional statement • Logically Equivalent • Statements that have the same truth value

32. Law of Detachment • Law of Detachment • If is true and is true, then is also true • If an angle is obtuse, then it cannot be acute • is obtuse • cannot be acute

33. Law of Syllogism • Law of Syllogism • If is true and are true, then • If Molly arrives at school early, she can get help in math. • If Molly gets help in math, then she will pass her test. • If Molly arrives at school early, the she will pass her test.

34. Truth Tables A table used to organize the truth values of statements Truth Value – The truth or falsity of a statement

35. Disjunction • Compound statement formed by joining two or more statements with or • , reads p or q • False only when both statements are false • True when one or both statements is true • Conjunction • Compound statement formed by joining two or more statements with and • , reads p and q • False when one or both statements is false • Both statements must be true for the conjunction to be true