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. Line A B 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
𝐴𝐵 means line AB or BA m A B m

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
𝐶𝐷 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 P A D L

7. Collinear Complementary Angles Coplanar Supplementary Angles
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
𝐸𝐹 or 𝐹𝐸 E F

9. 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 L 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 𝐴𝐵 𝐶𝐷 B D A C

12. Perpendicular Lines At a 90° angle.
Perpendicular lines have slopes that are negative reciprocals
Named by 𝐸𝐹⟘𝐺𝐻 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. 1 4 2 3

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. 1 4 2 3

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 Postulate Corollary Undefined Terms
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
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.
Five Key Elements
Given
Draw Diagrams
Prove
Statement
Reasons

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 𝐴𝐵 , then 𝐴𝑀 ≅ 𝑀𝐵 .
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 ∠𝑃𝑄𝑆, then 𝑚∠𝑃𝑄𝑅+𝑚∠𝑅𝑄𝑆=𝑚∠𝑃𝑄𝑆.
If 𝑚∠𝑃𝑄𝑅+𝑚∠𝑅𝑄𝑆=𝑚∠𝑃𝑄𝑆, then R is in the interior of ∠𝑃𝑄𝑆.

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

27. Reasoning Inductive Reasoning Deductive Reasoning Conjecture
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 Conclusion Counterexample Negation
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 (𝑝→𝑞) Converse (𝑞→𝑝)
Statement that can be written in if-then form
Converse (𝑞→𝑝)
Exchanging the hypothesis and conclusion
Inverse (~𝑝→~𝑞)
Negating the hypothesis and conclusion
Contrapositive (~𝑞→~𝑝)
Exchange & negate the hypothesis & conclusion

31. If-Then Statements Related Conditionals Logically Equivalent
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 𝑝→𝑞 is true and 𝑝 is true, then 𝑞 is also true
If an angle is obtuse, then it cannot be acute
∠𝐴 is obtuse
∠𝐴 cannot be acute

33. Law of Syllogism Law of Syllogism
If 𝑝→𝑞 is true and 𝑞→𝑟 are true, then 𝑝→𝑟
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 𝒑 𝒒
𝒑∧𝒒
𝒑∨𝒒 T F

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