Unit 2 Reasoning & Proof

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

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

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

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

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

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°

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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.

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

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.

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

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

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

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 (~)

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

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

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

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.

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

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