UNIT 2Reasoning & 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• 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

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

• 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• 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

• or

F

E

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 K

M

L

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

A

C

D

B

Perpendicular Lines• At a 90° angle. • Perpendicular lines have slopes that are negative

reciprocals• Named by

G

FE

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.

14

23

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.

14

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• 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

• 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.

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 Figures1. Use the origin as a vertex or center of the figure

2. Place at least one leg on an axis

3. Keep figure in 1st quadrant if possible

4. 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 , 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 .

Angles formed by Parallel lines• Transversals

• Corresponding• Alternate Interior• Alternate Exterior• Consecutive

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

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

• 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 ()

• 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

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

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

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.

Truth Tables

T T T T

T F F T

F T F T

F F F F

A table used to organize the truth values of statementsTruth Value – The truth or falsity of a statement

• 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