Inductive reasoning and conditional statements

1. Inductive reasoning

Conjecture: an unproven statement that is based on observation

Inductive reasoning: find a pattern in specific cases, then write a conjecture for the simple one

EXAMPLE: 

Observations: Ellen wears a pink shirt on Wednesdays.

Today is Wednesday.

Conjecture: Ellen will wear a pink shirt today.

Conditional statement

− a logical statement that has two parts:

• conclusion

• hypothesis

         If-then statement: This is just the basic idea of the rule. It's like saying, "If this happens, then do that." For example

 "If it's raining, then bring an umbrella."

Converse: This is when we flip the rule around. So, if the original rule is "If it's raining, then I'll bring an umbrella," the converse would be " But remember, this might not be true all the time.

If I'm bringing an umbrella, then it's raining." 

Inverse: This is when we say the opposite of both parts of the rule. For example, if the original rule is "If it's raining, then I'll bring an umbrella," the inverse would be 

"If it's not raining, then I won't bring an umbrella." 

Contrapositive: This is when we flip and negate both parts of the rule. So, using the same example, the contrapositive would be ..  And the cool thing is, if the original rule is right, then the contrapositive is also right. If the original rule is wrong, then the contrapositive is wrong too.

"If I'm not bringing an umbrella, then it's not raining."

If-then form − “if” part contains the hypothesis

− “then” part contains the conclusion

• If there’s pizza, then it must be Friday.

Converse: − exchange the hypothesis and conclusion

• If it’s friday, then there must be pizza

Inverse:− negate both the hypothesis and conclusion

• If  there's no pizza, then it’s not friday

Contrapositive:− first write the converse,

then negate both the hypothesis and conclusion

• If it’s not friday, then there is no pizza

Biconditional statements 

− statements that contain the

phrase “if and only if”

− the conditional statement and its

converse are both true 

iff− if and only if

2. POINT

− a position on a sheet of paper or a position in space

− A point has no size

− Normally represented by a small cross mark () or a

small dot ()

− Points are marked by uppercase letters

A b

LINE

− consists of an infinite number of points

<—--------—---------------------—--------—--------------------->

− arrows indicate that the line goes on in both

directions without end.

<—--------—---------------------—--------—--------------------->

A B

Line AB or line BA

<—--------—---------------------—--------—--------------------->

If theres no point its called Line ℓ

3. Collinear and Non-collinear

Collinear points − three or more points that lie on the same line

A B C

Non-collinear points − three or more points that don’t lie on the

same line

D

E F

Plane 

− consists of an infinite number of lines

− a flat surface that extends in all directions without end

—-—-—-—-—-

|         Q |

| E |

|   F           G |

—-—-—-—-—-

Plane Q

Plane FEG

Coplanar points

 − points that line in the same plane

Line Segments

− terminated line

Endpoint -> •—-----------------------------•  <- Endpoint

       G H

  GH or HG

Rays

• part of the line that has a fixed starting point but no end point

•—--------—---------------------—--------•—---------------------> HG

H         G

GH ≠ HG

<—--------—---------•------------—--------—---------------------• GH

      H     G

• If point C lies on  AB between A and B, then CA and CB are opposite rays.

<—--------•—-------------------•--—--------—---------•------------>

    A       C   B

Intersection

 − one or more points in common to two or more geometric figures

The intersection of two The intersection of two

distinct lines is a point. distinct plain is a line.

Angles – measuring

Protractor Postulate: Consider OB and a point A on one side of OB.

The rays of the form OA can be matched one to

one with the real numbers from 0 to 180.

The measure of ∠AOB is equal to the

absolute value of the difference between

the real numbers for OA and OB.

m∠AOB = 140°

Angles – classification

Acute: less than 90*   |        Right: exactly 90*    |    Obtuse: greater than 90*  |   Straight: 180*

Less than 180*

Angle Addition Postulate: If P is in the interior of ∠RST, then the

measure of ∠RST is equal to the sum of

the measures of ∠RSP and ∠PST .

If P is in the interior of ∠RST, then the

m∠RST = m∠RSP + m∠PST .

Congruent Angles

 − Two angles are congruent angles if they have the same measure.

Angle Bisector 

− a ray that divides an angle into two angles that are congruent.

RT bisects ∠SRU. So, ∠SRT ≅ ∠TRU.

Angle Pair Relationships – Definitions

Complementary angles − two angles whose sum of measures is 90°

Supplementary angles − two angles whose sum of measures is 180°

Adjacent angles − two angles that share a common vertex and side,

but have no common interior points

Angle Pair Relationships – Definitions

Linear pair − two adjacent angles whose noncommon sides are opposite rays

− angles in a linear pair are supplementary angles. They share a common vertex and one common side but do not overlap.

Vertical angles − two angles whose sides form two pairs of opposite

Rays They are opposite angles created by the crossing lines.

Polygons – Definitions

Plane Figure − a figure that lines in a plane

Polygon − a closed plane figure with the following properties:

• It is formed by three or more line segments, called sides.

• Each side intersects exactly two sides, one at each endpoint, so that no two sides with a common endpoint are collinear

Vertices

Vertex − each endpoint of a side of the polygon

Name a polygon by listing the vertices in consecutive order

ABCDE or CDEA or DEABC�

Convex vs. Concave

Convex− a polygon with no line that contains a side will not contain a point in the interior of the polygon

Concave − a polygon that is not convex

− nonconvex

Classifying Polygons

Regular Polygons

Equilateral polygon − a polygon with all sides congruent

Equiangular polygon− a polygon with all angles congruent

Regular polygon − a convex polygon that is both equilateral and

equiangular

DIAGONAL

Diagonal of a polygon − a segment that joins two nonconsecutive

vertices of a polygon