REASONING IN GEOMETRY

Chapter 1

PATTERNS AND INDUCTIVE REASONING

Section 1-1

Inductive Reasoning

When you make a conclusion based on a pattern of examples or past events

ConjectureA conclusion that you reach based on inductive reasoning

CounterexampleAn example that shows your conjecture is false

It only takes one counterexample to prove your conjecture false

ExamplesFind the next three terms of each sequence.

11.2, 9.2, 7.2, …….1, 3, 7, 13, 21, …….

……..

POINTS, LINES AND PLANES

Section 1-2

PointA basic unit of geometry

Has no sizeNamed using capital letters

LineA series of points that extends without end in two directions.

Named with a single lowercase letter or by two points on the line

Collinear and Noncollinear

Points that lie on the same line

Points that do not lie on the same line

RayHas a definite starting point and extends without end in one direction

Starting point is called the endpoint

Named using the endpoint first, then another point

Line SegmentHas a definite beginning and end

Part of a line Named using endpoints

Plane

A flat surface that extends without end in all directions

Named with a single uppercase script letter or three noncollinear points

Coplanar and Noncoplanar

Points that lie in the same plane

Points that do not lie in the same plane

POSTULATES

Section 1-3

PostulatesFacts about geometry that are accepted as true

Postulate 1-1

Two points determine a unique line

Postulate 1-2If two distinct lines intersect, then their intersection is a point.

Postulate 1-3Three noncollinear points determine a unique plane.

Postulate 1-4If two distinct planes intersect, then their intersection is a line.

CONDITIONAL STATEMENTS AND THEIR CONVERSES

Section 1-4

Conditional StatementWritten in if-then formExamples:If points are collinear, then they lie on the same line.

If a figure is a triangle, then it has three angles.

If two lines are parallel, then they never intersect.

HypothesisThe part following the if

If points are collinear, then they lie on the same line.

If a figure is a triangle, then it has three angles.

If two lines are parallel, then they never intersect.

ConclusionThe part following the then

If points are collinear, then they lie on the same line.

If a figure is a triangle, then it has three angles.

If two lines are parallel, then they never intersect.

ConverseA conditional statement is formed by exchanging the hypothesis and the conclusion in a conditional statement

Example

Statement: If a figure is a triangle, then it has three angles.

Converse: If a figure has three angles, then it is a triangle.

A PLAN FOR PROBLEM SOLVING

Section 1-6

PerimeterThe distance around a figure

FormulaAn equation that shows how certain quantities are related

AreaThe number of square units needed to cover the surface of a figure