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Chapter 1 Tools of Geometry

Goals: 1) learn to draw conclusions based on patterns 2) learn the building blocks for the structure of geometry

3) learn to measure line segments and angles 4) understand the idea of basic geometric constructions 5) learn to use the coordinate plane to represent geometric figures

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1-1 Patterns and Inductive Reasoning

New Vocabulary Inductive Reasoning ~ Conjecture ~ Counterexample

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I. Using Inductive Reasoning

Inductive Reasoning is reasoning based on patterns you observe.

Sequence Pattern 2 Subsequent Termsa. 3, 6, 12, 24,...

b.

Example 1

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II. Making a Conjecture

A conclusion you reach using inductive reasoning is called a Conjecture.

Example 2

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Not all conjectures turn out to be true. You can prove that a conjecture is false by finding one counterexample. A counterexample to a conjecture is an example proving the conjecture false.

III. Testing a Conjecture

Example 3 The first three odd prime numbers are 3, 5, and 7. Make a conjecture about the fourth odd prime number.

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1-2 Points, Lines, and Planes

New Vocabulary point ~ space ~ line ~ collinear points plane ~ coplanar ~ postulate ~ axiom

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PointPoint

Space

Line

Plane

Matching

A B

A

P

I. Undefined Terms and other definitions

* A Point is generally thought of as a location. A point has no size, but is represented by a dot and usually named by a capital letter.

Space is defined as the set of all points. Space is infinite in size, and can be thought of as a 3-D object with no boundaries.

* A line can be thought of as a series of points extending in opposite directions with no boundaries. You can name a line by stating any two points on that line.

Points that lie on the same line are called Collinear Points.

*A Plane is a flat surface that has no thickness containing many points and lines in both directions. A plane can be thought of as a 2-D object (length by width) with no boundaries.

Points and lines that lie on the same plane are called Coplanar.

A Postulate or Axiom is an accepted statement or fact.

Point

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Example 4

A

E

C

DB

1.

2.

3.

4.

5.

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Example 5

AB

CD

E F

GH

1.

2.

3.

4.

5.

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POSTULATESPostulate 1.1: Through any two points there is exactly one line.

Postulate 1.2: If two lines intersect, then they intersect in exactly one point.

Postulate 1.3: If two planes intersect, then they intersect in exactly one line.

Postulate 1.4: Through any 3 noncollinear points there is exactly one plane.

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1-3 Segments, Rays, Parallel Lines and Planes

New Vocabulary segment ~ ray ~ opposite ray ~ parallel lines skew lines ~ parallel planes

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MatchingSegment

Ray

Opposite Rays

Parallel Lines

Skew Lines

Parallel Planes

A B

A B

A B C

I. Definitions

A segment is the part of a line consisting of two endpoints and all of the points in between.

A ray is the part of a line consisting of one endpoint and all of the points on the line on one side of the endpoint.

Opposite rays are two collinear rays with the same endpoint. Opposite rays always form a line.

Parallel Lines are coplanar lines that do not intersect.

Skew lines are noncoplanar, therefore they are not parallel and still do not intersect.

Parallel Planes are planes that do not intersect.

AB

CD

E F

GH

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1-4 Measuring Segments and Angles

New Vocabulary coordinate ~ congruent segments ~ midpointangle ~ acute angle ~ right angleobtuse angle ~ straight angle ~congruent angles

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Postulate 1-5 The Ruler Postulate The distance between two points on a real number line can be found by calculating the absolute value of the difference of the two corresponding numbers.

I. Segments, Midpoints, and Measuring Segments

A coordinate is similar to a point. It can be thought of as a location, specifically on a number line or on a coordinate grid.

A B

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Postulate 1-6 The Segment Addition PostulateIf three points A, B, and C are collinear and B is between A and C, then AB + BC = AC.

Two segments with the same length are called congruent segments. You can determine if two segments are congruent by calculating the length of each segment and then comparing them.

A B C

Example 6 If EG = 100, find the value of x. Then find the value of EF and FG.

E F G

4x - 20 2x + 30

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A Midpoint of a segment is a point that divides the segment into two congruent segments. A midpoint of any segment is said to bisect the segment.

A B C

Example 7 Finding lengths

C is the midpoint of segment AB. Find AC, CB, and AB.

A C B

2x + 1 3x - 4

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An angle (<) is formed by two rays with the same endpoint. The rays are the sides of the angle, and the endpoint is the vertex.

II. Angles, Naming, and Measuring

A

E

D

Example 8 Naming Angles

A

B

D

C12

1.

2.

3.

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Postulate 1-7 The Protractor Postulate

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Matching

Acute Angle

Right Angle

Obtuse Angle

Straight Angle

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Postulate 1-8 Angle Addition Postulate

Angles with the same measure are Congruent Angles.

A

O

B

C A O C

B

Example 9 What is m <TSW if m<RST = 50 and m<RSW = 125?

R

S

T

W