Geometric StructureBasic Information

Introduction Instruction Examples Practice

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Introduction Instruction Examples Practice

Geometry is a way of thinking about and seeing the world. Geometry is evident in nature, art and culture. What geometric

objects do you see in this picture?

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Geometry is both ancient and modern.Geometry originated as a systematic study in the works of Euclid, through its synthesis with the work of Rene Descartes, to its present connections with computer and calculator technology.

What geometric objects do you see in this picture?

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The basic terms and postulates of geometry will be introduced as well as the tools needed to explore geometry.

What geometric term are you familiar with?

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Three building blocks of geometry are pointspoints, lineslines and planesplanes. They are considered building blocks because they are basic and undefined in terms of other figures.

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A pointpoint is the most basic building block of geometry.• has no size• indicates location• represented by a dot• named with a capital letter.

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j

Points A and Q.

CF or FC and j.

AQ

C F

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A lineline is a straight, continuous arrangement of infinitely many points.• has infinite length but no thickness.•extends forever in two directions.

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j

Points A and Q.

CF or FC and j.

AQ

C F

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A lineline is named with two identified points on the line with a line symbol (double-headed arrows) placed over the letters; or by a single, lower case script letter.

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j

Points A and Q.

CF or FC and j.

AQ

C F

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A planeplane has length and width but no thickness.• is like a flat surface the extends infinitely along its length and width.• represented by a four-sided figure drawn in perspective.

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Q

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A plane is named with a script capital letter, Q.It may also be named using three points (not on the same line) that lie in the plane, such as G, F and E.

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Q

G

E

F

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An axiomatic systemaxiomatic system is a way of organizing facts.• postulates postulates are accepted without proof• theorems theorems are truths that can be derived from postulates

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Mathematicians accept undefined terms and definitions so that a consistent system may be built.The theorems of an axiomatic system rest on postulates and other theorems.

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As with all axiomatic systems, geometry is connected with logic.This logic is typically expressed with convincing argument or proofproof.

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•Consider the model. Look at points A and E.•How many lines pass through these two points?•Complete the postulate:

Through any two Through any two points there is points there is

exactly one exactly one ______________.______________.

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•Consider the model. Look at points A, E and H.•How many planes pass through these three noncollinear points?•Complete the postulate:

Through any three noncollinear points there is exactly one

_______________.

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Collinear pointsCollinear points are points that lie on the same line.In the figure at the right, A, B and C are collinear.A, B and D are noncollinear.

Any Any twotwo points are points are collinearcollinear.

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D

A B C

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Coplanar pointsCoplanar points are points that lie in the same plane.In the figure at the right, E, F, G, and H are coplanar.E, F, G, and J are noncoplanar.

Any Any threethree points are points are coplanar.coplanar.

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When geometric figures have one or more points in common, they are said to intersectintersect.The set of points that they have in common is called their intersection.

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mn

S

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•Examine the geometric model at the right.•Specifically, identify the places where lines intersect each other.•Complete the theorem:

The intersection of The intersection of two lines is a two lines is a ___________.___________.

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point

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•Consider the model.•Specifically, identify the places in the diagram where planes intersect each other.•Complete the postulate:

The intersection of two planes is a _______________.

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line

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1. Through any two points there is exactly one _______.

2. Through any three noncollinear points there is exactly one ______.

3. The intersection of two lines is a _______.

4. The intersection of two planes is a ______.

Our First 3 Postulates and a Theorem

POINT

LINE

LINE

PLANE

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Example 1Back to main example page

1. Name the intersection of plane ABDC and plane YZDB.

2. How many lines drawn in the figure contain point Z? Enumerate.

3. How many planes drawn in the figure contain line BY? Enumerate.

4. True or false: Two planes intersect in exactly one point. Explain.

Example 2Classify each statement as truetrue or falsefalse. Explain each.

5.Two lines intersect in a plane.

6.Any three points are contained in exactly one line.

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Example 1Back to main example page

•Name the intersection of plane ABDC and plane YZDB.

•How many lines drawn in the figure contain point Z?

•How many planes drawn in the figure contain line BY?

•True or false: Two planes intersect in exactly one point.

line BD

Two – line DZ and YZ

plane BAXY and plane YZDB

False - line

2

Example 2Classify each statement as truetrue or falsefalse.

•Two lines intersect in a plane.

•Any three points are contained in exactly one line.

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False - point

False – only collinear points

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Introduction Instruction Examples Practice

With the foundational terms (point, line and plane) described, other geometric figures may be defined.

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“Let no one ignorant of geometry enter my door.” - Plato

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A segmentsegment is a part of a line that begins at one point and ends at another.• has two endpoints• named by its endpoints• a bar (no arrows) is drawn over the two capitalized letters

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MN or NM

M

N

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A ray ray is a is a part of a line that starts at a point and extends infinitely in one direction. • has one endpointendpoint• named with its endpoint first• a single arrow is drawn over the two capitalized letters.

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MN

M

N

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Opposite rays are two collinear rays that share a common endpoint.

and are opposite rays. This is page

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FE FG

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The length or measure of a segment is the distance between its endpoints.

e.g. the length of is PQ

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PQ

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Segment with equal length are said to be congruent ().

If AB = CD, then .

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CD AB

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B is between A and C iff they are collinear and AB + BC = AC.

The midpoint of a segment is the point that divides the segment into two congruent segments. In the figure, DE = EF.

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A segment bisector is a segment, ray, line or plane that intersects a segment at its midpoint.

A perpendicular bisector intersects the segment at the midpoint and is perpendicular to it.

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•An angleangle is a figure formed by two rays with a common endpoint.•The common endpoint is the vertex of the vertex of the angleangle.•The rays are the sides sides of the angleof the angle.•Angles are formed when lines, rays, or line segments intersect.

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S

R

T

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•An angle divides the plane into two regions

•Interior•Exterior

•If two points, one from each side of the angle, are connected with a segment, the segment passes through the interior of the angle.

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interior of angle

exterior of angle

S

R

T

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•An angle is named using three points.•The vertex must be the middle point of the name.•Write SRT or TRS.•Say “angle S R T” or “angle T R S.”•If there is no possibility of confusion, the angle may be named S or 1.

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1S

R

T