1/18/20161.1: the building blocks of geometry expectation: g1.1.6: recognize euclidean geometry as...

05/03/23 1.1: The Building Blocks of Geometry

1.1: The Building Blocks of Geometry

Expectation:G1.1.6:Recognize Euclidean geometry as an axiom

system. Know the key axioms and understand the meaning of and distinguish between undefined terms (e.g., point, line, and plane), axioms, definitions, and theorems.

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Undefined Terms

1.

2.

3.

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Points

Models of points

0 dimensional (have no size)

-Dots-Stars in the sky-Cities on a map

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PointsNaming points

- Use capital letters

A

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LinesPerfectly straight extending infinitely without thickness.

Models of Lines- roads- hallway- edge of a box

1 dimensional

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Naming Lines

l

Line l

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Naming Lines

A

B

Line AB or BA

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PlanesInfinite, flat surface without thickness.2 dimensional

Models of Planes-desktop-wall

-rooftop

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Drawing Planes

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Naming Planes

Plane R

R

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Naming Planes

Plane PQR, PRQ, RPQ, RQP, QRP or QPR

PR

Q

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Collinear Points Defn:Three or more points are collinear iff they are on the same line.

A B C

A, B and C are collinear.

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Coplanar Points

Defn: Four or more points are coplanar iff they are on the same plane.

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Coplanar Points

P, Q, R and S are coplanar points.

PR

Q S

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SegmentsDefn: A segment is a part of a line that begins at one point and ends at another. The points are called the endpoints of the segment.

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Segments

A B

AB

This is not a segment!!!

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Segments

A B

The segment with endpoints A and B: denoted AB or BA

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RaysDefn: A ray is a part of a line that starts at a point and extends infinitely in one direction. The point is called the endpoint of the ray.

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Rays

Y

Z

The ray with endpoint Y containing Z: denoted YZ (may not be denoted ZY or ZY).

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AnglesDefn: An angle is a figure formed by two rays with a common endpoint. The common endpoint is called the vertex of the angle and the two rays are called the sides of the angle.

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Angles

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Angles

vertex of the angle

sides of the angle

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Naming an AngleThere are 3 ways to name an

angle: ( represents an angle)1. The vertex.2. A point from each side and

the vertex (vertex is between the other points).

3. A number.

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Naming Angles

A

B

C

1

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Naming Angles

A

B

C

B

1

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Naming Angles

A

B

C

ABC or CBA

1

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Naming Angles

A

B

C

1

1

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Intersect and Intersections

If 2 figures share one or more points, then they intersect. The point or points they have in common is called their intersection.

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Postulates

A postulate is a statement that is accepted as being true without proof.

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Complete Activity 1.1 on page 12.

You may work in pairs.

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Intersecting Lines Postulate

If two lines intersect, then their intersection is a point.

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Intersecting Planes Postulate

If two planes intersect, then their intersection is a line.

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Unique Line PostulateThrough any two distinct points,

there is a unique (one and only one) line.

Sometimes this is stated in an alternate form: ”Two points determine exactly one line.”

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Unique Plane PostulateThrough any 3 noncollinear points

there is a unique plane.

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Flat Plane PostulateIf two points are in a plane, then the

line containing them is in the plane.

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Dimension PostulateGiven a line in a plane, there is at least

one point in the plane that is not on the line.

Given a plane in space, there is at least one point in space not on the plane.

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Assignment:

pages 14 – 15, # 9 – 48