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Warm Up On a number line, graph each inequality. 1. x ≥ 3 2. 2 ≤ x ≤ 6

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- Warm Up
- On a number line, graph each inequality.
- 1. x ≥ 3
- 2. 2 ≤ x ≤ 6
- 3. x < 1 OR x > 0

P15 # 1-

Warm Up

In a sequence of starts and stops, an elevator travels from the first floor to the fifth floor and then to the second floor. From there, the elevator travels to the fourth floor and then to the third floor. If the floors are 3 m apart, how far has the elevator traveled?

Warm Up

What is the area of the shaded figure?

Objectives

Define Geometry

Identify, name, and draw points, lines, segments, rays, and planes.

Apply basic facts about points, lines, and planes.

Vocabulary

undefined termpoint

lineplane

collinearcoplanar

segmentendpoint

rayopposite rays

postulate

Geometry (Greek) "measurement of earth or land"

The study of geometry can be broken into two broad types:

plane geometry and Solid geometry

plane geometry: which deals with only two dimensions, it deals with objects that are flat, such as triangles and lines, that can be drawn on a flat piece of paper.

Solid geometry: allows width, depth and height. The world around us is obviously three-dimensional, such as cubes and spheres.

The most basic figures in geometry are undefined terms, which cannot be defined by using other figures. The undefined terms are: Point, Line, and Plane which are the building blocks of geometry.

Point: A exact location or place. Usually represented by a dot. It is important to understand that a point is not a thing, No size.

We indicate the position of a point by placing a dot with a pencil. This dot may have a diameter of, say, 0.2mm, but a point has no size. Points are usually named by using an upper-case single letter.

If a set of points all lie in a straight line, they are called 'collinear'.

If a set of points all lie on the same plane, they are called 'coplanar'.

Point: A exact location or place. Usually represented by a dot. It is important to understand that a point is not a thing, but a place.

Collinear: When a set of points all lie in a straight line.

K

L

M

N

Points that lie on the same line are collinear. K, L, and M are collinear. K, L, and N are noncollinear.

Points that lie on the same plane are coplanar. Otherwise they are noncoplanar.

Line: A geometrical object that is straight, infinitely long and infinitely thin.

A line is one-dimensional. It has zero width. A straight line is the shortest distance between any two points on a plane.

Line: In the figure below, the line PQ passes through the points P and Q, and goes off in both directions forever, and is perfectly straight. A line, strictly speaking, has no ends.

PLANE: A flat surface that is infinitely large and with no thickness and extends forever.

It is difficult to draw planes, since the edges have to be drawn. When you see a picture that represents a plane, always remember that it actually has no edges, and it is infinitely large.

Plane:

Plane:

Possible answer: AE, BE, CE

TEACH! Example 1

Example 1: Naming Points, Lines, and Planes

A. Name four coplanar points.

A, B, C, D

B. Name three lines.

Line Segment:

A straight line which links two points without extending beyond them. A line segment is one-dimensional. It has a measurable length, but has zero width..

The word 'segment' typically means 'a piece' of something, and here it means the piece of a full line, which would normally extend to infinity in both directions.

Line Segment: See the figure below. The line segment PQ links the points P and Q. The points P and Q are called the 'endpoints' of the segment.

RAY:

A portion of a line which starts at a point and goes off in a particular direction to infinity.

One way to think of a ray is a line with one end. A ray starts at a given point and goes off in a certain direction forever, to infinity.

The point where the ray starts is called (confusingly) the endpoint. A ray has no measurable length, because it goes on forever in one direction.

A Ray On its way to infinity it may pass through one or more other points. In the figure above, the ray starts at A and also passes through B..

Opposite Rays:

Opposite rays are two rays that both start from a common point and go off in exactly opposite directions and form a straight line.

You may name a ray using the Endpoint and any other point which the ray passes through. Such as

Do not forget the ray crown

Opposite Rays: When the two rays are opposite, the points A,Q and B are collinear, and QA and QB form a single straight line through the common endpoint Q...

N

M

T

Example 2: Drawing Segments and Rays

Draw and label each of the following.

A. a segment with endpoints M and N.

B. opposite rays with a common endpoint T.

M

N

TEACH! Example 2

Draw and label a ray with endpoint M that contains N.

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

Postulates about points, lines, and planes help describe geometric properties.

XY

Example 3: Identifying Points and Lines in a Plane

Name a line that passes through two points.

TEACH! Example 3

Name a plane that contains three noncollinear points.

Possible answer: plane GHF

HINT

Use a dashed line to show the hidden parts of any figure that you are drawing. A dashed line will indicate the part of the figure that is not seen.

Example 4: Representing Intersections

A. Sketch two lines intersecting in exactly one point.

B. Sketch a figure that shows a line that lies in a plane.

TEACH! Example 4

Sketch a figure that shows two lines intersect in one point in a plane, but only one of the lines lies in the plane.

2. A point on BC.

Lesson Quiz: Part I

1. Two opposite rays.

3. The intersection of plane N and plane T.

4. A plane containing E, D, and B.

Lesson Quiz: Part II

Draw each of the following.

5. a line intersecting a plane at one point

6. a ray with endpoint P that passes through Q

CB and CD

2. A point on BC.

Possible answer: BD

Lesson Quiz: Part I

1. Two opposite rays.

Possible answer: D

3. The intersection of plane N and plane T.

4. A plane containing E, D, and B.

Plane T

Lesson Quiz: Part II

Draw each of the following.

5. a line intersecting a plane at one point

6. a ray with endpoint P that passes through Q