Points, Lines, and Planes

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(For help, go to the Skills Handbook, page 722.)

1. y = x + 5 2. y = 2x – 4 3. y = 2x y = –x + 7 y = 4x – 10 y = –x + 15

4. Copy the diagram of the four points A, B, C, and D. Draw as many different lines as you can to connect pairs of points.

GEOMETRY LESSON 1-2Points, Lines, and Planes

Solve each system of equations.

1. By substitution, x + 5 = –x + 7; adding x – 5 to both sides results in 2x = 2; dividing both sides by 2 results in x = 1. Since x = 1, y = (1) + 5 = 6. (x, y) = (1, 6)

2. By substitution, 2x – 4 = 4x – 10; adding –4x + 4 to both sides results in –2x = –6; dividing both sides by –2 results in x = 3. Since x = 3, y = 2(3) – 4 = 6 – 4 = 2. (x, y) = (3, 2)

3. By substitution, 2x = –x + 15; adding x to both sides results in 3x = 15; dividing both sides by 3 results in x = 5. Since x = 5, y = 2(5) = 10. (x, y) = (5, 10)

4. The 6 different lines are AB, AC, AD, BC, BD, and CD.

Solutions


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-A single location in space

-Indicated by an “infinitely small” dot

-Named by a capital letter

Example:

Point

GEOMETRY LESSON 1-1Patterns and Inductive Reasoning

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Definitions

Good to Know:

-A geometric figure is a set of points

-Space is defined as the set of all points

-A series of points that extend infinitely in two opposite directions

-Drawn with an arrow at each end, indicating that it extends indefintely

Example:

Line


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Definitions

A line can be named either one of two ways:

1. A line can be named by placing the line symbol ( ) over any two points that fall on the line.

Example:

Line


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Definitions

A line can be named either one of two ways:

2. A line can be named with a single, lowercase letter

Example:

Line


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Definitions

-All points that lie on the same line are said to be collinear

Example:

Collinear points


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Definitions

- Points C, A and B are collinear- Points C, A and D are NOT collinear

- Are points D, B and E collinear?- YES!

Any other set of three points do not lie on a line, so no other set of three points is collinear.

For example, X, Y, and Z and X, W, and Z form triangles and are not collinear.

In the figure below, name three points that are collinear and three points that are not collinear.

Points Y, Z, and W lie on a line, so they are collinear.


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-A flat surface with no thickness

-Comprised of an infinite number of lines

-Extends without end in the direction of all of its lines

Example:

Planes


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Definitions

A plane can be named in either of two ways:

1. A single capital letter

Example:

Planes


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Definitions

A plane can be named in either of two ways:

2. the combination of three of its non-collinear points

Example:

Planes


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Definitions

You can name a plane using any three or more points on that plane that are not collinear. Some possible names for the plane shown are the following:

plane RST

plane RSU

plane RTU

plane STU

plane RSTU

Name the plane shown in two different ways.


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Points and lines on the same plane are said to be coplanar

Coplanar


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Definitions

-line b, line c, and point K are coplanar

-line a and line b are NOT coplanar

-line a and point K are coplanar, but not on the plane shown!

As you look at the cube, the front face is on plane AEFB, the back face is on plane HGC, and the left face is on plane AED.

The back and left faces of the cube intersect at HD.

Planes HGC and AED intersect vertically at HD.

Use the diagram below. What is the intersection of plane HGC and plane AED?


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Points X, Y, and Z are the vertices of one of the four triangular faces of the pyramid. To shade the plane, shade the interior of the triangle formed by X, Y, and Z.

Shade the plane that contains X, Y, and Z.


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Through any two points, there is exactly ONE linePostulate 1-1:


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Postulates

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 three noncollinear points, there is exactly one plane.

1. Name three collinear points.

2. Name two different planes that contain points C and G.

3. Name the intersection of plane AED and plane HEG.

4. How many planes contain the points A, F, and H?

5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line.

Use the diagram at right.


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1. Name three collinear points.

2. Name two different planes that contain points C and G.

3. Name the intersection of plane AED and plane HEG.

4. How many planes contain the points A, F, and H?

5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line.

Use the diagram at right.

D, J, and H

planes BCGF and CGHD

HE

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Sample: Planes AEHD and BFGC never intersect.


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Points, Lines, and Planes

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