Chapter 1

Chapter 1

Section 1.1

Identify Points, Lines, and Planes

Warm Up Problems

Solve the inequality. Graph the solution.

126 x

3154 x

To define it or not to define it

• Definition: Using known words to define a new word

• Undefined terms: When a term is not given a definition– Points, Lines, and Planes are all undefined

terms– No formal definitions, but there is agreement

about what they mean

Undefined terms

• Point– Something with no dimension

• Line– Something that extends forever in one

dimension

• Plane– Something that extends forever in two

dimensions

•

Points• Points do not have actual size.

• How to Sketch:

Using dots

• How to label:

Use capital letters

Never name two points with the same letter (in the same sketch).

A

B AC

Lines

• Lines extend indefinitely and have no thickness or width.• How to sketch : using arrows at both ends.

• How to name: 2 ways(1) small script letter – line n(2) any two points on the line -

• Never name a line using three points - , , , , ,AB BC AC BA CA CB

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nA

BC

ABC�������������� �

Planes

• A plane is a flat surface that extends indefinitely in all directions.

• How to sketch: Use a parallelogram (four sided figure)

• How to name: 2 ways

(1) Capital script letter – Plane M(2) Any 3 non collinear points in the plane - Plane: ABC/ ACB / BAC /

BCA / CAB / CBA

A

BC

Horizontal Plane

M

Vertical Plane Other

SOLUTION

EXAMPLE 1 Name points, lines, and planes

b. Name three points that are collinear.Name four points that are coplanar.

a. Other names for PQ are QP and line n. Other names for plane R are plane SVT and plane PTV.

a. Give two other names for PQ and for plane R.

EXAMPLE 1 Name points, lines, and planes

b. Points S, P, and T lie on the same line, so they are collinear. Points S, P, T,and V lie in the same plane, so they are coplanar.

GUIDED PRACTICE for Example 1

ANSWER TS, PT; point V

1. Use the diagram in Example 1. Give two other names for ST . Name a point that is not coplanar with points Q, S, and T.

Defined Terms

• Collinear Points– Points that lie on the same line

• Coplanar Points– Points that lie in the same plane

• Intersect– Two or more geometric figures are said to intersect if

they have one or more points in common

• Intersection– Set of points the figures have in common

Defined Terms

• Line Segment– A piece of a line cut off by two end points

• Ray– All the points on one side of a given point

called the endpoint or initial pointEnd

Point

Symbolization

AB

••A BLines

Use two points on the line and draw a

line above them

Rays

A

•B

ABUse the end point

and any other point on the ray then

draw a ray on top

Opposite Rays: Two rays that have the same end point but extend in opposite directions

EXAMPLE 2 Name segments, rays, and opposite rays

b. Name all rays with endpoint J . Which of these rays are opposite rays?

SOLUTION

a. Another name for GH is HG .

b. The rays with endpoint J are JE , JG , JF , and JH . The pairs of opposite rays with endpoint J are JE and JF , and JG and JH .

a. Give another name for GH .

GUIDED PRACTICE for Example 2

3. Are HJ and JH the same ray ? Are HJ and HG the same ray? Explain.

ANSWER FE

ANSWER

No; the rays have different endpoints.Yes; points J and G lie on the same side of H.

2. Give another name for EF

Intersection of Figures

The intersection of two figures is the set of points that are common in both figures.

The intersection of two lines is a point.

m

n

P

Continued…….

Line m and line n intersect at point P.

3 Possibilities of Intersection of a Line and a Plane

(1) Line passes through plane – intersection is a point.

(2) Line lies on the plane - intersection is a line.

(3) Line is parallel to the plane - no common points.

Intersection of Two Planes is a Line.

P

R

A

B

Plane P and Plane R intersect at the line AB�������������� �

EXAMPLE 3 Sketch intersections of lines and planes

a. Sketch a plane and a line that is in the plane.

b. Sketch a plane and a line that does not intersect the plane.

c. Sketch a plane and a line that intersects the plane at a point.

SOLUTION

a. b. c.

EXAMPLE 4 Sketch intersections of planes

Sketch two planes that intersect in a line.

SOLUTION

Draw: a second plane that is horizontal. Shade this plane a different color. Use dashed lines to show where one plane is hidden.

STEP 2

Draw: the line of intersection.STEP 3

Draw: a vertical plane. Shade the plane.

STEP 1

GUIDED PRACTICE for Examples 3 and 4

4. Sketch two different lines that intersect a plane at the same point.

Use the diagram at the right.

5. Name the intersection of PQ and line k.

ANSWER Point M

ANSWER

GUIDED PRACTICE for Examples 3 and 4

6. Name the intersection of plane A and plane B.

7. Name the intersection of line k and plane A.

ANSWER Line k

ANSWER Line k

HW #1 p. 876: 1-6

p. 5-8: 1-16, 17-27 odd, 40-44, 47, 50,

53, 56