1.2 Points, Lines and 1.2 Points, Lines and PlanesPlanesGeometryGeometry

Objectives/Assignment:Objectives/Assignment: Understand and use the basic Understand and use the basic

undefined terms and defined terms undefined terms and defined terms of geometry.of geometry.

Sketch the intersections of lines and Sketch the intersections of lines and planes.planes.

Using Undefined terms and Using Undefined terms and definitiondefinitionA definition uses known words to

describe a new word. In geometry, some words such as point, line and plane are undefined terms or not formally defined.

Using Undefined terms and Using Undefined terms and definitiondefinition

• A point has no dimension. It is usually represented by a small dot.

A

Point A

Using Undefined terms and Using Undefined terms and definitiondefinition• A line extends in one

dimension. It is usually represented by a straight line with two arrowheads to indicate that the line extends without end in two directions. In this book, lines are always straight lines.

A

B

l

Line l or AB

Using Undefined terms and Using Undefined terms and definitiondefinitionA plane extends in two

dimensions. It is usually represented by a shape that looks like a tabletop or wall. You must imagine that the plane extends without end even though the drawing of a plane appears to have edges.

A

BC

M

Plane M or plane ABC

A few basic concepts . . . A few basic concepts . . . Must be commonly understood Must be commonly understood

without being defined. One such without being defined. One such concept is the idea that a point lies concept is the idea that a point lies on a line or a plane.on a line or a plane.

Collinear pointsCollinear points are points that lie on are points that lie on the same line.the same line.

Coplanar pointsCoplanar points are points that lie on are points that lie on the same plane.the same plane.

Ex. 1: Naming Collinear and Ex. 1: Naming Collinear and Coplanar PointsCoplanar Pointsa.a. Name three Name three

points that are points that are collinearcollinear

Solution: Solution: D, E and F lie on the D, E and F lie on the

same line, so same line, so they are collinear.they are collinear.

G

D E F

H

Ex. 1: Naming Collinear and Ex. 1: Naming Collinear and Coplanar PointsCoplanar Pointsb.b. Name four points Name four points

that are coplanar.that are coplanar.

Solution: Solution: D, E, F, and G lie on the D, E, F, and G lie on the

same plane, so they same plane, so they are coplanar. Also are coplanar. Also D, E, F, and H are D, E, F, and H are coplanar; although, coplanar; although, the plane containing the plane containing them is not drawn.them is not drawn.

G

D E F

H

Ex. 1: Naming Collinear and Ex. 1: Naming Collinear and Coplanar PointsCoplanar Pointsc.c. Name three points Name three points

that are not that are not collinear.collinear.

Solution: Solution: There are many There are many

correct answers. correct answers. For instance, For instance, points H, E, and G points H, E, and G do not lie on the do not lie on the same line.same line.

G

D E F

H

More . . . More . . . Another undefined Another undefined

concept in geometry is concept in geometry is the idea that a point the idea that a point on a line is between on a line is between two other points on two other points on the line. You can use the line. You can use this idea to define this idea to define other important terms other important terms in geometry.in geometry.

Consider the line AB Consider the line AB (symbolized by AB). (symbolized by AB).

l

Line l or AB

More . . . More . . . The The line segmentline segment

or segment AB or segment AB (symbolized by AB) (symbolized by AB) consists of the consists of the endpoints A and B, endpoints A and B, and all points on and all points on AB that are AB that are between A and B.between A and B.

l

Line l or ABA

A

B

BSegment AB

More . . . More . . . The ray AB The ray AB

(symbolized by AB) (symbolized by AB) consists of the consists of the initial point A and initial point A and all points on AB all points on AB that lie on the that lie on the same side of A as same side of A as point B.point B.

l

Line l or ABA

A

B

B

Ray AB

More . . . More . . . Note that AB is the Note that AB is the

same as BA and AB same as BA and AB is the same as BA. is the same as BA. However, AB and However, AB and BA are not the BA are not the same. They have same. They have different initial different initial points and extend points and extend in different in different directions.directions.

l

Line l or ABA

A

B

B

Ray BA

More . . . More . . . If C is between A and If C is between A and

B, then CA and CB are B, then CA and CB are opposite rays.opposite rays.

Like points, segments Like points, segments and rays are collinear and rays are collinear if they lie on the same if they lie on the same line. So, any two line. So, any two opposite rays are opposite rays are collinear. Segments, collinear. Segments, rays and lines are rays and lines are coplanar if they lie on coplanar if they lie on the same plane.the same plane.

l

Line l or ABA

C

B

Ex. 2: Drawing lines, segments Ex. 2: Drawing lines, segments and raysand rays Draw three noncollinear points J, K, Draw three noncollinear points J, K,

and L. Then draw JK, KL and LJ.and L. Then draw JK, KL and LJ.

J

K

L

Draw J, K and LThen draw JK

Ex. 2: Drawing lines, segments Ex. 2: Drawing lines, segments and raysand rays Draw three noncollinear points J, K, Draw three noncollinear points J, K,

and L. Then draw JK, KL and LJ.and L. Then draw JK, KL and LJ.

J

K

L

Draw KL

Ex. 2: Drawing lines, segments Ex. 2: Drawing lines, segments and raysand rays Draw three noncollinear points J, K, Draw three noncollinear points J, K,

and L. Then draw JK, KL and LJ.and L. Then draw JK, KL and LJ.

J

K

L

Draw LJ

Ex. 3: Drawing Opposite Ex. 3: Drawing Opposite RaysRays Draw two lines. Label Draw two lines. Label

points on the lines and points on the lines and name two pairs of name two pairs of opposite rays.opposite rays.

Solution: Points M, N, Solution: Points M, N, and X are collinear and X are collinear and X is between M and X is between M and N. So XM and XN and N. So XM and XN are opposite rays.are opposite rays.

P

MQ

N

X

Ex. 3: Drawing Opposite Ex. 3: Drawing Opposite RaysRays Draw two lines. Label Draw two lines. Label

points on the lines and points on the lines and name two pairs of name two pairs of opposite rays.opposite rays.

Solution: Points P, Q, Solution: Points P, Q, and X are collinear and X are collinear and X is between P and X is between P and Q. So XP and XQ and Q. So XP and XQ are opposite rays.are opposite rays.

P

MQ

N

X

Goal 2: Sketching Goal 2: Sketching Intersections of Lines and Intersections of Lines and PlanesPlanes Two or more geometric intersect if Two or more geometric intersect if

they have one or more points in they have one or more points in common. The intersection of the common. The intersection of the figures is the set of points the figures figures is the set of points the figures have in common.have in common.

Ex. 4: Sketching Ex. 4: Sketching intersectionsintersections Sketch the figure Sketch the figure

described.described.a.a. A line that A line that

intersects a plane in intersects a plane in one pointone point

Draw a plane and a Draw a plane and a line.line.

Emphasize the point Emphasize the point where they meet. where they meet.

Dashes indicate Dashes indicate where the line is where the line is hidden by the planehidden by the plane

Ex. 4: Sketching Ex. 4: Sketching intersectionsintersections Sketch the figure Sketch the figure

described.described.b.b. Two planes that Two planes that

intersect in a lineintersect in a line Draw two planes.Draw two planes. Emphasize the line Emphasize the line

where they meet.where they meet. Dashes indicate Dashes indicate

where one plane is where one plane is hidden by the other hidden by the other plane.plane.