Lesson 1-5 Postulates and Theorems Relating Lesson 1-5 Postulates and Theorems Relating Points, Lines, and Planes (page 22) Essential Question How are the relationships of geometric figures used in real life situations?

__________________ and _____________________. The phrases, “exactly one” and “one and only one” ⇒ __________________ and _____________________. existence uniqueness

2 3 4 Postulate 5 A line contains at least _____ points; A plane contains at least _____ points not all in one line; Space contains at least _____ points not all in one plane. 3 4

Postulate 6 Through any two points there is exactly _____ line. 1

1 1 Postulate 7 Through any three points there is at least _____ plane ; Through any three noncollinear points there is exactly _____ plane. 1 1

2 Postulate 8 If _____ points are in a plane, then the line that contains the points is in that plane.

line Postulate 9 If two planes intersect, then their intersection is a ________. line

THEOREMS: statements that can be ___________. proved NOTE: Writing proofs will be covered in Chapter 2.

Theorem 1-1 If two lines intersect, then they intersect in exactly 1 point. NO PROOF - see classroom exercise #1

Theorem 1-2 Through a line and a point not in the line there is exactly 1 plane. NO PROOF - see written exercise #20

Theorem 1-3 If two lines intersect, then exactly 1 plane contains the lines. NO PROOF - see postulate 5 & theorem 1-1

Relationships between points … must Two points _________ be collinear. Three points _________ be collinear or noncollinear. Three points _________ be coplanar. Three noncollinear points determine a _________. may must plane

Relationships between points … may Four points _________ be coplanar or noncoplanar. Four noncoplanar points determine __________. Space contains at least ____ noncoplanar points. space 4

Three ways to determine a plane … (1) 3 noncollinear points determine a plane. Example:

Three ways to determine a plane … A line and a point not on the line determine a plane. Example:

Three ways to determine a plane … (3) 2 intersecting lines determine a plane. Example:

Relationships between two lines in the same plane … intersect Two lines are either parallel or they _____________ in exactly one point. Examples:

Relationships between a line and a plane … A line and a plane are either parallel, or Examples:

Relationships between a line and a plane … A line and a plane are either parallel, or they _____________ in exactly one point, or Examples: intersect

Relationships between a line and a plane … A line and a plane are either parallel, or they _______________ in exactly one point, or the plane _______________ the line. Examples: intersect contains

Relationships between two planes … intersect Two planes are either parallel or they ______________ in a line. Examples:

What is the number of points of intersection for 3 lines? Draw the relationships to determine your answer. NO points 1 point 2 points 3 points

Written Exercises on pages 25 & 26 RECOMMENDED: 1 to 11 odd numbers Assignment Written Exercises on pages 25 & 26 RECOMMENDED: 1 to 11 odd numbers REQUIRED: 13, 14, 15, 16, 19 How are the relationships of geometric figures used in real life situations?

Prepare for a test on Chapter 1 Points, Lines, Planes, and Angles Assignment Worksheet on Lesson 1-5 Prepare for a test on Chapter 1 Points, Lines, Planes, and Angles How are the relationships of geometric figures used in real life situations?