6.1 Relating Lines to Planes Objective: After studying this section, you will be able to understand basic concepts relating to planes, identify four methods of determining a plane, and apply two postulates concerning lines and planes.

m C Plane surface Not a plane surface Preliminary concepts to understand B A B

Remember If points, lines, segments, and so forth, lie in the same plane, we call them coplanar. Points, lines, segments, and so forth that do not lie in the same plane are called noncoplanar. P B T m R S A V A, B, S, T, and V are coplanar points AB and ST are coplanar lines AB and ST are coplanar segments A, B, S, T, and R are noncoplanar points AB, ST, and RP are noncoplanar lines AB, ST, and RP are noncoplanar segments

P B T m R S A V Definition The point of intersection of a line and a plane is called the foot of the line. Point V is the foot of the line RP in plane m.

Four ways to determine a plane PostulateThree noncollinear points determine a plane A B C k

TheoremA line and a point not on the line determine a plane P

TheoremTwo intersecting lines determine a plane

TheoremTwo parallel lines determine a plane

Two postulates concerning lines and planes PostulateIf a line intersects a plane not containing it, then the intersection is exactly one point. A B k

PostulateIf two planes intersect, their intersection is exactly one line. k A B m

Example 1 m A B n P S W V R m n = U A, B, and V determine plane _____ Name the foot of RS in m AB and RS determine plane ____ AB and point ____ determine plane n Does W lie in plane n? Line AB and line ____ determine plane m A, B, V, and ____ are coplanar points A, B, V, and ____ are noncoplanar points If R and S lie in plane n, what can be said about RS?

Given: A, B, and C lie in plane m Prove m A B C P

Summary In your own words describe how you can determine a plane. Homework: worksheet