Opener Consider the pattern: 1, 9, 25, 49, … 1.Describe the pattern, predict next 3. 2.If any of the #’s are divided by 4, what is the remainder? 3.Write a conjecture based on 2. 4.Is there a variable expression that always represents an odd number? (ex: 2x will be a even number)

Goal 1 Using Undefined Terms and Definitions. Goal 2 Sketching Intersections of Lines and Planes Sect. 1.2 Points, Lines, and Planes

USING UNDEFINED TERMS AND DEFINTIONS Points: In geometry, points do not have any actual size, though they sometimes represent objects that do have size such as stars or cities. A point is usually named by a capital letter (point A) In the coordinate plane, a point can also be named by an ordered pair, A(1,-4).

USING UNDEFINED TERMS AND DEFINTIONS Lines: A line has no thickness but its length goes on forever in two directions. A line is a straight arrangement of points. There are an infinite number of points on a line. A line has no thickness A line goes on forever in two directions. Arrows represent the fact that the line extends in both directions forever

Labeling Lines A line is often named by a lowercase script letter. If the names of two points such as points A and B, as in the diagram, then the line can be denoted by line AB or line BA USING UNDEFINED TERMS AND DEFINTIONS

Collinear Points: points that lie on the same line Points A, B, and C are collinear Non Collinear Points - Points that do not lie on the same line. Points A, B and H are non-collinear.

USING UNDEFINED TERMS AND DEFINTIONS Any two points determine a line Remember:

Planes - another undefined term Planes are often modeled by flat surfaces such as window panes or walls. Unlike these surfaces, a plane has no thickness and extends indefinitely in all directions. You should be familiar with the coordinate plane (x-y axis) already. USING UNDEFINED TERMS AND DEFINTIONS

Planes are often modeled using four sided-figures like the one below. A plane is defined and named by three noncollinear points, thus this could be called Plane ABC. Planes can also be named by upper case script letters. This could be called plane B USING UNDEFINED TERMS AND DEFINTIONS B

COPLANAR POINTS -- Points that lie in the same plane. - Points A, B, & C are coplanar. - All points plotted in the Cartesian Plane are coplanar. NON-COPLANAR POINTS – Points not on same plane. Point D does not lie in plane ABC, thus the point D is not coplanar with A, B & C. Points B, C & D define a new Plane BCD. Points A, B & D define a new Plane ABD. Points A, C & D define a new Plane ACD. USING UNDEFINED TERMS AND DEFINTIONS

SKETCHING INTERSECTIONS OF LINES AND PLANES THE INTERSECTION OF TWO PLANES -- The intersection of two unique planes is a line. - Could two planes intersect in any other way? - If they could what would the intersection be?

Any three non-collinear points determine a plane. SKETCHING INTERSECTIONS OF LINES AND PLANES Remember:

USING UNDEFINED TERMS AND DEFINTIONS LINE SEGMENT Line Segment AB, written, consists of points A and B and all points between A and B. The measure of, written AB (without the bar over the letters), is the distance between A and B. Thus, the measure of a line segment is the same as the distance between its two endpoints.

USING UNDEFINED TERMS AND DEFINTIONS Rays: You can think of a ray as a laser beam. It begins at one point and continues in one direction. Definition: Part of a line that consists of a point, (called an initial point ), and all points on the line that extend in one direction. Note: is not the same as.

USING UNDEFINED TERMS AND DEFINTIONS OPPOSITE RAYS - Any given point on a line determines exactly two rays, called opposite rays. This point is the common endpoint of the opposite rays. In the figure below and are opposite rays, and P is the common endpoint. P is between Q and S.

SKETCHING INTERSECTIONS OF LINES AND PLANES Example 1 A. Name 4 coplanar points B. Name 4 points that are not coplanar C. Name 3 collinear points

Example 2 Draw three collinear points A, B, and C. Draw point D which is not collinear with A, B, and C. Draw and. SKETCHING INTERSECTIONS OF LINES AND PLANES

Example 3 Name two pairs of opposite rays in the figure

Example 4 SKETCHING INTERSECTIONS OF LINES AND PLANES Sketch a line and a plane that do not intersect

SKETCHING INTERSECTIONS OF LINES AND PLANES Example 5 Sketch two planes that do not intersect and a line that intersects each plane in one point.

Homework even