Section 1-2 (cont): Points, Lines, and Planes SPI 32A: Identify properties of plane figures Objectives: Understand basic terms of geometry Understand basic postulates of geometry Vocabulary Postulates or Axioms: An accepted statement of fact Starting point to prove theorems using deductive reasoning Inductive versus Deductive Reasoning Inductive: examine examples, observe a pattern, and assume pattern will never end Deductive: uses accepted facts (postulates, etc) to reason in a step-by-step fashion until a conclusion is reached.

Example of using Postulate 1-1 Through any two points there is exactly one line. Postulate 1-1 B Line m is the only line that passes through points A and B A Example of using Postulate 1-1 When you graph a linear equation, such as y = 2x + 1, you plot two points and then draw a line though the two points. 1. Plot the y-intercept 2. Plot the slope 3. Draw a line through the two points

Example of using Postulate 1-2 If two lines intersect, then they intersect exactly in one point. Postulate 1-2 B E C A D and intersect at C Example of using Postulate 1-2 In algebra, one way to solve a system of two equations is to graph the two equations. The solution to the system of equations is the single point where the two lines intersect. Solve the systems of equations: y = 2x + 1 and y = -3/2 x + 3 1. Graph the first equation (slope-intercept) 2. Graph the second equation (slope-intercept) 3. They intersect at only one point.

Postulate 1-3 Postulate 1-4 If two PLANES intersect, then they intersect exactly in one LINE. Postulate 1-3 R T W S Plane RST and Plane STW intersect at Through any three noncollinear points there is exactly one plane. Definition Noncollinear: points that do not lie on the same plane Postulate 1-4 The 3 legged stand will always be stable. As long as the feet on the stand do not lie in one line, the feet of the three legs will lie exactly in one plane.

Use Postulates 1-1 through 1-4 to explain each situation. 1. A land surveyor can always find a straight line from the point where she stands to any other point. Through any two points there is exactly one line. Postulate 1-1 2. A carpenter knows that a line can represent the intersection of two flat walls. If two planes intersect, then they intersect in exactly in one line. Postulate 1-3 3. A furniture maker knows that a three-legged table is always steady, but a four legged table will sometimes wobble. Through any three noncollinear points there is exactly one plane. Postulate 1-4

Navigation of Ships Rescue teams use Postulates 1-1 and 1-2 to determine the location of a distress signal. In the diagram, a ship at Point A receives a signal from the northeast. A ship at point B receives the same signal from due west. Draw the diagram & plot the location of the distress signal. Explain how the two Postulates help locate the distress signal. By Post. 1-1 points D and B determine a line and points A and D determine a line. The distress signal is on both lines and by Post. 1-2, there can only be one distress signal. B W D NE A