2. The Building Blocks <ul><li>Point </li></ul><ul><li>Plane </li></ul><ul><li>Line </li></ul><ul><li>These 3 objects are used to make all of the other objects that we will use in Geometry </li></ul><ul><li>What do you think it means to be a “Building block of Geometry? What might one be? </li></ul>
3. Point <ul><li>The most basic building block </li></ul><ul><li>Has no size </li></ul><ul><li>Only has a Location </li></ul><ul><li>Representation </li></ul><ul><ul><li>Shown by a Dot </li></ul></ul><ul><ul><li>Named with a single Capital letter </li></ul></ul><ul><li>Ex: </li></ul>• What would a real world example be? = “Point P”
4. Line <ul><li>A straight, arrangement of infinitely many points. </li></ul><ul><li>Infinite length, but no thickness </li></ul><ul><li>Extends forever in 2 directions </li></ul><ul><li>Named by any 2 points on the line with the line symbol above the letters (order does not matter </li></ul><ul><li>Ex: </li></ul>= “Line AB” or “Line BA” <ul><li>Real World Example? </li></ul>
5. Plane <ul><li>An imaginary flat surface that is infinitely large and with zero thickness </li></ul><ul><li>Has length and width, but no thickness </li></ul><ul><li>It is like a flat surface that extends infinitely along its length and width </li></ul><ul><li>Represented by a 4 sided figure, like a tilted piece of paper </li></ul><ul><ul><li>This is really only part of a plane </li></ul></ul><ul><li>Named with a Capital Cursive letter </li></ul><ul><li>Ex: </li></ul>= “Plane P” <ul><li>Real World Example? </li></ul>
6. Explaining the Objects <ul><li>Can be difficult </li></ul><ul><li>Early Mathematicians attempted to: </li></ul><ul><li>Ancient Greeks </li></ul><ul><ul><li>“ A point is that which has no part. A line is a breathless length.” </li></ul></ul><ul><li>Ancient Chinese Philosophers </li></ul><ul><ul><li>“ The line is divided into parts, and that part which has no remaining part is a point.” </li></ul></ul>
7. What’s the Problem?
8. Definitions <ul><li>A definition is a statement that clarifies or explains the meaning of a word or phrase </li></ul><ul><li>It is impossible to define “point,” “line,” and “plane” without using words or phrases that need to be defined. </li></ul><ul><ul><li>Therefore we refer to these building blocks as “Undefined” </li></ul></ul><ul><li>Despite being undefined, these objects are the basis for all geometry </li></ul><ul><ul><li>Using the terms “point,” “line,” and “plane,” we can define all other geometry terms and geometric figures </li></ul></ul>
9. Definitions <ul><li>Collinear – Lie on the same line </li></ul><ul><ul><li>Example – Points A and B are “Collinear” </li></ul></ul>
10. Definitions <ul><li>Coplanar – Lie on the same plane </li></ul><ul><ul><li>Example – Point A, Point B, and Line CD are “Coplanar.” </li></ul></ul>
11. Definitions <ul><li>Line Segment – Two points (called endpoints) and all of the points between them that are collinear. </li></ul><ul><ul><li>In other words, a portion of a line </li></ul></ul><ul><ul><li>Represent a Line Segment by writing its endpoints with a bar over the top </li></ul></ul><ul><ul><li>Example: </li></ul></ul>
12. Definitions <ul><li>Ray – Begins at a single point and extends infinitely in one direction </li></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><li>You need 2 points to name a ray, the first is the endpoint , and the second is any other point that the ray passes through . </li></ul></ul>
13. Definitions <ul><li>Congruent – equal in size and shape </li></ul><ul><ul><li>We mark 2 congruent segments by placing the same number of slash marks on them. </li></ul></ul><ul><ul><li>The symbol for congruence is and you say it as “is congruent to.” </li></ul></ul><ul><ul><li>Example: </li></ul></ul>
14. Definitions <ul><li>Bisect – Divide into 2 congruent parts </li></ul><ul><li>Midpoint – the point on the segment that is the same distance from both endpoints. </li></ul><ul><ul><li>The midpoint bisects the segment </li></ul></ul>
15. Definitions <ul><li>Parallel Lines – 2 lines that never intersect </li></ul><ul><ul><li>We mark 2 lines as parallel by placing the same number of arrow marks on them. </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><li>To write this as a statement, we would write </li></ul></ul>
16. Definitions <ul><li>Perpendicular Lines – 2 lines that intersect at a Right Angle (90°). </li></ul><ul><ul><li>We mark 2 lines as Perpendicular by placing a small square in the corner where they cross </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><li>To write this as a statement, we would write: </li></ul></ul>
17. <ul><li>Things you may Assume </li></ul><ul><ul><li>You may assume that lines are straight, and if 2 lines intersect, they intersect at 1 point. </li></ul></ul><ul><ul><li>2) You may assume that points on a line are collinear and that all points & objects shown in a diagram are coplanar unless planes are drawn to show that they are not coplanar. </li></ul></ul>
18. <ul><li>Things you may NOT Assume </li></ul><ul><ul><li>You may not assume that just because 2 lines, segments, or rays look parallel that they are parallel – they must be marked parallel </li></ul></ul><ul><ul><li>You may not assume that 2 lines are perpendicular just because they look perpendicular – they must be marked perpendicular </li></ul></ul><ul><ul><li>Pairs of angles, segments, or polygons are not necessarily congruent, unless they are marked with information that tells you that they are congruent. </li></ul></ul>