building blocks of geometry
TRANSCRIPT
Building Blocks of Geometry
The Building Blocks
• Point
• Plane
• Line
• These 3 objects are used to make all of the other objects that we will use in Geometry
• What do you think it means to be a “Building block of Geometry? What might one be?
Point
• The most basic building block• Has no size• Only has a Location• Representation– Shown by a Dot– Named with a single Capital letter
• Ex:
• What would a real world example be?
= “Point P”
Line
• A straight, arrangement of infinitely many points.
• Infinite length, but no thickness• Extends forever in 2 directions• Named by any 2 points on the line with the
line symbol above the letters (order does not matter
• Ex: = “Line AB” or “Line BA”
• Real World Example?
Plane• An imaginary flat surface that is infinitely large and with zero
thickness• Has length and width, but no thickness• It is like a flat surface that extends infinitely along its length and
width• Represented by a 4 sided figure, like a tilted piece of paper
– This is really only part of a plane• Named with a Capital Cursive letter• Ex:
P= “Plane P”
• Real World Example?
Explaining the Objects
• Can be difficult• Early Mathematicians attempted to: • Ancient Greeks• “A point is that which has no part. A line is a breathless
length.”
• Ancient Chinese Philosophers• “The line is divided into parts, and that part which has
no remaining part is a point.”
What’s the Problem?
Definitions
• A definition is a statement that clarifies or explains the meaning of a word or phrase
• It is impossible to define “point,” “line,” and “plane” without using words or phrases that need to be defined.• Therefore we refer to these building blocks as “Undefined”
• Despite being undefined, these objects are the basis for all geometry• Using the terms “point,” “line,” and “plane,” we can
define all other geometry terms and geometric figures
Definitions
• Collinear – Lie on the same line– Example – Points A and B are “Collinear”
Definitions
• Coplanar – Lie on the same plane– Example – Point A, Point B, and Line CD are
“Coplanar.”
Definitions• Line Segment – Two points (called endpoints) and all of the points
between them that are collinear.– In other words, a portion of a line– Represent a Line Segment by writing its endpoints with a bar over the
top– Example:
Definitions• Ray – Begins at a single point and extends infinitely in one direction– Example:
– 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.
Definitions• Congruent – equal in size and shape– We mark 2 congruent segments by placing the same number of slash marks on them.– The symbol for congruence is and you say it as “is congruent to.”– Example:
Definitions
• Bisect – Divide into 2 congruent parts
• Midpoint – the point on the segment that is the same distance from both endpoints.• The midpoint bisects the segment
Definitions• Parallel Lines – 2 lines that never intersect– We mark 2 lines as parallel by placing the same number of arrow marks on them.– Example:
– To write this as a statement, we would write
Definitions• Perpendicular Lines – 2 lines that intersect at a Right Angle (90°).– We mark 2 lines as Perpendicular by placing a small square in the corner where they cross– Example:
– To write this as a statement, we would write:
• Things you may Assume1) You may assume that lines are straight, and if 2
lines intersect, they intersect at 1 point.
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.
• Things you may NOT Assume1) You may not assume that just because 2 lines,
segments, or rays look parallel that they are parallel – they must be marked parallel
2) You may not assume that 2 lines are perpendicular just because they look perpendicular – they must be marked perpendicular
3) Pairs of angles, segments, or polygons are not necessarily congruent, unless they are marked with information that tells you that they are congruent.