BUILDING BLOCKS OF GEOMETRY THE BUILDING BLOCKS WHAT
BUILDING BLOCKS OF GEOMETRY
THE BUILDING BLOCKS • WHAT DO YOU THINK IT MEANS TO BE A “BUILDING BLOCK OF GEOMETRY? WHAT MIGHT ONE BE? • POINT • LINE • PLANE • THESE 3 OBJECTS ARE USED TO MAKE ALL OF THE OTHER OBJECTS THAT WE WILL USE IN GEOMETRY
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: = “POINT P” • WHAT WOULD A REAL WORLD EXAMPLE BE?
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: • REAL WORLD EXAMPLE? = “PLANE P”
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. ”
Where are you?
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 • 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 CONGRUENT TO. ” • EXAMPLE: AND YOU SAY IT AS “IS
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.
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
THINGS YOU MAY ASSUME 1) 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 ASSUME 1) YOU MAY NOT ASSUME THAT JUST BECAUSE 2 LINES, SEGMENTS, OR RAYS LOOK PARALLEL THAT THEY ARE PARALLEL – THEY MUST BE MARKED PARALLEL 1) YOU MAY NOT ASSUME THAT 2 LINES ARE PERPENDICULAR JUST BECAUSE THEY LOOK PERPENDICULAR – THEY MUST BE MARKED PERPENDICULAR 1) PAIRS OF ANGLES, SEGMENTS, OR POLYGONS ARE NOT NECESSARILY CONGRUENT, UNLESS THEY ARE MARKED WITH INFORMATION THAT TELLS YOU THAT THEY ARE CONGRUENT.
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