GEOMETRY Means earth measurement. Early peoples used their knowledge of geometry to build roads, temples, pyramids, and irrigation systems. Means earth measurement. Early peoples used their knowledge of geometry to build roads, temples, pyramids, and irrigation systems.

GEOMETRY The more formal study of geometry today is based on an interest in logical reasoning and relationships rather than in measurement alone. The more formal study of geometry today is based on an interest in logical reasoning and relationships rather than in measurement alone.

GEOMETRY Euclid (300 b.c.) organized Greek geometry into a thirteen-volume set of books named The Elements, in which the geometric relationships were derived through deductive reasoning. Thus, the formal geometry studied today is often called Euclidean geometry. This geometry is also called plane geometry because the relationships deal with flat surfaces. Euclid (300 b.c.) organized Greek geometry into a thirteen-volume set of books named The Elements, in which the geometric relationships were derived through deductive reasoning. Thus, the formal geometry studied today is often called Euclidean geometry. This geometry is also called plane geometry because the relationships deal with flat surfaces.

GEOMETRY Geometry has undefined terms, defined terms, postulates (assumptions that have not been proven, but have “worked” for thousands of years), and theorems (relationships that have been mathematically and logically proven). Geometry has undefined terms, defined terms, postulates (assumptions that have not been proven, but have “worked” for thousands of years), and theorems (relationships that have been mathematically and logically proven).

GEOMETRY This presentation will deal with undefined terms and defined terms in geometry. This presentation will deal with undefined terms and defined terms in geometry.

Undefined Terms Provide the starting place for geometry Provide the starting place for geometry The simplest concepts in geometry that all other concepts are based upon. The simplest concepts in geometry that all other concepts are based upon. Words that do not have formal definitions, but their meanings are accepted by all. Words that do not have formal definitions, but their meanings are accepted by all.

GEOMETRY’S UNDEFINED TERMS POINTLINEPLANE

POINT A point is a location, or a place.

POINT A point is not a physical object – you can not touch it, feel it, or even see it.

POINT A point has no size.

POINT A point can be represented by a dot. The size that you make the dot is not important because a point has no size. Make it just large enough to see.

POINT A point is named by using an “upper-case” “block” letter. EXAMPLE: P This figure would be called “point P”.

POINT EXAMPLE 2: X This figure would be called “point X”.

LINE A line is a set of points that extends infinitely in a straight path in 2 opposite directions. What does “infinitely” mean ? NEVER-ENDING (Goes on forever.)

LINE EXAMPLE : Why does it have arrows on both ends ? TO SHOW THAT IT EXTENDS INFINITELY IN BOTH (OPPOSITE) DIRECTIONS

NAMING A LINE There are 2 ways: 1. Use the names of (any) two points on the line. Example: P X What name(s) can we give to this line? Line PX or Line XP PX or XP

NAMING A LINE There are 2 ways: 2. Use a “lower-case” “cursive” letter (located near one of the arrows). Example: What name can we give to this line? Line m m

NAMING A LINE There are 2 ways: 2. Use a “lower-case” “cursive” letter (located near one of the arrows). 1. Use the names of (any) two points on the line. EXAMPLE : line PX (OR USE A “SYMBOL” FOR LINE AND WRITE IT ABOVE THE LETTERS.) PX EXAMPLE: line m or m

How many names does this line have? R G K h There are 7 different names that you could use to name this line. They are: RG, GR, RK, KR, GK, KG, h HOW MANY DID YOU GET ???

Let’s try again. How many names does this line have? F N X t There are 7 different names that you could use to name this line. They are: FN, NF, FX, XF, NX, XN, t DID YOU DO BETTER THIS TIME ?

ONE MORE TIME. How many names does this line have? T Y N There are 12 different names that you could use to name this line. They are: TC, CT, TY, YT, TN, NT, CY, YC, CN, NC, YN, NY C GOT IT ?????

PLANE A plane is a “flat surface” – a set of points that extends infinitely in 2 dimensions. What does “dimension” mean ? Length, Width, Height. A plane is infinitely long and infinitely wide, but it has no height (or depth). No thickness.

PLANE A plane can be drawn (represented) by a four sided figure known as a parallelogram.

PLANE A plane can be named by a single “upper-case” “block” letter (written near a corner of the plane). Q What is the name of this plane? Plane Q or □Q

PLANE A plane can also be named using three points on the plane (three capital letters). What is the name of this plane? Plane TRN or Plane RNT or Plane NTR. R N T

What is the name of this plane? D B C F ANSWER: PLANE F Or PLANE BCD

COLLINEAR POINTS Points that lie on the same line Points that lie on the same line Points A, B, C, D are collinear. Points A, B, C, D are collinear. Points X, Y, Z are non-collinear (not on the same line) Points X, Y, Z are non-collinear (not on the same line)

COPLANAR POINTS Points that lie on the same plane Points that lie on the same plane Points A, B, C are coplanar Points A, B, C are coplanar

COPLANAR POINTS Points P, Q, R are coplanar Points P, Q, R are coplanar Points P, Q, R, S are non- coplanar (not on the same plane Points P, Q, R, S are non- coplanar (not on the same plane

SOME OF GEOMETRY’S DEFINED TERMS LINE SEGMENT RAY OPPOSITE RAYS INTERSECTION

LINE SEGMENT A part of a line that consists of two endpoints and all points between them. A part of a line that consists of two endpoints and all points between them. Name a segment by Name a segment by its endpoints. its endpoints. AB or BA AB or BA Notice the symbol above the letters have no arrows.

RAY A part of a line that consists of one endpoint and all the points of the line on one side of the endpoint A part of a line that consists of one endpoint and all the points of the line on one side of the endpoint Name a ray by its Name a ray by its endpoint and another endpoint and another point on the ray. point on the ray. AB (not BA) AB (not BA) Notice the symbol above the letters has an arrow on only one end pointing away from the endpoint.

OPPOSITE RAYS Two rays that share the same endpoint and form a line. Two rays that share the same endpoint and form a line. Point B is the Point B is the common endpoint. common endpoint. BA and BC are BA and BC are opposite rays. opposite rays. Opposite rays form a straight line or straight angle. A straight angle = 180 ⁰. Opposite rays form a straight line or straight angle. A straight angle = 180 ⁰.

INTERSECTION Geometric figures intersect if they have one or more points in common Geometric figures intersect if they have one or more points in common An intersection can be either a point or a line An intersection can be either a point or a line

INTERSECTION Two lines intersect at a single point. Two lines intersect at a single point. Line LM and Line GH intersect at Point D. Line LM and Line GH intersect at Point D.

INTERSECTION A line and a plane intersect at a single point. A line and a plane intersect at a single point. Line b intersects Plane M at Point Y. Line b intersects Plane M at Point Y.

INTERSECTION Two planes intersect at a line. Two planes intersect at a line. Plane m intersects Plane n at Line l. Plane m intersects Plane n at Line l.