Chapter 1

Understanding Points, Lines, and Planes

Geometry

The branch of mathematics that studies the shapes of objects and their sizes, properties and relationships, and spatial reasoning.

Geo - “earth” metry - “measure”– Geometry literally means earth measure

We will study the logical structures of geometry and its relationships to algebra, probability and triogonometry.

Terms MUST KNOW!

*Point – names a location

Has no size.

Use capital letters EX: P, Point P, Pt P, Not: •P (u do not need the • to name a pt)

• P

• Q

*Line – straight path that never ends **

Symbol

to name: AB

BA or line l A B l

*Plane – flat surface that extends forever

To name use the Script capital letter or 3 points not on the same line.

EX: P or ABCP

AB

C

Terms Continued….

Segment – part of a line consisting of 2 pts and all pts in between **

Symbol

CD or DCnot CD (must have segment above letters)

Endpoint – pt at one end of a segment or the starting pt of a ray C or D

look at the above pictures points C and D are endpts

Ray – part of a line that starts at an endpt & extends forever in 1 direction **

Symbol

AB or CDnot BA (the endpt must be the first pt listed)

C D

A B

CD

Terms Continued…

Opposite Ray – 2 rays with a common endpt that form a line

BA & BC

Collinear Pts on the same line

3 non-collinear pts determine a plane

Coplanar Pts on the same plane

If 2 pts are in a plane, then the line containing them is in the plane

A B C

* The most basic figures, points, lines, and planes are “undefined terms” in geometry. This means they cannot be defined using other figures.

** 2 letters without a symbol refers to distance.

you must use the correct symbol to represent either a line, segment, or ray!!

ONLY HONORS NEEDS TO WRITE THIS SLIDE!

The Geometry we are familiar with and the kind we will study is called Euclidean geometry, named after the Greek mathematician, Euclid.

Other geometries, called non-Euclidean– Spherical Geometry – geometry of the sphere. More

suited to our earth. Planes are the surfaces of a sphere and all lines are circles on the sphere

– Hyperbolic Geometry – geometry on a circular plane– Coordinate Geometry – (analytic geometry) uses the

coordinate system (x,y) to study properties of lines, segments, etc.

Examples

1. Name the two planes.2. What is the intersection of the two planes?3. Which plane contains points G, E and D? 4. Which plane contains point H?5. What points are not in both planes?6. Name a point not in plane N.7. Point F is the intersection of _ ____ and ______.

M & N

AB

MN

G, E, D, C, H, WG

AB CH

Examples

True or False8. B, E, and D are coplanar9. G, A, and F are collinear10. H and E determine a line11. C & H determine plane N12. ray AF and ray FB are opposite rays13. Line AB and Plane M have exactly 3 points of intersection.

True

False

True

False

False

False

Optical Illusions

Your eyes see images and send them to your brain to interpret, but sometimes the brain can be tricked and your spatial reasoning is tested.

Optical Illusion Examples

Are the petals moving?

you will probably find it hard to believe that squares A and B are actually the same shade of gray

Honors Only!!

Slope formula y2-y1 = m

x2-x1

y = mx + b y – y1 = m(x – x1) Parallel lines have SAME slope DIFF y- intercepts Perpendicular lines’ slopes are opposite reciprocals

example : 3 and -1/3 Oblique lines have DIFFERENT slopes, not opposite

reciprocals