Transcript angle - rreidymath

OTCQ 091709
Are the integers a closed set under division?
OTCQ 091709
Are the integers a closed set under division?
No
-1/2, ¾, -9/7 …
Remember a + or - integer has a decimal of
.0000….
Aim 1-4 and 1-5 How do we
define midpoints, bisectors, rays
and angles?
NY GG 31, GG 32, GG 33, GG
35, GG42, GG 43
Objective1 SWBAT define
midpoints, bisectors, rays and
angles?
Geometry
• Point
– An exact location in
space
• Line
– An endless collection
of points along a
straight path
• Line segment
– Part of a line that has
two endpoints
.
A
Geometry
Ray
Part of a line that has
one endpoint and
extends endlessly in
the other direction
Plane
An endless, flat
surface that is named
by any three points not
on the same line.
.B
.C
.A
Lines
• A line has infinite length, zero width, and zero
height. Any two points on the line can be used to
name it. The symbol ↔ written on top of two letters
is used to denote that line. A line may also be
named by one small letter l.
Line Segment
• A line segment does not extend forever. A
line segment has two endpoints. We write
the name of a line segment with endpoints
A and B as AB .
B
A
Midpoint and Bisector of a
Segment
• MIDPOINT-A point on the line segment
that cuts the segment into two congruent
pieces.
• BISECTOR OF A SEGMENT-A line,
segment, ray, or plane that intersects the
segment at its midpoint.
Ray
• We may think of a ray as a "straight" line that
begins at a certain point and extends forever in
one direction.
• The point where the ray begins is known as its
endpoint.
• We write the name of a ray with endpoint A and
passing through a point B as
.
Angles
An angle is formed when two rays have
the same endpoint.
This endpoint is called the vertex.
The two rays that form the angle are called
sides.
Angles, Cont.
• We name an angle by using a point on
each ray and the vertex. The angle below
may be specified as angle ABC or as
angle CBA; you may also see this written
as ABC or as CBA.
Practice
Name each figure
1.
3.
2.
4.
5.
Angles
• There are four types of angles
– Right angle
– Straight angle
– Acute angle
– Obtuse angle
Right Angles
• Forms a square
corner
• Forms a 90 degree
angle.
90
degrees
– Use a corner of your
index card to make
sure you a have a right
angle
Straight Angle
• Forms a straight line
• Angle is 180 degrees
180 degrees
– Use the edge of your
index card to make
sure you have a
straight angle.
Acute Angles
• Forms an angle that
is less than a right
angle
• Angle is less than 90
degrees
Obtuse Angles
• Form an angle that is
more than a right
angle
• Angle is more than
90 degrees
Review
Name each picture
• Ray
• Parallel lines
1.
2.
• Line
3.
Review
Name each picture
• Acute angle
4.
5.
6.
• Line segment
• Point
Degrees: Measuring Angles
• We measure
the size of
an angle
using
degrees with
the
Protractor
Obtuse or Acute?
1.
3
2.
4.
Postulates
• A statement that is accepted without proof.
• Usually these have been observed to be
true but cannot be proven using a logic
argument.
Postulates Relating Points,
Lines, and Planes
• Postulate 5: A line contains at least two points; a plane
contains at least three points not all in one line; space
contains at least four points not all in one plane.
Postulates Relating Points,
Lines, and Planes
• Postulate 6: Through any two points there
is exactly one line.
Postulates Relating Points,
Lines, and Planes
• Postulate 7: Through any three points
there is at least one plane (if collinear),
and through any three non-collinear points
there is exactly one plane.
Postulates Relating Points,
Lines, and Planes
• Postulate 8: If two points are in a plane,
then the line that contains the points is in
that plane.
A
.
B
.
Postulates Relating Points,
Lines, and Planes
• Postulate 9: If two planes intersect, then
their intersection is a line.
Theorems
• Theorems are
statements that have
been proven using a
logic argument.
• Many theorems
follow directly from
the postulates.
Theorems Relating Points,
Lines, and Planes
• Theorem 1-1: If two lines intersect, then
they intersect in exactly one point.
• Theorem 1-2: Through a line and a point
not in the line there is exactly one plane.
• Theorem 1-3: If two lines intersect, the
exactly one plane contains the lines.
Points
• A point represents position only; it has
zero size (that is, zero length, zero width,
and zero height).
Types of Points
• Points that lie on the
same line are called
collinear points. If there
is no line on which all of
the points lie, then they
are non-collinear points.
Collinear>
Non-collinear>
Types of Lines
• PARALLEL LINES- two lines that are
always the same distance apart, and will
never intersect. Parallel can be
abbreviated as ||. An example of parallel
lines is on the Italian flag. Lines a and b on
the flag are parallel.
Types of Lines
• SKEW LINES - two lines that do not
intersect, and are not parallel. Skew lines are
always non-coplanar. An overpass on a
highway is an excellent example of skew
lines.
• This only occurs when you consider lines in 3
dimensional space.
Plane
• A plane may be considered as an infinite set of
points forming a connected flat surface extending
infinitely far in all directions.
• A plane has infinite length, infinite width, and zero
height (or thickness). It is usually represented in
drawings by a four-sided figure.
• A single capital letter is used to denote a plane.
The word plane is written with the capital letter so
as not to be confused with a point.
Relationships to Planes
• COPLANAR - on the same plane. Points or
objects may not be collinear, but if they lie in
the same plane they are coplanar.
NONCOPLANAR - any number of points not
lying in the same plane.