Transcript 1.1 Points, Lines and Planes

1.1 Points, Lines and Planes
Undefined Terms
There are three undefined terms in
Geometry.
They are Points, Lines and Planes.
They are considered undefined
because they have only been
explained using examples and
descriptions.
Points
Points are simply locations.
Drawn as a “dot.”
Named by using a Capital Letter
No size or shape.
Verbally you say “Point P”
P
B
A
Line
l
A line is a collection of an infinite number
of points (named or un-named).
Points that lie on the line are called
Collinear.
Collinear Points are points that are on the
same line.
Draw a line with arrows on each end to
signify that it is infinite in both directions.
Name by either two points on the line or
lower case “script letter”
Line (Continued)
A line has only one dimension
(length).
It has no width or depth.
Postulate – There exists exactly one
line through two points.
To plot a point on a number line,
you’ll need only one number.
A
l
B
Plane
A plane is a flat surface made up of
an infinite number of points.
Points that lie on the same plane are
said to be Coplanar.
Planes are named by using a capital,
script letter or three non-collinear
points.
K
R
Plane P P
F
Plane RFK
Plane (Continued)
Although a plane looks like it is a
quadrilateral, it is in fact infinitely
long and wide.
Planes (Coordinate Plane) have two
dimensions – so you need two
numbers to plot a point. P(x,y)
Space
Space is a boundless, three
dimensional set of all points. Space
can contain, points, lines and planes.
In chapter 13 you will see that you’ll
need three numbers to plot a point in
space. P(x,y,z)
Describing What you see!
There are key terms such as:
Lies in,
Contains,
Passes through,
Intersection,
See Pg 12.
1.2 Linear Measure and Precision
Introduction
Lines are infinitely long.
There are portions of lines that are finite.
In other words, they have a length.
The portion of a line that is finite is
called a Line Segment.
A line segment or segment has two
distinct end points.
A
B
AB
Betweenness
Betweenness of points is the
relationships among three collinear
points.
We can say B is between A and C and
you should think of this picture.
C
A
B
Notice that B is between but not in exact middle.
Example
L
M
N
Find the length of LN
or LN=?
From this picture we can always write this equation:
LM + MN = LN.
So, if LM = 3 and MN = 5, we can say that LN = 8.
What if LM = 2y, MN = 21 and LN = 3y+1?
Then we can write….. 2y + 21 = 3y + 1
From this equation we can solve for y and
substitute that value to find LN.
Congruence of Segments
Segments can be Congruent if they
have the same measurement.
We have a special symbol for
congruent. It is an equal sign with a
squiggly line above it.

AC  FG
Hint: Shapes can be congruent, measurements
can only be equal. So if you’re talking about a
shape, you say congruent or not congruent!
Congruence
Congruence can not be assumed!
Don’t think, that just because it looks
like the same length, it is.
Short cut… we can use congruent
marks to show that segments are
congruent.
A
C
Q
P
Precision (H)
The precision of a measurement
depends on the smallest unit of
measure available on the measuring
tool.
The precision will always be ½ the
smallest unit of measure of the
measuring device.
Precision (H)
1
2
3
4
5
6
Here to find the length we would have to say it
is four units long b/c it is closer to 4 than 5.
The precision of this measuring device is ½ the
smallest unit of measure, 1”, or the precision is
± 1/2.
We can say the measurement is 4 ± 1/2
So the segment could be as small as 3 ½” or as
big as 4 ½” and still be called 4”.
Precision (Con’t)
1
2
3
4
5
6
Here we have the same segment but a
different, more accurate measuring device.
The units are broken down into ¼’s.
The segment is closer to 4 ¼ than 4 ½.
The precision is ½ of ¼, or 1/8th.
So the length is 4 ¼ ± 1/8th.
Smallest 4 1/8th
Largest 4 3/8th.
1.3 Distance and Midpoints
Distance
The coordinates of the two endpoints
of a line segment can be used to find
the length of the segment.
The length from A to B is the same as
it is from B to A.
Thus AB = BA (This stands for the
measurement of the segment)
Distance (length) can never be
negative.
Midpoint
Definition - The midpoint of a segment
is the point ½ way between the
endpoints of the segment.
If B is the Midpoint (MP) of AC
then, AB = BC.
The midpoint is a location, so it can be
positive or negative depending on where
it is.
A
B
C
One Dimensional
A
C
-3
-2
B
-1
0
1
2
D
3
4
If point A was at -3 and point B was at 2,
then AB=5 b/c the formula for AB = |A – B|
The MP formula is (A+B)/2 (-3+2)/2 = -1/2
What if point C was at -2 and D was at 4,
what is CD?
CD = |4 – (-2)| or | -2 – 4| = 6
MP is (4 + (-2))/2 = 1
Two Dimensional
We designate points on a plane using
ordered pair P(x,y).
We plot them on the Cartesian Coordinate
plane just as you did in Alg I.
Again, distances can not be negative
because lengths are not negative.
Midpoints can be either positive or
negative b/c it is simply a location.
d
 x2  x1 
2
 ( y2  y1 )
2
 x1  x2 y1  y2 
MP  
,

2 
 2
Distance (Shortcut)
5
(2, 5)
6
(8, 10)
Find the distance between
these two points.
d=√(8 – 2)2 + (10 – 5)2 = √36 + 25 = √61
Or use the Pythagorean theorem.
Create a right triangle.
d2 = 62 + 52 = 36 + 25 = 61
so d = √61
1.4 Angle Measure
Another Portion of a Line
We already talked about segments, now let
us talk about Rays.
A ray is a portion of a line that has only
one end point. It is infinite in the other
direction.
A ray is named by using the end point and
any other point on the ray.
Z
X
ZX
XZ
Opposite Rays
If you chose a point on a line, that
point determines exactly two rays
called Opposite Rays.
These two opposite rays form a line
and are said to be collinear rays.
C
A
B
BA
BC
Angles
Angles are created by two noncollinear rays that share a common end
point.
C
<CED or <DEC
E
D
Angles are named by using one letter
from one side, the vertex angle, and
one letter from the other side.
An angle consists of two sides which
are rays and a vertex which is a point.
Interior vs. Exterior
Exterior
C
E
D
Exterior
Interior
Exterior
Classifications of Angles
Right Angle – An angle with a
measurement of exactly 90°
m<ABC=90°
Acute Angle – An angle with a
measurement more than 0° but less
than 90° 0° < m<ABC < 90°
Obtuse Angle – An angle with a
measurement more than 90° but less
than 180° 90° < m<ABC < 180°
Congruence of Angles
Angles with the same measurement are
said to be congruent.
C
A
25°
E
25°
D
G
m<ACE = 25° and m<DCG = 25°…
since the two angles have the same
measurement we can say that they’re
congruent.
ACE  DCG
Angle Bisector
An angle bisector is a Ray that divides
an angle into two congruent angles.
A
D
P
H
If DP is an angle bisector….
Then <ADP is congruent to <PDH.
1.5 Angle Relationships
Angle Pairs
Adjacent Angles
Adjacent Angles – Are two angles that
lie in the same plane, have a common
vertex, and a common side but no
common interior points.
A
C
<ABC and <CBD are
Adjacent Angles.
They don’t have to be
equal.
D
B
Common Side?
Common Vertex? B
No Common Interior Point?
BC
Vertical Angles
Vertical Angles – Are two non-adjacent
angles formed by intersecting lines.
A
C
B
Two Intersecting
Lines?
E
D
<ABD and <CBE are non-adjacent angles formed
by intersecting lines. They are Vertical Pair.
What else?
<ABC and <DBE are also Vertical Pair.
Linear Pair
Linear Pair – Is a pair of adjacent angles
whose non-common sides are opposite
rays.
P
L
M
N
Are <LMP and PMN are Adjacent? Yes!
Are Ray’s ML and MN the NonYes!
Common Sides?
Are Ray’s ML and MN Opposite Rays? Yes!
<LMP and <PMN are Linear Pair!
Complementary Angles
Complementary Angles – Are two
angles whose measures have a sum
of 90°
Do you see the word Adjacent in the
definition? No!
25°
1
2
<1 and <2 are Comp.
65°
2
1
Supplementary Angles
Supplementary Angles – Are two
angles whose measures have a sum
of 180°
Do you see the word Adjacent in the
definition? No!
1
L
2
M
25°
P
N
<1 and <2 are Supp.
2
155°
1
Perpendicular Lines
Perpendicular Lines intersect to form
four right angles.
Perpendicular Lines intersect to form
congruent, adjacent angles.
Segments and rays can be
perpendicular to lines or to other line
segments or rays.
The right angle symbol indicates that
the lines are perpendicular.
Assumptions
Things that can be assumed.
Coplanar, Intersections, Collinear,
Adjacent, Linear Pair and Supplementary
Things that can not be assumed.
Congruence, Parallel, Perpendicular,
Equal, Not Equal, Comparison.
1.6 Polygons
Polygon
Polygon – A closed figure whose sides
are all segments and they only
intersect at the end points of the
segments.
Polygons are named by using
consecutive points at the vertices.
Example – A triangle with points of A,
B and C is named ΔABC.
Concave vs. Convex
Concave – A polygon is concave when
at least one line that contains one of
the sides passes through the interior.
Convex – A polygon is convex when
none of the lines that contains sides
passes through the interior.
Concave
Convex
Classification by Sides
Polygons are classified by the number
of sides it has.
3 – Triangle
5 – Pentagon
7 – Heptagon
9 – Nonagon
11 – Undecagon
4 – Quadrilateral
6 – Hexagon
8 – Octagon
10 – Decagon
12 – Dodecagon
Any polygon more than 12 – then “NGon. Example 24 sides is a 24-gon.
Regular Polygon
Regular Polygon – Is a polygon that is
equilateral (all sides the same
length), equiangular (all angles the
same measurement) and convex.
Examples:
Triangles – Equilateral Triangle
Quadrilateral - Square
Perimeter
Perimeter – The sum of the lengths of
all the sides of the polygon.
May have to do distance formula for
coordinate geometry problem.
See example #3.