Download Geometry –3-1

Geometry – 3-1
Name
Date
Key Terms
a
Parallel lines Coplanar lines that do not intersect
We write a║b or put extra arrowheads on parallel lines
b
Parallel planes Planes that do not intersect
Plane ABC║Plane EFG
A
B
C
D
G
Skew lines Two non coplanar lines AC and DF
Transversal
E
F
A line that intersect two other coplanar lines at two distinct points
Below -- Line l is a transversal with respect to lines m and n
Angles formed when a transversal cuts two lines
(angles are position specific)
Exterior angles 1 , 2 , 7 , 8
(the outside angles)
l
m
Interior angles 3 , 4 , 5 , 6
(the inside angles)
P
a
i
r
s
o
f
a
n
g
l
e
s
1
3
Consecutive interior angles 4 and 6
3 and 5
Consecutive exterior angles 1 and 7
2 and 8
2
n
4
5
7
6
8
Alternate interior angles
4 and 6
3 and 6
Alternate exterior angles
1 and 8
2 and 7
Corresponding angles
(in the same spot)
1 and 5 ; 2 and 6 ; 3 and 7 ; 4 and 8
Geometry – 3-2
Name
Date
Key Terms
p
If two parallel lines are cut by a transversal, then:
1
If m║n, then
a.
3
2
m
4
Corresponding angles are congruent
1  5 and 2  6
3  7 and 4  8
5
7
b
Alternate interior angles are congruent
c.
Alternate exterior angles are congruent
d.
Consecutive interior angles are supplementary
4 and 6 are supplementary
2 and 8 are supplementary
6
n
8
4  5 and 3  6
1  8 and 2  7
Theorem 3.4 If a transversal is perpendicular to one of two parallel lines,
then it is perpendicular to the other parallel line.
a
If a║b and c  a, then c  b
b
Summary – If we have a transversal cutting two parallel lines, we know:
1.
Corresponding angles are congruent
2.
Alternate interior angles are congruent
3.
Alternate exterior angles are congruent
4.
Consecutive interior angles are supplementary
5.
If the transversal is perpendicular to one of the parallel lines,
then it is perpendicular to the other
c
Geometry – 3-5
Name
Date
Key Terms
Converse of a postulate or a theorem
Exchanging the hypothesis (IF) and conclusion (THEN) of a conditional statement.
Postulate 3.5 If given a line and a point not on the line,
A
●
then there is exactly one line through the given point
parallel to the given line.
p
Given point A not on line m
There is only one line p through A that is parallel to m
m
p
Two lines are cut by a transversal are parallel, if:
1
3
a.
Corresponding angles are congruent
If 1  5 or 2  6
3  7 or 4  8
5
7
Then
b
c.
d.
e.
n
m║n
Alternate interior angles are congruent
If 4  5 or 3  6
Then
m║n
Alternate exterior angles are congruent
If 1  8 or 2  7
Then
m║n
both lines are perpendicular to the transversal
If
a  b and a  c , then b║c
a
b
c
m
6
8
Then
m║n
Consecutive interior angles are supplementary
If 4 and 6 are supplementary
2 and 8 are supplementary
2
4
Geometry – 3-3
Name
Date
Key Terms
Slope =
rise change in y y2  y1 y



m
run change in x x2  x1 x
 - Greek symbol “delta” which means “change”
Rate of change describes how a quantity changes over time
No slope
Positive slope
Negative slope
0 slope
What is the slope of the line with the points?
A (2, 1) and B (0, -3)
m
1  (3) 4
 2
20
2
Postulate 3.2 Two non-vertical lines are parallel, iff they have the same slope.
Points C (0, 2) and D (-1, 0) Slope of CD =
20
2
  2 , therefore AB ║ CD
0  (1) 1
Postulate 3.3 Two non vertical lines are perpendicular iff the product of their slopes is -1.
(or if the slopes are negative reciprocals – change sign and flip fraction)
Slope of AC 
2 1
1
1


02 2
2
AC  CD and AC  AB
Slope of BD 
0  (3) 3

 3
1  0 1
BD is not perpendicular or parallel to any of these segments
Geometry – 3-4
Name
Dated
Key Terms
Slope-intercept form An equation in the form y  mx  b ,
where m is the slope and b is the y-intercept
y  2x  3
1
y   x4
2
y4
slope is 2 and y-intercept is +3
1
2
slope is  and y-intercept is −4
slope is 0 and y-intercept is +4 (horizontal line through +4)
Given the slope is -3 and the y-intercept is 5, the equation of the line is y  3x  5
Point-slope form
An equation in the form y  y1  m( x  x1 )
where m is the slope and ( x1 , y1 ) is any point on the line
Write the equation of the line that has a slope of
2
and passes through (9, 1)
3
2
( x  9)
3
2
y 1  x  6
3
2
y  x5
3
y 1 
Given two points find the equation of a line
(1, 4) and (-2, -2)
first find the slope m 
4  (2) 6
 2
1  (2) 3
Now with the slope (of 2) and either point, use the point slope formula
Using (1, 4)
Using (-2, -2)
y  4  2( x  1)
y  (2)  2( x  (2))
y  4  2x  2
y  2  2x  4
y  2x  2
y  2x  2
Geometry – 3-6
Name
Date
Key Terms
Equidistant The same or equal distance
Distance between a point and a line
A
The shortest distance between a point and
a line is the length of the segment perpendicular
from the point to the line.
m
B
If AB  m , then AB is the distance from A to m
Distance between two parallel lines
The length of any segment perpendicular to both lines
d
d
Two coplanar lines are parallel if they are everywhere equidistant.
The distance (d) between the parallel lines is same everywhere
Theorem 3.9 In a plane, if two lines are equidistant
a
from a third line, the lines are parallel.
If the distance between a and b is the same as the
distance between c and b,
then a║c
b
c