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Review for Final
Equations of lines
General Angle Relationships
Parallel Lines and Transversals
Construction
Transformations
Proofs
Equations of Lines
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•
•
•
•
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Given two points writing the equation of a line
Slope intercept y  mx  b
Point Slope of a line y  y  m( x  x )
rise y  y
m


1. Find the slope
run x  x
2. Calculate the y-int
Substitute in the slope and one set of ordered
pairs (x,y) you should them be able to solve for b
• 3. Write the equation of a line
1
1
2
1
2
1
Parallel and Perpendicular Lines
• Parallel Lines - slopes are the same need to
calculate a new y-int
Sub in same slope, sub in different x and y
values and solve for new b
• Perpendicular lines - slopes are opposite
reciprocals, which means flip the fraction and
change the sign to the opposite of what the
original equation was
Perpendicular lines can have the same y-int,
but you need to calculate it like all the other
equations
Horizontal and Vertical Lines
• Zero slope is a slope when the rise of the line
is zero
Lines with a zero slope are horizontal
y=number
• Undefined slope is a slope when the run is
zero
Lines with an undefined slope are vertical
x=number
Examples
Given the following
x1 y1
x2 y 2
ordered pairs find the (2,5)
(4,7)
equation of the line.
y2  y1 7  (5) 12
m


 2
(2, -5) and ( -4,7)
x2  x1
42 6
y  mx  b
 5   2 * 2  b
 5  4  b
1  b
y  2 x  1
Example
Given the following equation of the line find the
following: (solve the equation for y to find the
slope)
3x  2 y  8
 2 y  3 x  8
3
y  x4
2
1. Equation of a line parallel and through (-2,2)
2. Equation of a line perpendicular and through (3,1)
Parallel
y  mx  b
3
2   (2)  b
2
2  3  b
5b
3
y  x5
2
Perpendicular
y  mx  b
2
m
3
2
1  
(3)  b
 3 
 1  2  b
1 b
2
y
xb
3
Midpoint
Know how to find the midpoint of a segment
 x1  x2 y1  y2 
,


2 
 2
Know how to work backwards to find the other
endpoint
x1  x2
 midpo int x
2
y1  y2
 midpo int y
2
Examples
Given the following endpoints of a segment find
the midpoint.
A(5,1) and B(-3,-7)
 5  (3) 1  (7) 
,


2 
 2
2 6
 ,

2 2 
(1,3)
Example
Given the one endpoint of a segment A(2,4) and
the midpoint of the segment B(-1,3) Find the
other endpoint.
2 x
 1
2
2  x  2
x  4
4 y
3
2
4 y  6
y2
General Angles
Know the relationships
Vertical Angles
Linear Pairs
Supplementary (supplement)
Complementary (complement)
When do angles add to 360
Example
Example
x  90  4 x  10  180
5 x  80  180
5 x  100
x  20
Parallel Lines and Transversal
If lines are parallel this is true
Corresponding Angles congruent
1 and 5
2 and 6
3 and 7
110 and 4
Alternate Interior Angles congruent
3 and 5
2 and 4
Alternate Exterior Angles congruent
1 and 7
110 and 6
Same Side Interior Angles –
supplementary
3 and 4
2 and 5
110
3
k
1
2
names
of lines
4
7
5
l
6
Lines are parallel
Example
a
c
b
4x+28
19.
17.
d
23.
22.
18.
How are 4x+28 and 5x+8
related, walk your way
around to prove
20.
21.
5x+8
4x+48 corresponds to 19
19 is Alternate Exterior
angle to 5x+8
Makes angles congruent
4 x  28  5 x  8
28  x  8
20  x
4(20)  28  108
Find all missing angles and explain why
you know that angle in relation to other
angles
Example
R
70
50
2
50
3
S
1
T
1  130( LinearPair )
2  70  50  180
2  60
2  3  50  180
3  70
Constructions
Know cheat sheet – what are the main
constructions and what ideas deal with what type
Altitude is perpendicular line from vertex to side
opposite
Median is a segment from vertex to midpoint of
opposite side, need to construct perpendicular
bisector to find midpoint
Perpendicular Bisector constructs a line that is
equidistant from the endpoints of the segment
Angle Bisector constructs a ray that is equidistant
from the sides of the angle
Example
Draw a triangle and construct the altitude from
one vertex and the median from the other
Median
Example
Mark 2 points on your paper.
Find a path that is equidistant from both point
no matter where on the path you are
This would be the perpendicular bisector because it is equidistant
from the two points
Transformations
Know basic ideas of transformation
Know transformation rules
Do the given transformation
Identify the transformation
Transformations
Translation – slide left, right up or down, add or
subtract to the x or y coordinate (x+num, y+num)
Reflect – mirror image over a line
over x-axis (x,-y)
over y-axis (-x,y)
over line y=x (y,x)
Rotate around the origin 180 (-x,-y)
Some of the transformations can be doubles of
other know how to combine them.
Example
• Part A: On your grid, draw the triangle J’K’L’, the image of triangle JKL
after it has been reflected over the y-axis. Be sure to label your vertices
•
• Part B: On your grid, draw the triangle J’’K’’L’’, the image of J’K’L’ after it
has been reflected over the line y=x. Be sure to label your vertices
Part
PartAB
66
6
4
44
2
22
-5
-5-5
5
55
-2
-2-2
Example
Write the rule for the given transformation.
(x,y)- ( ___, ___), show some work on how
you came to that conclusion.
Original
(5,-7)
8
6
4
2
-10
-5
5
-2
-4
New
10
Proofs
Know set up of two column proof
1st Givens
2nd Prove other parts congruent need at least 3
3rd State triangles congruent
4th CPCTC of other parts of triangle
Proofs
Know cheat sheet and key terms
Visual – Vertical angles and Reflexive sides
Given information used for reasons
Midpoint
Angle Bisector
Segment Bisector
Perpendicular
Segment is a perpendicular Bisector
Parallel sides
Example
Given: BD is a bisector of < ABC
DB is perpendicular to AC
S
Prove: BD bisects AC
B
BD Bisects <ABC
G
BD is perpendicular
to AC
G
BD=BD
<ABD=<CBD
<ADB=<CDB
A
D
C
R
Reflexive
Def Bisect
Def Perpendicular
ABD=CBD
ASA
AD=DC
CPCTC
BD bisects AC
AD=DC
Example
Are the triangles congruent, give conjecture if they are give reason if they are
not.