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Review for Final
Equations of linesGeneral Angle Relationships
Parallel Lines and TransversalsConstruction
TransformationsProofs
Equations of Lines
• Given two points writing the equation of a line• Slope intercept • Point Slope of a line • 1. Find the slope • 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
bmxy
)( 11 xxmyy
12
12
xx
yy
run
risem
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 horizontaly=number
• Undefined slope is a slope when the run is zero
Lines with an undefined slope are verticalx=number
ExamplesGiven the following ordered pairs find the equation of the line.(2, -5) and ( -4,7)
12
1
45
2*25
26
12
24
)5(7
)7,4()5,2(
12
12
2211
xy
b
b
b
bmxy
xx
yym
yxyx
Example
Given the following equation of the line find the following: (solve the equation for y to find the slope)
1. Equation of a line parallel and through (-2,2)2. Equation of a line perpendicular and through (3,-1)
42
3
832
823
xy
xy
yx
Parallel
52
3
5
32
)2(2
32
xy
b
b
b
bmxy
Perpendicular
bxy
b
b
b
m
bmxy
3
2
1
21
)3(3
21
3
2
Midpoint
Know how to find the midpoint of a segment
Know how to work backwards to find the other endpoint
2,
22121 yyxx
xmidpoxx
int2
21 ymidpo
yyint
221
Examples
Given the following endpoints of a segment find the midpoint.A(5,1) and B(-3,-7)
)3,1(
2
6,
2
2
2
)7(1,
2
)3(5
Example
Given the one endpoint of a segment A(2,4) and the midpoint of the segment B(-1,3) Find the other endpoint.
4
22
12
2
x
x
x
2
64
32
4
y
y
y
General Angles
Know the relationshipsVertical AnglesLinear PairsSupplementary (supplement)Complementary (complement)When do angles add to 360
Example
Example
20
1005
180805
18010490
x
x
x
xx
Parallel Lines and TransversalIf lines are parallel this is trueCorresponding Angles - congruent1 and 5 2 and 63 and 7 110 and 4Alternate Interior Angles - congruent3 and 5 2 and 4Alternate Exterior Angles - congruent1 and 7 110 and 6Same Side Interior Angles – supplementary3 and 4 2 and 5
k
l
110
7 6
54
3 2
1
namesof lines
Lines are parallel
Example a b c d 21. 23.
22. 5x+8
20.
19.
18.
17.
4x+28
How are 4x+28 and 5x+8 related, walk your way around to prove
4x+48 corresponds to 1919 is Alternate Exterior angle to 5x+8Makes angles congruent
10828)20(4
20
828
85284
x
x
xx
Find all missing angles and explain why you know that angle in relation to other angles
Example
170
50
2
3
50
S
R
T
703
1805032
602
18050702
)(1301
LinearPair
Constructions
Know cheat sheet – what are the main constructions and what ideas deal with what typeAltitude is perpendicular line from vertex to side oppositeMedian is a segment from vertex to midpoint of opposite side, need to construct perpendicular bisector to find midpointPerpendicular Bisector constructs a line that is equidistant from the endpoints of the segmentAngle 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 transformationKnow transformation rulesDo the given transformationIdentify 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 vertices6
4
2
-2
-5 5
6
4
2
-2
-5 5
Part A 6
4
2
-2
-5 5
Part B
Example
Write the rule for the given transformation. (x,y)- ( ___, ___), show some work on how you came to that conclusion.
8
6
4
2
-2
-4
-10 -5 5 10
Original
New
(5,-7)
Proofs
Know set up of two column proof1st Givens2nd Prove other parts congruent need at least 33rd State triangles congruent4th CPCTC of other parts of triangle
Proofs
Know cheat sheet and key termsVisual – Vertical angles and Reflexive sidesGiven information used for reasons
MidpointAngle BisectorSegment BisectorPerpendicularSegment is a perpendicular BisectorParallel sides
Example
Given: BD is a bisector of < ABCDB is perpendicular to AC
Prove: BD bisects AC
DA C
B
AD=DCBD bisects AC
CPCTCAD=DC
ASAABD=CBD
ReflexiveDef BisectDef Perpendicular
BD=BD<ABD=<CBD<ADB=<CDB
G
G
BD is perpendicularto AC
BD Bisects <ABC
RS
ExampleAre the triangles congruent, give conjecture if they are give reason if they are not.