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.