Download Keys GEO SY14-15 Openers 2-24

Geometry Opener(s) 2/24
2/24
It’s National Tortilla Chip Day, National Trading Card
Day, Mexican Flag Day and World Spay Day!!! Happy
Birthday Floyd Mayweather, Jr., Billy Zane, Michelle
Shocked, Steve Jobs, Joe Lieberman, Michel Legrand
and Winslow Homer!!!
Agenda
1. Opener (8) Emilyoz.com/tedoz.com
2. [Period 7: Slope √ (5)]
3. [All other periods: Discussion: What’s the same? (5)]
4. Notes 1: Finding Slope (10)
5. Models 1: Slope (5)
6. Practice 1: Slope (5)
7. Notes 2:  and || Slopes (10)
8. Models 2:  and || Slopes (5)
9. Practice 2:  and || Slopes (5)
10. Models 3:  and || Slopes (5)
11. Practice 3:  and || Slopes (5)
12. Models 4:  and || Slopes (5)
13. Practice 4:  and || Slopes (5)
14. HW: Wksht. 3-2, p. 134 & Wksht. 3-3, p. 138 (12)
15. HW ?s/Catch-up: Text ?s, p. 142, #16-38 even [Honors:
#39-46 all]; ICCE  Proofs for Text ?s, p. 137, #33 & 35
(5)
16. HW √ (5)
17. Exit Pass (5)
Standard(s)
 CCSS-HSG-CO.C.9: Prove theorems about lines and
angles…when a transversal crosses parallel lines, alternate
angles are congruent and corresponding angles are congruent.
 CCSS-HSG-GPE.B.5: Use the slope criteria for parallel and
perpendicular lines and use them to solve geometric problems.
Essential Question(s)




How Do I (HDI) find missing  measures using ICCE 
concepts?
HDI prove missing  measures using ICCE  theorems?
HDI differentiate between a gradual and steep rise visually?
HDI differentiate between a gradual and steep rise
arithmetically?
Objective(s)





Students will be able to (SWBAT) correlate anti-blobbiness
with geometry.
SWBAT name and identify || lines, transversals and ICCE
angles.
SWBAT determine missing ICCE angle measures, using
postulates/theorems and/or algebra.
SWBAT find slope using formulaic methods.
SWBAT find slope using linear graphs.
2/24
What to do today:
1. Do the opener.
2. [Period 7: √ HW]
3. Discuss picture similarities. [Not Period 7]
4. Take some slope notes.
5. Record some slope models.
6. Practice finding slope.
7. Take some /|| slope notes.
8. Record some /|| slope models.
9. Practice finding /|| slope.
10. Record some /|| models/practice.
11. Record and work on HW.
12. Ask ?s about HW/Catch up on HW.
13. √ HW.
14. Do the exit pass.
TODAY’S OPENER
Find x and y.
THE LAST OPENER
Find x and y.
Exit Pass (12/11 – 13/14)
For numbers 1-6 find the coordinates of each
image.
1. Rx-axis (A)
2. Ry-axis (B)
3. Ry = x (C)
4. Rx = 2 (D)
5. Ry= -1 (E)
6. Rx = -3 (F)
The Last Exit Pass
HOMEWORK Period 1
Finish Wksht. 3-2, p. 134 and 3-3, p. 138
HOMEWORK Period 7
Finish Wksht. 3-2, p. 134 and 3-3, p. 138
HOMEWORK Period 3
Finish Wksht. 3-2, p. 134 and 3-3, p. 138
HOMEWORK Period 5 and 8
Finish Wksht. 3-2, p. 134 and 3-3, p. 138
HOMEWORK Period 2A
Finish Wksht. 3-2, p. 134 and 3-3, p. 138
Extra Credit
Period 1 Period 2A Period 3
Israel H. (5x)
Jose C. (6x)
Mirian S. (4x)
Perla S. (2x)
Melissa A.
Israel A.
Benito E.
Amal S. (3x)
Stephanie L.
(4x)
Alexis S. (3x)
Evelyn A. (3x)
Daniela G.
Yesenia M.
Yazmin C. (2x)
Joe L.
Safeer A.
Kevin G.
Isabel G.
Roxana M.
1.
Jaime A. (5x)
Nadia L. (2x)
Anthony P.
Griselda Z. (3x)
Jaclyn C. (3x)
Brandon S. (6x)
Mayra C.
Jacob L. (2x)
Leo G.
Alejandra G.
Mayra G.
Rodrigo F.
Anthony C.
Sonia T. (2x)
Amanda S. (4x)
Josue A. (6x)
Arslan A.
Angie H.
Paulina G. (2x)
Alicia R.
Ricardo D.
Rosie R.
Ronny V.
Gaby O.
Period 5
Period 7
Period 8
Antonio B. (5x)
Rogelio G. (6x)
Eraldy B. (2x)
Anthony G.
(3x)
Alex A. (3x)
Brianna T.
Jose B. (7x)
Carlos L. (3x)
Anadelia G.
(3x)
Jose D. (3x)
Cesar H. (2x)
Saul R.
Alex A.
Jose C.
Adriana H. (6x)
Jackie B. (4x)
Jose R. (6x)
Julian E. (4x)
Jocelyn C. (9x)
Jenny Q.
Ruby L. (3x)
Ana R. (4x)
V. Limon
Zelexus R.
Kamil L.
Diego P.
Alfredo F.
Lilliana F. (3x)
Christian A. (2x)
Liz A. (2x)
Jessica T. (2x)
Jorge C. (2x)
Gerardo L. (2x)
Val R.
Gerardo L.
Esme V. (2x)
Alejandra P.
Cynthia R.
Xavier G. (3x)
Maria M. (2x)
Fernando V. (2x)
Watch a portion of RHYTHmetric video (“What are the 3 ‘laymen’s’ terms for 3 of our 4 transformations?”).
http://www.youtube.com/watch?v=NKtJd1hkI9k
http://www.youtube.com/watch?v=V9uYcnjlAks
http://www.youtube.com/watch?v=X1xPZjItmDk%20
http://www.shodor.org/interactivate/activities/Transmographer/
Your Name
Your Period
Parallel Lines and
Transversals
2/17/15
FIGURE
OBJECT
My Grandfather’s Stool
ICCE s
Alternate Interior s: 4 & 6
2
1
3
4
Consecutive Interior s: 4 & 5
Corresponding s: 2 & 6
6
5
Alternate Exterior s: 1 & 7
8
7
Statements
1.
Reasons
1. Given
2. Def. of TR
3. Def. of Cor. s
4. Cor. s Post.
Prove: x = 16°
5. Def. of 
6. Sub. Prop.
7. Subt. Prop.
8. Sub. Prop.
9. Add. Prop.
10. Sub. Prop.
Statements
1.
Reasons
1. Given
2. Def. of TR
3. Def. of Alt. Int. s
4. Alt. Int. s Thm.
5. Def. of 
Prove: m1 = 67°
6. Sub. Prop.
7. Def. of Alt. Int. s
8. Alt. Int. s Thm.
9. Def. of 
10. Sub. Prop.
11.  Add. Post.
12. Sub. Prop.
13. Sub. Prop.
Prove: x = 34°
Use Def. of Linear Pair, Supplement
Thm. and Angle Add. Post.
Given: m6 = 23°
Prove: m3 = 67°
Prove: m1 = 113°
Honors
1
2
4
5
6
7
9
8
3
Fingernail
Car hood
Streetlamp
Mount Fuji
Airplane wing
High heel
Slanting Roof
Brad Pitt’s nose
Horizon
B
A
Everything You Always Wanted To Know About Lines But Were Afraid To Ask
What’s slope?
a. It’s the steepness of a road or a mountain side OR A LINE.
b. It’s the ratio of change between 2 y-coordinates and 2 xcoordinates.
c. In formulas and graphs, it’s the variable ‘m’.
You need 2 points first. Then you can choose 1 of 2 methods:
So how do I find slope?
The Formula Methods
The Counting Method
1. Call one point (x1, y1)
2. Call the other (x2, y2)
3. Plug them into
𝒚 −𝒚
m = 𝒙𝟏 − 𝒙 𝟐
1. Plot the points in a graph.
2. Connect the points.
3. From 1 point to the other,
count the up and down
squares. Put this number on
the top of your ratio.
4. From the other point, count the
right to left squares. Put it on
the bottom.
5. Simplify and change the sign
to negative if the line FALLS to
the right.
𝟏
𝟐
4. Calculate
5. OR put one point on top
of the other and subtract:
(x1, y1)
-(x2, y2)
6. Your y value goes on top;
Your x value goes on bottom.
Slope-Intercept Form
Point-Slope Form
If you know the slope…
If you know the slope…
m
and you know the y-intercept (where the line
crosses the y-axis)…
(0, b)
you end up with the equation…
y = mx + b
m
and you know one point on the line…
(x1, y1)
you end up with the equation…
y – y1 = m(x – x1)
m = 3/2
m = -2
m = 5/4
Equations of
b=4
b=6
(4,6)
Lines?
lines have opposite sign/reciprocal slopes
that multiplied together = -1.
WHEN YOU ARE GIVEN a point and a
equation…
1. Find the slope in the given equation.
2. FLIP it…and change the sign.
3. Plug the slope and point in the POINT-SLOPE
formula.
4. Simplify or change to slope-intercept form.
(3,1)
y = -3x + 2
(4,7)
(3, 6) (6, 2)
m = -13/5
m=3
m=1
(5, -7)
(1, 6)
(2, 5)
Equations of || Lines?
|| lines have the SAME slope.
WHEN YOU ARE GIVEN a point and a ||
equation…
1. Find the slope in the given equation.
2. Use that same slope in your new equation.
3. Plug the slope and point in the POINT-SLOPE
formula.
4. Simplify or change to slope-intercept form.
𝟏
(-4, 2)
y=𝟐x+5
(4, 1) (2, -3)
(1, -1)
What’s a Line?
Y = mx + b
or
y - y1 = m(x – x1)
D – Denominators don’t contain variables.
O – Operations with variables don’t include ÷.
V – Variables don’t get multiplied together.
E – Exponents don’t include 2 or 3 or 4 or a square root…√)…ONLY 1!!!!!!!!!!!!!
3yx = 7
y = 10 – 5x2
6y = -3x – 15
𝟐
y=𝒙
Yes or No
Yes or No
Yes or No
Yes or No
How Do I Graph?
If you see an x and a y…
1. Get y on one side by itself.
2. Divide EVERYTHING by the y coefficient.
3. Plot the solitary number on the y axis.
4. Turn the x coefficient into a fraction.
5. From the plotted point, count up the numerator…count right the denominator.
(If the fraction is NEGATIVE, count up the numerator…count left the denominator.)
6. Connect the points with a double-arrowed line!
If you only see an x or only see a y…
1. Divide both sides by the coefficient…plot a vertical line if x…plot a horizontal line if y.
Let’s try graphing…
(0, 3) (5, 0)
(2, 3) (5, 7)
(2, 8) (2, -8)
(1.5, -1), (3, 1.5)
Let’s try graphing…
Passes through (8, -6)
|| to 2x – y = 4
Passes through (6, 1)
|| to line with x-intercept -3
and y-intercept 5
Passes through (2, -2)
to x + 5y = 6
Passes through (-2, 1)
to y = 4x – 11
Let’s try graphing…
|| to 4x - 3y = -6 and
passing through (0, -1)
 to 5x + y = 10 and
passes through (5,1)
|| to y = -2x – 5 and
passing through (0, 3)
 to 4x = 3
Let’s try graphing…
|| to y = 2x + 4 and passing  to 2x + 7y = 14
through (0, -2)
|| to y = -2x – 4 and
passing through (0, 2)
 to 6x + 2y = 6
Let’s try graphing…
(1, 3), m = 2
(2, -4), m = -1
Passes through (0, 1)
Passes through (4, -5)
with m =
1
3
with m = 2
What’s Slope? What’s So Special About || &
Lines?
What’s Slope? Recap
What’s Special About || Lines? Recap
PARALLEL LINES HAVE THE SAME
If you have 2 points…
(x1, y1) and (x2, y2)
SLOPE!!!
Then put one point on top of the other
and subtract:
Are these lines || ?
(x1, y1)
-(x2, y2)
(4,3) (1,-3)
(1,2) (-1,3)
You end up with a slope ‘fraction’.
(2,8) (-2,2)
(0,9) (6,0)
The first difference goes on the bottom
(3,9) (-2,-1)
y = 2x*
and the second difference goes on the
top:
 Make an x/y table and find values
𝑇ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (𝑦1 − 𝑦2)
for x = 0 and x = 1. Then determine
the slope.
𝑇ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (𝑥1 − 𝑥2)
What’s the slope?
(1, 5)
(-1, -3)
(8, -5)
(4, -2)
(4, 5)
(2, 7)
What’s Special About
Lines?
Recap
PERPENDICULAR LINES HAVE
OPPOSITE SIGNS/RECIPROCAL
SLOPES!!!! When Their Slopes Are
Multiplied, They Equal -1!!!!
Are these lines
?
(-2,0) (0, 5) (2,0) (0,-5)
(1,0) (0,3)
(3,0) (0,1)
(-2,-3) (2,5) x + 2y = 10*
 Do the one hand on the x; one
hand on the y method, graph &
determine slope…or change to y
= x + # and determine slope.
What About Graphing || &
Recap
Lines?
1. Plot the point it passes through.
2. If ||, use the slope of the line it’s
parallel to…in fraction form.
3. If
, use the slope of the line it’s
perpendicular to…in fraction form…
FLIP it, then change its sign.
4. From the plotted point, count up the
numerator…count right the
denominator.
5. (If the fraction is NEGATIVE, count
up the numerator…count left the
denominator.)
6. Connect the points with a doublearrowed line!
TRANSFORMATION GRAPHIC ORGANIZER
REFLECTIONS TRANSLATIONS ROTATIONS
METHODS
1.
2.
3.
4.
Mira
Formulas
Tracing Paper
Fold on reflection line and
hold up to the light
5. Ruler and measuring
6. Compass and  bisector (1
big arc on reflection line; 2
little arcs from
intersections)
METHODS
1. Formulas
2. Counting units on graph


Flips
Orientation changes
Isometric; an isometry
Lengths, angles, betweenness
and collinearity preserved
Pre-image  Image
Formula…over the x-axis:
CHARACTERISTICS






(x, y)  (x, -y)

Notation:
R(x-axis)(ABC)=A’B’C’
1. Formulas (around origin)
2. Tracing Paper (draw axes)
3. Spinning (always turn
paper
3. Input/output tables (to figure
paper in opposite direction
out the formula)
of CW or CCW arrow)
4. Tracing paper
4. SFA (Subtract spin point
/formula/add spin point)
5. Compass and protractor
(line to spin point; measure
angle; line away from spin
point)
CHARACTERISTICS




METHODS

Slides
Orientation the same
Isometric; an isometry
Lengths, angles, betweenness and
collinearity preserved
Pre-image  Image
2 reflections over || lines = 1
translation
Formula…3 right and 3 down:
(x, y)  (x+3, y-3)

Notation:
T<3, -3>(ABC)=A’B’C’
CHARACTERISTICS








Turns/Spins
Orientation changes
Isometric; an isometry
Lengths, angles, betweenness and
collinearity preserved
Pre-image  Image
A rotation of 270 around the
origin = 1 reflection
Formula…180° CW (clockwise)
around origin: (x, y)  (-x,-y)
Notation:
r(180°, O)(ABC)=A’B’C’
FORMULAS
 (x, y)  (x, -y): over x



axis
(x, y)  (-x, y): over yaxis
(x, y)  (-x, -y): over
the origin
(x, y)  (y, x): over
x=y
(x, y)  (-y, -x): over
-x=y
FORMULAS




x
-2
3
-4
y
3
-6
3
A flip over the origin
(x, y)  (-x, -y)





+x is right
-x is left
+y is up
-y is down
 r(-270,O)(x, y) = (y, -x) {CCW}
EXAMPLES
EXAMPLES
A slide 3 right and 3 down…
(x, y)  (x + 3, y – 3)
x’ y’
x” y”
r(90,O)(x, y) = (y, -x) {CW}
r(180,O)(x, y) = (-x, -y) {CW}
r(270,O)(x, y) = (-y, x) {CW}
r(-90,O)(x, y) = (-y, x) {CCW}
r(-180,O)(x, y) = (-x, -y)
{CCW}
EXAMPLES
A flip over the x-axis…
(x, y)  (x, -y)
FORMULAS
x
-4
-1
-4
y
5
5
0
A slide 1 left and 2 up
(x, y)  (x - 1, y + 2)
x’ y’
x” y”
A spin 180° CW…
(x, y)  (-x, -y)
x y
1 4
5 2
3 -1
A 90° CCW (counterclockwise) spin
around the origin
(x, y)  (-y, x)
x’ y’
x” y”
Today’s Worksheets Rubrics
Line of Symmetry Wksht.
#2. 2 pts.
50 %  at least 3 pts.
#3. 2 pts.
64%  at least 6 pts.
#4. 2 pts. (Correctly identifying 5 letters with 76%  at least 9 pts.
lines of symmetry; correctly identifying 5
88%  at least 12 pts.
letters with NO lines of symmetry)
93%  at least 15
#5. 6 pts. (1 for each image)
100%  18
#6. 4 pts. (1 for each image WITH the
specified # of lines of symmetry)
#7. 2 pts. (Yes with an example to support
‘yes’ or no with counterexample to support
‘no’)
Reflections Wksht.
#1. 8 pts. (1 for each ? or gap fill)
#2. 7 pts. (1 for each instruction or request to
write a statement)
#3. 4 pts. (1 for each image)
#4. 5 pts. (1 for each instruction; 2 for the
entire path)
50 %  at least 4 pts.
64%  at least 8 pts.
76%  at least 12 pts.
88%  at least 16 pts.
93%  at least 20
100%  24
Reflections on a Coordinate Plane Wksht.
#2. 8 pts. (1 for each table entry and ?)
50 %  at least 6 pts.
#3. 7 pts. (1 for each table entry and ?)
64%  at least 12 pts.
#4. 9 pts. (1 for each table entry, ? and
76%  at least 18 pts.
labeling the image)
88%  at least 24 pts.
#5. 12 pts. (1 for each set of pre-image & 93%  at least 30
image coordinates; 2 for each set of
100%  36
formulaic coordinates [in the boxes!])
YOUR PROOF CHEAT SHEET
IF YOU NEED TO WRITE A PROOF ABOUT
ALGEBRAIC EQUATIONS…LOOK AT THESE:
Reflexive
Property
Symmetric
Property
Transitive
Property
Addition & Subtraction
Properties
Multiplication &
Division Properties
Substitution
Property
Distributive
Property
IF YOU NEED TO WRITE A PROOF ABOUT
LINES, SEGMENTS, RAYS…LOOK AT
THESE:
For every number a, a = a.
Postulate 2.1
For all numbers a & b,
if a = b, then b = a.
For all numbers a, b & c,
if a = b and b = c, then a = c.
For all numbers a, b & c,
if a = b, then a + c = b + c & a – c = b – c.
For all numbers a, b & c,
if a = b, then a * c = b * c & a ÷ c = b ÷ c.
For all numbers a & b,
if a = b, then a may be replaced by b in any
equation or expression.
For all numbers a, b & c,
a(b + c) = ab + ac
Postulatd 2.2
Postulate 2.3
Postulate 2.4
Postulate 2.5
Postulate 2.6
Postulate 2.7
The Midpoint
Theorem
IF YOU NEED TO WRITE A PROOF ABOUT THE
LENGTH OF LINES, SEGMENTS, RAYS…LOOK
AT THESE:
Reflexive
Property
Symmetric
Property
Transitive
Property
Addition & Subtraction
Properties
Multiplication &
Division Properties
Substitution
Property
Segment Addition
Postulate
Through any two points, there is exactly ONE
LINE.
Through any three points not on the same
line, there is exactly ONE PLANE.
A line contains at least TWO POINTS.
A plane contains at least THREE POINTS not on
the same line.
If two points lie in a plane, then the entire line
containing those points LIE IN THE PLANE.
If two lines intersect, then their intersection is
exactly ONE POINT.
It two planes intersect, then their intersection
is a LINE.
If M is the midpoint of segment PQ, then
segment PM is congruent to segment MQ.
IF YOU NEED TO WRITE A PROOF ABOUT
THE MEASURE OF ANGLES…LOOK AT
THESE:
AB = AB
(Congruence?)
If AB = CD,
then CD = AB
If AB = CD and CD = EF,
then AB = EF
If AB = CD,
then AB  EF = CD  EF
If AB = CD,
then AB */ EF = CD */ EF
If AB = CD,
then AB may be replaced by CD
If B is between A and C, then AB + BC = AC
If AB + BC = AC, then B is between A and C
Reflexive
Property
Symmetric
Property
Transitive
Property
Addition &
Subtraction
Properties
Multiplication &
Division Properties
Substitution
Property
m1 = m1
(Congruence?)
If m1 = m2,
then m2 = m1
If m1 = m2
and m2 = m3, then m1 = m3
If m1 = m2,
then m1  m3 = m2  m3
DEFINITION OF
CONGRUENCE
Whenever you change from
 to = or from = to .
If m1 = m2,
then m1 */ m3 = m2 */ m3
If m1 = m2,
then m1 may be replaced by m2
IF YOU NEED TO WRITE A PROOF ABOUT ANGLES IN GENERAL…LOOK AT THESE:
Postulate 2.11
The  Addition
Postulate
Theorem 2.5
The Equalities Theorem
If R is in the interior of PQS,
then mPQR + mRQS = mPQS.
THE CONVERSE IS ALSO TRUE!!!!!!
Q
Congruence of s is

Reflexive, Symmetric & Transitive
P
R
S
Theorem 2.8
Vertical s
Theorem
If 2 s are vertical, then they are .
(1  3 and 2  4)
IF YOU NEED TO WRITE A PROOF ABOUT COMPLEMENTARY or SUPPLEMENTARY ANGLES
…LOOK AT THESE:
Theorem 2.3
Supplement
Theorem
If 2 s form a linear pair,
then they are
supplementary s.
Theorem 2.4
Complement
Theorem
If the non-common sides of
2 adjacent s form a right ,
then they are complementary s.
Theorem 2.12
 Supplementary
Right s Therorem
Theorem 2.6
R The  Supplements
Theorem
S
P Q
Q
P
If 2 s are  and supplementary, then each
 is a right .
Theorem 2.7
The  Complements
R Theorem
S
Theorem 2.13
 Linear Pair Right
s Therorem
s supplementary to the
same  or to  s are .
(If m1 + m2 = 180 and
m2 + m3 = 180, then 1  3.)
s complementary to the
same  or to  s are .
(If m1 + m2 = 90 and
m2 + m3 = 90, then 1  3.)
If 2  s form a linear pair, then they
are right s.
YOUR PROOF CHEAT SHEET (continued)
IF YOU NEED TO WRITE A PROOF ABOUT RIGHT ANGLES or PERPENDICULAR LINES…LOOK AT THESE:
Theorem 2.9
Perpendicular lines
Theorem 3-4
If a line is  to the 1st of two || lines,
Perpendicular Transversal Theorem
4 Right s Theorem
intersect to form 4 right s.
then it is also  to the 2nd line.
Theorem 2.10
Postulate 3.2
All right s are .
2 non-vertical lines are  if and only if the PRODUCT of their
Right  Congruence Theorem
Slope of  Lines
slopes is -1. (In other words, the 2nd line’s slope is the 1st line’s
slope flipped (reciprocal) with changed sign.)
Theorem 2.11
Perpendicular lines
Postulate 3.2
If 2 lines are  to the same 3rd line, then thhose 2
 Adjacent Right s Theorem
form  adjacent s.
 and || Lines Postulate
lines are || to each other.
Theorem 4-6
Theorem 4-7
If the 2 legs of one right  are  to
If the hypotenuse and acute  of one right
Leg-Leg (LL) Congruence
Hypotenuse-Angle
the corresponding parts of another
 are  to the corresponding parts of
(HA) Congruence
right , then both s are .
another right , then both s are .
Theorem 4-8
Postulate 4-4
If the hypotenuse and one leg of one right
If the leg and acute  of one right  are
Leg-Angle (LA)  to the corresponding parts of another
Hypotenuse-Leg (HL)
 are  to the corresponding parts of
Congruence
Congruence
another right , then both s are .
right , then both s are .
IF YOU NEED TO WRITE A PROOF ABOUT ICCE ANGLES or PARALLEL LINES…LOOK AT THESE:
Postulate 3.1
If 2 || lines are cut by a
Postulate 3.4
If 2 lines are cut by a transversal
Corresponding Angles
transversal, then each pair of CO
Corresponding Angles/|| Lines
so that each pair of CO s is ,
Postulate (CO s Post.)
s is .
Postulate (CO s/|| Lines Post.) then the lines are ||.
Theorem 3.1
If 2 || lines are cut by a
Theorem 3.5
If 2 lines are cut by a transversal so
Alternate Interior Angles
transversal, then each pair
Alternate Exterior Angles/|| Lines
that each pair of AE s is , then the
Theorem (AI s Thm.)
of AI s is .
Theorem (AE s/|| Lines Thm.)
lines are ||.
Theorem 3.2
If 2 || lines are cut by a
Theorem 3.6
If 2 lines are cut by a transversal
Consecutive Interior Angles
transversal, then each pair
Consecutive Interior Angles/|| Lines so that each pair of CI s is
Theorem (CI s Thm.)
of CI s is supplementary.
Theorem (CI s/|| Lines Thm.)
supplementary, the lines are ||.
Theorem 3.3
If 2 || lines are cut by a
Theorem 3.7
If 2 lines are cut by a transversal so
Alternate Exterior Angles transversal, then each pair
Alternate Interior Angles/|| Lines that each pair of AI s is , then the
Theorem (AE s Thm.)
of AE s is .
Theorem (AI s/|| Lines Thm.)
lines are ||.
Postulate 3.2
2 non-vertical lines have the same
Postulate 3.5
If you have 1 line and 1 point NOT on that
Slope of || Lines
slope if and only if they are ||.
|| Postulate
line, ONE and only ONE line goes through
that point that’s || to the 1st line.
Theorem 6.6
Theorem 6.4
A midsegment of a  is || to one
In ACE with ̅̅̅̅̅
𝑩𝑫 || ̅̅̅̅
𝑨𝑬 and
 Midsegment Thm.

Proportionality
Thm.
intersecting
the
other 2 sides in distinct
side of the , and its length is ½
̅̅̅̅
𝑩𝑨 ̅̅̅̅
𝑫𝑬
the length of that side.
points, = .
̅̅̅̅
𝑪𝑩
̅̅̅̅
𝑪𝑫
Postulates and Theorems to IDENTIFY CONGRUENT TRIANGLES: SSS, ASA, SAS or AAS
Postulates and Theorems to IDENTIFY SIMILAR TRIANGLES: AA, SSS or SAS
Linear Equation in
Slope-Intercept
Form
Linear Equation in
Point-Slope Form
y = mx + b
m = slope, b = yintercept
y – y1 = m(x – x1)
m = slope,
(x1, y1) = 1 point on
the line
Linear Equation in Standard Form
Ax + By = C
I – Numbers and coefficients can only be Integers. (No fractions or decimals.)
P – The x coefficient must be Positive. (A > 0)
O – Zero can only appear beside a variable Once. (If A = 0, then B ≠ 0)
D – Numbers and coefficients can only be Divisible by 1. (GCF = 1)
S – Variables can only be on the Same side of the equal sign.
CI s: 2 inside || lines on SAME side of transversal.
CO s: 1  inside || lines & 1  outside || lines, on OPPOSITE sides of transversal.
AI s: 2 inside || lines on OPPOSITE sides of transversal.
AE s: 2 outside || lines on OPPOSITE sides of transversal.
AE
CO
AI
CO
CI
AE
AI/
CI