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Geometry Opener(s) 12/16 (12/21-12/13)
12/16
It’s the first day of Las Posadas, Barbie and
Barney Backlash Day, National ChocolateCovered Anything Day and Zionism Day!!! Happy
Birthday Benjamin Bratt, William Perry, Liv
Ullmann, Philip K. Dick, Margaret Mead, Noel
Coward, Wassily Kandinsky and Jane Austen!!!
12/16
What to do today:
1. Do the opener.
2. Create a HDI Prove graphic organizer.
3. Use whiteboards to write out HW proofs.
4. √ HW.
5. Record some proof models.
6. Practice proof writing.
7. Begin homework.
TODAY’S OPENER
Given: IMS and IEL
are equilateral.
Prove: IMS  IEL
Agenda
1. Opener (8)
2. HDI Prove Recap: Graphic Organizer? (25)
3. HW ?s/CX (22)
4. Model: Overlapping s and CPCTC (5)
5. Practice: Chapters 4-4 & 4-5 (25)
6. HW Assignment: Wkshts. 5 & 6 (15)
Standard(s)
 CCMS-HSS-IC: Making Inferences and Justifying
Conclusions
 CCSS-M-G-CO.C.9: Prove theorems about lines and
angles
Essential Question(s)
 How Do I (HDI) use equality and congruency properties
and definitions to write an ‘adult’ proof?
 HDI use givens to determine congruencies?
 HDI use pictures to determine congruencies?
 HDI use parts congruencies to notate a geometric
figure?
 HDI use notations and parts congruencies to prove
figures’s overall congruence?
Objective(s)
 Students will be able to (SWBAT) correlate antiblobbiness with geometry.
 SWBAT use algebraic and geometric equality and
congruency properties to write a proof.
 SWBAT list triangle congruencies by reading givens.
 SWBAT list triangle congruencies by analyzing pictures.
 SWBAT notate triangle congruencies with arcs and
ticks.
 SWBAT use triangle postulates and theorems to prove
congruence.
http://en.origami-club.com/fruit/corn/anime-corn/index.html
M
L
I
S
E
If not possible, what
corresponding parts need to be congruent to
make the conclusion true?
THE LAST OPENER





Get a proof block board for your pair. It may be in front of
your desks!
Get a set of proof blocks.
Get a marker and eraser for your row from Mr. Keys.
Transfer your first homework problem to the board using
proof blocks and copy it onto the landscape paper.
Transfer your first homework problem to the board using a
t-chart and copy it onto the portrait paper.
Exit Pass (12/11 – 13/14)
The Last Exit Pass
Here are 2 givens. Write a conjecture for each. Draw a figure to illustrate
your conjecture.
1st Given: ⃗⃗⃗
IS is one side of TIS.
2nd Given: HIJK is a square
HOMEWORK Period 1 & 7
 ce Wkshts. 5 & 6
HOMEWORK Period 3, 5 and 8
 ce Wkshts. 5 & 6
HOMEWORK Period 2A
 ce Wkshts. 5 & 6
Extra Credit
Period 1 Period 2A Period 3
Alex H. (4x)
Amal S. (2x)
Evelyn A. (6x)
Israel A. (3x)
Israel H. (6x)
Joe L.
Jose C. (6x)
Lesslye P.
Mirian S. (5x)
Oscar
Perla
Stephanie L.
(2x)
Victor C. (3x)
Yessenia M.
(3x)
Yazmin C.
Ana O. (2x)
Anthony C. (5x)
Brandon S. (2x)
Fabian
Gabino G. (2x)
Gabriel M. (4x)
Jaclyn
Jaime A. (2x)
Josh P.
Leo G.
Nadia L.
Noemi O.
William M. (4x)
Abrahan
Alicia R. (3x)
Amanda S.
Angie H. (2x)
Anthony C.
Arslan (3x)
Cesar O.
Claudia
Enrique G. (2x)
Gaby
Jocelyn J.
Josue A. (2x)
Kassandra G.
(3x)
Michelle S. (2x)
Paulina G. (5x)
Ricardo D. (2x)
Ronny V. (4x)
Rosie R.
Sierra (3x)
Period 5
Period 7
Adrian O.
Alex A. (7x)
Anthony G.
(5x)
Brianna T. (6x)
Eraldy B. (6x)
Jesus H.
Jonathan A.
Jose B. (3x)
Jose C. (6x)
Jose D. (3x)
Jose G. (2x)
Liz L. (2x)
Maria M.
Rogelio G. (6x)
Solai L. (2x)
Tony B. (4x)
Adriana H. (3x)
Alfredo F. (4x)
Brenda (2x)
Carmen A.
Diego P. (7x)
Gabriela G. (5x)
Gustavo C. (3x)
Jackie B. (3x)
Jocelyn C. (6x)
Jose R.
Julian E.
Kamil L. (3x)
Liliana F.
Vanessa T. (3x)
Vicente L. (2x)
Zelexus R. (2x)
Period 8
Alejandra P.
Andrea N. (2x)
Bianca (2x)
Brian H. (3x)
Cynthia
Esmerelda V.
(2x)
Fernando
Gerardo L.
Jessica T. (2x)
Jorge C.
Jose G.
Kevin A.
Liliana R.
Santi
Saul
Stephanie E.
Valeria R. (3x)
Yuritzi
Triangle Congruence Worksheet #1
For each pair of triangles, tell which postulates, if any, make the triangles congruent.
1. ABC  EFD
2. ABC  CDA
______________
______________
C
B
D
A
C
F
A
B
3. ABC  EFD
D
E
4. ADC  BDC ______________
______________
C
F
C
B
A
5. MAD  MBC
D
A
E
B
D
6. ABE  CDE ______________
______________
D
D
C
C
E
A
A
M
B
B
Triangle Congruence Worksheet #2
I.
For each pair of triangles, tell which postulate, if any, can be used to prove the triangles congruent.
1. AEB  DEC ______________
2. CDE  ABF ______________
A
E
D
C
C
F
B
E
A
D
3. DEA  BEC ______________
A
B
4. AGE  CDF ______________
B
E
D
C
5. RTS  CBA ______________
6. ABC  ADC ______________
B
T
S
C
A
R
A
B
C
D
NAME___________________________________________________
PERIOD___________
1. Did your teacher finish adding all the tick marks and arcs to the 2 triangles? Could you add some more? If
so, draw them in!
2. Look at the 2 triangles and answer:
a) Are they congruent? Yes or No
b) Write the triangle congruency statement.
c) Give the postulate or theorem that makes them congruent OR state what additional congruent sides or
angles are necessary to make them congruent.
1.
D
2.
C
3.
O
A
E
E
T
E
L
A
R
W
V
B
a. ______________
a. ______________
a. ______________
b. _____   _____
b. _____   _____
b. _____   _____
c. ______________
c. ______________
c. ______________
4. Given: GEF  GHF
5. Given: W  S
6.
I
W
S
H
L
U
G
E
a. ______________
a. ______________
a. ______________
b. _____   _____
b. _____   _____
b. _____   _____
c. ______________
c. ______________
c. ______________
7. Given: IMS & IEL
are equilateral.
8.
M
C
F
9.
H
L
I
T
S
E
P
A
A
B
D
E
M
a. ______________
a. ______________
a. ______________
b. _____   _____
b. _____   _____
b. _____   _____
c. ______________
c. ______________
c. ______________
3. Using the given postulate, tell which parts of the pair of triangles should be shown congruent in order to make the
triangles congruent.
1.
ASA
2.
SAS
3.
C
SAS
B
F
F
E
A
B
D
A
B
A
D
C
E
D
C
_______  ________
4.
_______  ________
ASA
5.
_______  ________
AAS
6.
D
P
P
SSS
C
S
T
A
R
R
Q
_______  ________
S
_______  ________
B
Q
_______  ________
11-3
NOTES: CONDITIONALLY SPEAKING
STATEMENT
FORMED BY…
Conditional
a given hypothesis
(p) and
__________________
SYMBOLS
& SHORT
FORMS
pq
If p then q
EXAMPLES
If 2 s have the
same measure,
_______________
p implies q
Converse
Inverse
Contrapositive
What’s New?
switching the
hypothesis (into the
conclusion’s place)
and _____________
_________________
negating (adding
‘not’ or  to) both
the hypothesis and
________________
________________
negating (adding
‘not’ or  to) both
the hypothesis and
_______________
_______________
qp
If 2 s are ,
_______________.
If q then p
q implies p
p  q
If not p then
not q
If 2 s do NOT
have the same
measure,
_______________
Not p implies
not q
q  p
If not q then
not p
Not q implies
not p
If 2 s are NOT ,
_______________
What do I need to Depending on the proof, you’ll need properties.
write a proof?
For algebraic proofs (as opposed to geometric
proofs), you’ll need the equality properties…
Property
Example
Reflexive
Property
For every number a,
a = a.
Symmetric
Property
For all numbers a & b,
if a = b, then b = a.
Transitive
Property
For all numbers a, b & c,
if a = b and b = c, then a = c.
Addition &
Subtraction
Properties
Multiplication &
Division
Properties
Substitution
Property
Given:
3(x – 2) = 42
Given:
𝟐
5 – 𝟑z = 1
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.
Prove:
x = 16
Prove:
z=6
What do I need to
write a
GEOMETRIC
proof?
Since geometry involves variables, numbers and
operations just like algebra, most of the algebraic
properties and equalities can be transformed into
geometric properties and equalities!!!
SEGMENTS
Reflexive
Property
ANGLES
AB = AB
m1 = m1
Symmetric
Property
If AB = CD,
then CD = AB
If m1 = m2,
then m2 = m1
Transitive
Property
If AB = CD and CD = EF,
then AB = EF
Addition &
Subtraction
Properties
Multiplication &
Division
Properties
Substitution
Property
Given:
Point C is the
midpoint of ̅̅̅̅
𝐀𝐁
and B is the
midpoint of ̅̅̅̅
𝐂𝐃
Given:
m2 = 60
2  10
If m1 = m2
and m2 = m3,
then m1 = m3
If AB = CD,
If m1 = m2,
then AB  EF = CD  EF
then m1  m3 =
m2  m3
If AB = CD, then
If m1 = m2,
AB x/÷ EF = CD x/÷ EF
then m1 x/÷ m3 =
m2 x/÷ m3
If AB = CD,
If m1 = m2,
then AB may be
then m1 may be
replaced by CD
replaced by m2
Prove:
̅̅̅̅
𝐀𝐂 ≅ ̅̅̅̅
𝐁𝐃
Prove:
m10 = 60
Given:
Prove:
mACB = mABC
mXCA = mYBA
A
X
C
B
Y
Proof Rubric
Givens are stated
Conjecture is stated
Exemplifying diagrams are drawn
Diagrams match conjecture
Diagrams are labeled
Verbal explanation of diagrams is written
Correct analysis of congruency or presence of
counterexample(s)
1 pt.
1 pt.
3 pts.
2 pts.
2 pts.
3 pts.
3 pts.
Conjecture & Counterexample Notes
11/21/14
Let’s come up with some rules of congruence…
1. #3, 6 and 9???
2. Look at #1, 2, 4, 5, 7 and 8. Can you come up with one more rule? Is
anything different from #3, 6 and 9?
The Success Project Rubric
Less than 10 pieces of 59-%
evidence in improper
format.
10 pieces of evidence
in proper format.
Details?
60+%
10 pieces of evidence
in proper format with
conclusion. Details?
70+%
10 pieces of evidence
in proper format with
conclusion PLUS
POSTER. Details?
80+%
10 pieces of evidence 90%+
in proper detailed
format with conclusion
PLUS POSTER… with
illustrations.
Name
Plane
Line
Ray
Segment
Point
Collinear
Noncollinear
Coplanar
NonCoplanar
Congruent
Definition
A flat 2-dimensional (length
and width) surface that
extends forever above, below,
to the right and to the left and
is defined by 3 points.
A straight 1-dimensional
(length) set of points that
extends forever in 2 directions
(left and right OR up and
down) and is defined by 2
points.
A straight 1-dimensional
(length) set of points that
extends forever in 1 direction
(left OR right OR up OR down)
and is defined by 2 points, one
of which is an initial end point.
A straight 1-dimensional
(length) set of points that does
not extend forever and is
delimited by 2 endpoints.
A 0-dimensional dot that
simply defines a location.
An adjective that describes
points contained on the same
line.
An adjective that describes
points NOT contained on the
same line.
An adjective that describes
usually non-collinear points
contained in the same plane.
An adjective that describes
points NOT contained in the
same plane.
Two geometric figures that
have the same SIZE and
SHAPE.
Figure
Notation
10-4
NOTES: Don’t Dis -Dis (or ‘Midpoints Part 2’)
How many words can you write down that begin with the prefix ‘dis-‘,
meaning apart or away or having a reversing force?
disable
disarrange
disembark
disinterred
disagree
disapprove
disentangle
dislike
disadvantage
disbelieve
disinfect
dislocate
disallow
discomfort
disinformation
dislodge
disappear
discontented
disinterest
disloyal
DISTANCE
X
Y
Given: A Number Line with 2 endpoints
distance
Objective: Find the
Step 1. Write the number values as a difference.
Step 2. Put absolute value signs around the difference.
Step 3. Subtract the number values and take the absolute
value.
Formula:
| 𝑿 − 𝒀 | or | 𝒀 − 𝑿 |
Title: Finding the length of a number line segment
(X, Y)
(X, Y)
(X, Y)
Given: A Coordinate Plane with 2 endpoints Objective: Find the distance
Step 1. Subtract the two x-coordinates…x1 and x2 .
Step 2. Square the difference.
Step 3. Subtract the two y-coordinates…y1 and y2 .
Step 4. Square the difference.
Step 5. Add the two differences.
Step 6. Take the square root of that sum.
Formula: √(𝐱 𝟐 − 𝐱 𝟏 )𝟐 + (𝐲𝟐 − 𝐲𝟏 )𝟐
Title: Finding the length of a coordinate plane segment
What’s new?
Why does this work???
9-26
NOTES: The Bible of Bi-
How many words can you write down that begin with the prefix ‘bi-‘?
1. bipolar
2. bisexual
3. binoculars
4. bifocals
5. biannual,
bimonthly,
biweekly,
bicentennial
6. binary
7. bipolar
8. bisexual
9. binoculars
10.
bifocals
11.
biannual,
bimonthly,
biweekly,
bicentennial
12.
binary
X
Given: A Number Line with 2 endpoints
13.
bipolar
14.
bisexual
15.
binoculars
16.
bifocals
17.
biannual,
bimonthly,
biweekly,
bicentennial
18.
binary
Y
Objective: Find the midpoint
Step 1. Add the number values for the two segment endpoints.
Step 2. Divide by 2.
Formula:
𝑿+𝒀
𝟐
Title: Finding the midpoint of a number line segment
A
B
Given: A Coordinate Plane with 2 endpoints
Step 1.
Step 2.
Step 3.
Step 4.
Formula:
Objective: Find the midpoint
Add the X coordinates of the segment endpoints.
Divide by two.
Add the Y coordinates of the segment endpoints.
Divide by two.
𝑿𝟏 + 𝑿𝟐 𝒀𝟏 + 𝒀𝟐
(
,
)
𝟐
𝟐
Title: Finding the midpoint of a coordinate plane segment
What’s new?
NOTES: Rays, Angles & Protractors
Name
Definition
2. Angle ()
An initial point or endpoint with an
infinite number of points extending in
one direction.
2 rays that share the same endpoint.
3.  side
Each ray that makes up the angle.
4. Degree
1/360th of a full circle.
1. Ray
5. Degree measure How open/closed the angle is in
degrees.
Figure
10/7
Notation
⃖⃗⃗⃗⃗
𝐼𝐹
BED
⃖⃗⃗⃗⃗
𝐼𝐹
°
23°
6.  arc
The curved line inside an angle that
distinguishes its interior from its
exterior.
A curved line touching the
two rays of an angle.
7. Vertex of an 
The endpoint of the two sides of an
angle.
A single uppercase printed
letter: the middle letter of
a named angle. (B in
ABC)
8. Interior of an 
The inside of an angle, denoted by the
arc. (yellow)
9. Exterior of an 
The outside of an angle, denoted by
the ABSENCE of an arc. (green)
10.Right 
An angle formed by 2 perpendicular
rays, its measure equal to 90°.
mABC = 90°
11.Acute 
An angle formed by 2 rays, its
measure less than 90°.
mABC < 90°
12.Obtuse 
An angle formed by 2 rays, its
measure greater than 90°.
180 > mABC > 90°
13.Reflex 
An angle formed by 2 rays, its
measure greater than 180°.
360 > mABC > 180°
14.Straight 
An angle formed by 2 opposite rays,
its measure equal to 180°.
mABC = 180°
15.Congruent s
2 angles that have the same degree
measure.
ABC ≅ DEF
16. Bisector
A ray with an endpoint that is
identical to the vertex of an angle and
which divides that angle in half.
17.Adjacent s
18.Complementary
s
19.Supplementary
s
20.Vertical s
21.Linear Pair
⃗⃗⃗⃗⃗
BG
Copying & Bisecting a Segment
Copying a Segment
1.
2.
3.
4.
5.
Draw a segment.
Label its endpoints I and F.
Draw another non-collinear point.
Label it E.
Draw an arc in your segment
from 1 endpt. to the other. (Don’t
change your compass’s width!)
6. Draw the same arc starting from
point E.
7. Label a point on the second arc N.
8. Connect E to N.
Bisecting a Segment
1. Place your compass point on E.
2. Place your pencil point at least
̅̅̅̅ and draw a
halfway across 𝑬𝑵
half-circle arc. (Don’t change your
compass’s width!)
3. Place your compass point on N.
4. Draw the exact same halfcircle arc. (Don’t change your
compass’s width!)
5. Label the arc intersection points
T and H.
6. Connect T to H.
Copying & Bisecting an Angle
Copying an Angle
1. Draw a ray.
2. Label its endpoint U.
3. Draw another non-collinear ray
that shares the 1st ray’s endpt.
4. Draw another separate ray…
far away from the angle.
5. Label its endpt. L
6. Draw an arc in your  from 1
side to the other. (Don’t
change your compass’s width!)
7. Label the intersection points B and G.
8. Draw the same arc on your
3rd ray.
9. Label its intersection point Y.
10.Draw a 2nd arc in your  with compass pt.
on G and pencil point on B. (Don’t change
your compass’s width!)
11.Draw the same arc on your 3rd ray.
12.Label the intersection point F.
13.Connect L to F.
Bisecting an Angle
1. Draw an angle.
2. Label its vertex A.
3. Draw an arc in your angle
from one side to the other.
4. Label the intersection pts. W and P.
5. Draw an arc with your compass pt.
on W, at least ½ -way across the
interior. (Don’t change your
compass’s width!)
6. Draw the same arc from your P.
7. Label the intersection point of
those 2 arcs S.
8. Connect A to S.
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