Download Keys GEO Openers 4-15

Opener(s) 4/15
4/15
It’s Tax Day…AND Jackie Robinson Day!!! Happy
Birthday Emma Watson, Seth Rogen, Samantha
Fox, Elizabeth Montgomery, Roy Clark, Harold
Washington, Sir Neville Marinner, Bessie Smith,
Nikita Khrushchev and Leonardo Da Vinci!!!
4/15
What to do today:
1. Do the opener.
2. Ask questions about HW.
3. Practice some Isosceles  algebra and proofs.
4. Work on your homework.
5. Do the exit pass.
OPENER
Agenda
 Opener (5)
 HW Questions (5)
 Classwork: Me - #15-18, p. 219; You - #19-22,
p. 219; Me - #23-24, p. 220, You - #25-26, p.
220; Me - # 29, p. 220; You - #30 & 40, p. 220
(20)
 Individual work 1: Wksht. 4-6, pgs. 215-16 (15)
 Exit pass (5)
Essential Question(s)
 How do I prove that triangles are congruent?
Objective(s)
 Students will be able to (SWBAT) use the SSS
postulate to prove 2 s are .
 SWBAT use the SAS postulate to prove 2 s
are .
 SWBAT use the ASA postulate to prove 2 s
are .
 SWBAT use the AAS postulate to prove 2 s
are 
 SWBAT determine 2 s’s incongruence based
on inadequate side or  information.
 GHF is equilateral with mF = 3x + 4,
mG = 6y and mH = 19z + 3. Draw a picture
of the triangle, label it and find x, y and z.
OUR LAST OPENER
Exit Pass
Our Last Exit Pass
Homework
 Wksht. 4-6, pgs. 215-16
Agenda writer:
Opener answerer:
ACHIEVE Manual distributor:
Period
8
Timekeeper:
Presenter:
Filer:
Tools Distributor:
Student Aide:
Notes: Some Basic  PTCs (Postulates, Theorems and/or Corollaries)
Theorem 4-1:
The sum of the measures of the s of m1 + m2 +
 Sum Theorem
a  is 180.
m3 = 180
( Sum Thm.)
Theorem 4-2:
If 2 s of one  are  to 2 s of a 2nd
If A  F and
rd
rd
3  Theorem
, then the 3 s of both s are .
C  D, then
rd
(3  Thm.)
B  E
Theorem 4-3:
The measure of an ext.  of a  = the mYZP = mX
Exterior  Theorem
sum of the measures of the 2 remote
+ mY
(Ext.  Thm.)
int. s.
Corollary 4-1:
The acute s of a right  are
mG + mJ =
Acute  Right  Corollary
complementary.
90
Corollary 4-2:
A  can have, at most, 1 obtuse or
1 Obtuse/Right  Corollary right 
Definition:
2 s are  if and only if (iff) their
In s CAT and DOG, if
Congruent Triangles corresponding parts are .
C  D, A  O
(CPCTC)
T  G and
̅̅̅̅
̅̅̅̅  ̅̅̅̅
The corresponding parts of 2 s are
𝐂𝐀  ̅̅̅̅
𝐃𝐎, 𝐀𝐓
𝐎𝐆
̅̅̅̅  𝐆𝐃
̅̅̅̅ then
congruent if and only if the s are
𝐓𝐂
congruent.
CAT  DOG
Theorem 4-4:
 congruence is reflexive. DOG  DOG
Properties of 
 congruence is
If CAT  PAW
Congruence
symmetric.
then PAW  CAT
(Refl. Prop. / Sym.
 congruence is transitive. If CAT  PAW and
Prop. / Trans. Prop.)
PAW  DOG
then CAT  DOG
Postulate 4.1:
2 s are  if the sides of one  are congruent
BUY  CAR
nd
Side-Side-Side
to the sides of a 2 .
Congruence (SSS)
Postulate 4.2:
Side-Angle-Side
Congruence (SAS)
Postulate 4.3:
Angle-Side-Angle
Congruence (ASA)
Postulate 4.5:
Angle-Angle-Side
Congruence (AAS)
2 s are  if 2 sides and the included angle of EAT  BUN
one  are  to the corresponding 2 sides and
included angle of another .
2 s are  if 2 angles and the included side of FAT  FLY
1  are  to 2 corresponding angles and the
included side of the other .
2 s are  if 2 angles and a non- included side PUR  PLE
of 1  are  to the 2 corresponding angles and
non-included side of the other 
ISOSCELES TRIANGLES
Isosceles  s Thm. If 2 sides of a 
(Thm. 4.9)
are , then the s
opposite those
sides are .
RIGHT TRIANGLES
Theorem 4-6:
2 RIGHT s are 
Right  Leg-Leg
if the legs of one
Congruence
 are  to the
(LL)
legs of the other
.
Isosceles  Sides
Thm.
(Thm. 4.10)
If 2 s of a  are
, then the sides
opposite those s
are .
Theorem 4-7:
Right  HypotenuseAngle Congruence
(HA)
Equilateral /
Equiangular
Corollary
(Cor. 4.3)
A  is equilateral if Theorem 4-8:
Right  Leg-Angle
it is equiangular
Congruence
AND
(LA)
A  is equiangular
if it is equilateral.
Equilateral 
Measures Corollary
(Cor. 4.4)
Each angle of an
equilateral 
measures 60.
Postulate 4.4:
Right  HypotenuseLeg Congruence
(HL)
2 RIGHT s are 
if the hyp. and an
acute  of one 
are  to the hyp.
and
corresponding
acute  of the
other .
2 RIGHT s are 
if 1 leg and 1
acute  of one 
are  to the
corresponding leg
and
corresponding
acute  of the
other .
2 right s are  if
the hyp. and a leg
of one  are  to
the hyp. and
corresponding leg
of the other .
13 Must-Follow Proof Steps
1. If you have givens,
2. If you have tick marks or arcs
marked in the figure,
3. WHATEVER…
4. If you see single or double
arrows in the figure,
5. If you have a || GIVEN,
6. If you have a bisect GIVEN,
7. If you have a median GIVEN,
8. If you have a midpoint
GIVEN/STATEMENT,
9. If you are asked to prove 2 s
are ,
10. If you are asked to prove 2
parts are ,
11. If you have a postulate
Reason that says ASA, SSS,
SAS or AAS,
12. If you have  segments that
are part of larger segments,
13. If you have  angles that are
part of linear pairs,
Write them EXACTLY the same in the
Statements column and say GIVEN in
the Reasons column.
Write them as Statements in the
Statements column and GIVENS in the
Reasons column.
As you list GIVENS, place tick and arc
marks in your figure!!!
Write it as a PARALLEL Statement and
GIVEN.
Write it as a   Statement with an AI or
CO  Postulate Reason.
Write it as a   Statement with a Def. of
Bisect Reason.
Write it as a midpoint Statement with a
Def. of Median Reason.
Write it as a  Segments Statement with
a Def. of Midpoint Reason.
Your last Reason must be ASA, SSS,
SAS or AAS.
Your last Reason must be CPCTC.
You MUST have 3 Statements in your
proof that match the 3 letters of your
postulate Reason!!!
 Write a  segment GIVEN
 Insert a Reflexive statement/reason
 Add the Reflexive segment to both
sides
 Use the Segment Addition Postulate
 Write a  angle GIVEN
 Insert a supplementary s statement
 Follow it with a Definition reason
 Use the ‘s Supp. to  s are ’ Thm.
A Sample Coordinate Plane
Large Groups
Alfredo
Angela
Mildred
Lesly
Lucia
Areli
Demetrius
Rolando
Janene
Salina
Angelo
Brian
Group 1
David
Mani
Jailene
Group 2
Anarely & Marco
Tony & Jasmine
Mag & Marcella
Steven
Group 3
Josefina
Edgar
Jessica
Group 4
Jen & Susana
Nataly & Cruz
Gab & Alejandra
Group 5
Javier
Sandra
Elizabeth
Groups of Three
Group 1
Lesly
Rolando
Mildred
Group 2
Maggie
Steve
Natalie
Group 3
Jasmine
Mani
Anarely
Group 4
Angela
Lucia
Brian
Group 5
Salina
Alfredo
Josefina
Group 6
Alejandra
Jessica
David
Group 7
Group 9
Cruz
Edgar
Elizabeth
Group 8
Jailene
Angelo
Marcela
Group 10
Sandra
Javier
Demetrius
Group 11
Jenny
Marco
Anthony
Areli
Gabino
Janeen
Group 12
Groups of Three and Four
Group 1
Lesly
Rolando
Mildred
Anthony
Group 2
Maggie
Marcela
Natalie
Demetrius
Group 3
Jasmine
Mani
Anarely
Angela
Group 4
Areli
Gabino
Brian
Group 5
Salina
Cruz
Josefina
Group 6
Alejandra
Jessica
David
Lucia
Group 7
Sandra
Javier
Jenny
Marco
Group 8
Jailene
Angelo
Edgar
Elizabeth
Group 9
Alejandro
Janeen
Steve
Alfredo
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
4 Right s Theorem
Perpendicular lines
intersect to form 4 right s.
Theorem 2.10
Right  Congruence Theorem
All right s are .
Theorem 2.11
 Adjacent Right s Theorem
Perpendicular lines form 
adjacent s.
Theorem 3-4
Perpendicular
Transversal Theorem
Postulate 3.2
Slope of  Lines
Postulate 3.2
 and || Lines
Postulate
If a line is  to the 1st of two || lines, then it is
also  to the 2nd line.
2 non-vertical lines are  if and only if the
PRODUCT of their slopes is -1. (In other words,
the 2nd line’s slope is the 1st line’s slope flipped
(reciprocal) with changed sign.)
If 2 lines are  to the same 3rd line, then those
2 lines are || to each other.
IF YOU NEED TO WRITE A PROOF ABOUT ICCE ANGLES or PARALLEL LINES…LOOK AT
THESE:
Postulate 3.1
Corresponding Angles
Postulate (CO s Post.)
If 2 || lines are cut by a transversal,
then each pair of CO s is .
Theorem 3.1
Alternate Interior
Angles Theorem (AI s
Thm.)
Theorem 3.2
Consecutive Interior
Angles Theorem (CI s
Thm.)
Theorem 3.3
Alternate Exterior
Angles Theorem (AE s
Thm.)
Postulate 3.2
Slope of || Lines
If 2 || lines are cut by a transversal,
then each pair of AI s is .
If 2 || lines are cut by a transversal,
then each pair of CI s is
supplementary.
If 2 || lines are cut by a transversal,
then each pair of AE s is .
2 non-vertical lines have the same
slope if and only if they are ||.
Postulate 3.4
Corresponding Angles/||
Lines Postulate (CO s/||
Lines Post.)
Theorem 3.5
Alternate Exterior
Angles/|| Lines Theorem
(AE s/|| Lines Thm.)
Theorem 3.6
Consecutive Interior
Angles/|| Lines Theorem (CI
s/|| Lines Thm.)
Theorem 3.7
Alternate Interior Angles/||
Lines Theorem (AI s/||
Lines Thm.)
Postulate 3.5
|| Postulate
Linear Equation in SlopeIntercept Form
Linear Equation in PointSlope Form
y = mx + b
m = slope, b = y-intercept
y – y1 = m(x – x1)
m = slope,
(x1, y1) = 1 point on the line
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.
If 2 lines are cut by a transversal so
that each pair of CI s is
supplementary, then the lines are ||.
If 2 lines are cut by a transversal so
that each pair of AI s is , then the
lines are ||.
If you have 1 line and 1 point NOT on
that line, ONE and only ONE line goes
through that point that’s || to the 1st
line.
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.
AE
CO
CI
AE
If 2 lines are cut by a transversal so
that each pair of AE s is , then the
lines are ||.
Linear Equation in Standard
Form
AI
CO
If 2 lines are cut by a transversal so
that each pair of CO s is , then the
lines are ||.
AI/
CI