Download It`s the day photographer Alberto Korda took his iconic photo of Che

Opener(s) 3/5
3/5
It’s the day photographer Alberto Korda took his
iconic photo of Che Guevara!!! Happy Birthday
Zhou Enlai, Rex Harrison, Elaine Paige, Andy Gibb
and Eva Mendes!!!
3/5
What to do today:
1. Do opener.
2. Check your homework.
3. Do the exit pass.
4. Note taking and  drawing.
5. Start homework.
OPENER
Find the measure of each angle if
m4 = m5.
Agenda
 Opener and questions (5)
 Homework check and questions (10)
 Exit Pass (5)
 Note-taking: Basic  Thms. (20)
 Proof check: #40, p. 190 (5)
 Individual Work: HW – Text ?s, p. 190, #42 +
#44 (5)
OUR LAST OPENER
m2 = 60. m5 = 70. Find the
other  measures.
Essential Question(s)
 How do I classify triangles?
 How do I determine missing s in a ?
Objective(s)
 Students will be able to (SWBAT) give a name
to a triangle based on its sides.
 SWBAT give a name to a triangle based on its
angles.
 SWBAT determine a missing triangle side
length or angle measure using algebra and a
triangle’s classification.
 SWBAT determine a missing angle measure
based on the “A  = 180” theorem.
3
1
2
4
5
6
7
Exit Pass
Look at the 2 skaters and the house roof in your homework.
Look at every  measure that’s OUTSIDE a  (an EXTERIOR )
and the 2 ‘far-away’  measures inside the  it touches (the
REMOTE INTERIOR s). Do you notice a relationship? What is
it?
Our Last Exit Pass
FGH is equilateral with FG = x + 5, GH = 3x – 9 and FH = 2x -2.
Find x and the measure of each side of the .
Homework
 Text ?s, p. 190, 42 + 44
Period
8
Agenda writer: Alfredo (4x), Jenny (2x)
Opener answerer: Angela
ACHIEVE Manual
distributor:
Timekeeper: Demetrius (11x)
Presenter:
Filer: Demetrius, Alfredo (2x), Edgar,
Steven, Jailene (3x), Areli, Jenny,
Jessica, Brian (2x), Alejandra, Rolando
Tools Distributor: Steven (2x), Angela,
Salina, Brian (2x), Sandra, Jessica,
David
Notes: Some Basic  PTCs (Postulates, Theorems and/or Corollaries)
Theorem 4-1:
 Sum Theorem
( Sum Thm.)
The _______ of the
Theorem 4-2:
3rd  Theorem
(3rd  Thm.)
If 2 s of one  are
3/5
m1 + m2 + m3 = 180
____________ of the
s of a  is 180.
___________ to 2 s of a
If A  F and
C  D, then
B  E
2nd , then the
_____________ s of
both s are .
Theorem 4-3:
Exterior  Theorem
(Ext.  Thm.)
The measure of an
mYZP = mX + mY
_______________  of a
 = the sum of the
measures of the 2
_____________
_____________ s.
Corollary 4-1:
Acute  Right  Corollary
The ____________ s of
a ______________  are
complementary.
Corollary 4-2:
1 Obtuse/Right  Corollary
A  can have, at most, 1
____________ or
____________ .
mG + mJ = 90
Notes: Some Basic  PTCs (Postulates, Theorems and/or Corollaries)
3/5
Theorem 4-1:
 Sum Theorem
( Sum Thm.)
The sum of the measures
of the s of a  is 180.
m1 + m2 + m3 = 180
Theorem 4-2:
3rd  Theorem
(3rd  Thm.)
If 2 s of one  are  to 2
s of a 2nd , then the 3rd
s of both s are .
If A  F and
C  D, then
B  E
Theorem 4-3:
Exterior  Theorem
(Ext.  Thm.)
The measure of an ext. 
of a  = the sum of the
measures of the 2 remote
int. s.
mYZP = mX + mY
Corollary 4-1:
Acute  Right  Corollary
The acute s of a right 
are complementary.
mG + mJ = 90
Corollary 4-2:
1 Obtuse/Right  Corollary
A  can have, at most, 1
obtuse or right 
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