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Transcript
Opener(s) 2/28
2/28
It’s Rare Disease Day!!! Happy Birthday nylon, Ben
Hecht, Linus Pauling, Vincente Minnelli, Bugsy
Siegel, Zero Mostel, Frank Gehry, Tommy Tune
and Robert Sean Leonard, !!!
=
Agenda
 Opener and questions (5)
 Feedback: Check classwork and homework (7)
 Lecture: You schmooze…you kaffeeklatsch,
you lose (3)
 Cognitive Tutor (25)
 Individual Work 1: Wksht. 4-2. p. 189 or
homework (5)
 Exit pass (5)
2/28
What to do today:
1. Do opener.
2. Collect classwork from yesterday.
3. Check homework.
4. Discuss some of your sad, d- and f-producing
work ethic toward classwork.
5. Cognitive Tutor (I will put in CT grades over
the weekend!).
6. Work on homework.
7. Do exit pass.
OPENER
Find the measure of each numbered
angle.
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.
OUR LAST OPENER
A  has 3 coordinates: (2,2), (3,9)
and (-5, 3). Use the distance formula to
find the length of each side (leave it in
square root form!) then classify the  by
sides.
Exit Pass
Look at 6-13 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, 28-38 + 40
Period
8
Agenda writer: Alfredo (2x), Jenny (2x)
Opener answerer:
ACHIEVE Manual
distributor:
Timekeeper: Demetrius (8x)
Presenter:
Filer: Demetrius, Alfredo, Edgar,
Steven, Jailene, Areli, Jenny, Jessica,
Brian, Alejandra
Tools Distributor: Steven, Angela,
Salina
I should see 2
(TWO)
calculations!!!
I should see 4
(FOUR)
calculations!!!
I should see 3
(THREE)
calculations!!!
I should see 1
(ONE)
calculations!!!
I should see 4
(FOUR)
calculations!!!
I should see 2
(TWO)
calculations!!!
I should see 2
(TWO)
calculations!!!
I should see 4 (FOUR)
calculations!!!
I should see 6 (SIX)
calculations!!!
I should see 8 (EIGHT)
calculations!!!
I should see 4 (FOUR)
calculations!!!
I should see
an algebraic
equation!!!
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