Opener(s) 1/4
1/4
It’s the eleventh day of Christmas…11 pipers
piping!!! Happy Birthday Julia Ormond, Michael
Stipe and Jacob Grimm of The Brothers Grimm!!!
1/4
What to do today:
1. Do opener.
2. Understand quiz mistakes.
3. Check winter break homework.
4. Write algebraic and geometric proofs.
OPENER
Agenda
 Opener (5-10)
 Lecture: More translating words into
symbols (5-10)
 Quiz Feedback: Question and Answer (5)
 HW Check: Winter Break √ (10)
 Notes: Proof Cheat Sheet (5)
 Practice: Seven proofs from 756-758 (1020)
Essential Question(s)
 How do I transform word problems into
algebraic equations?
 How do I prove a conjecture is true?
Objective(s)
 Students will be able to (SWBAT) translate
words into mathematical symbols.
 SWBAT solve % word problems.
 SWBAT set up a proof.
 SWBAT prove a conjecture using
definitions, properties, postulates and
theorems.
Find your new seat then begin
working on your 2nd percentage story
problem worksheet.
OUR LAST OPENER
1 and X are complementary,
2 and X are complementary,
m1 = 2n + 2, m2 = n + 32
X
Find n.
Find m1.
What is m2?
Find mX.
1
2
Exit Pass
What conclusion can you make about the sum of
m1 and m4 if m1 = m2 and m3 = m4?
Explain.
3
2
1
4
Our Last Exit Pass
If E is between Y and S, write one true and one false
statement about the segments that they form. Then explain
why your true statement is true and your false statement is
false.
Homework

Solve 5 proofs from pages 756-758:
a. Lesson 2-1: #3 & #4 (Write the conjecture then prove it.)
b. Lesson 2-6: #4
c. Lesson 2-7: #9 & #10
d. Lesson 2-8: Choose #1, #2 or #3, solve for x then write a
proof with your solution as the last line & #8
Period
8
Agenda writer: Alfredo (3x), Steven
(4x), Jenny(4x), Mani, Angela
Opener answerer: Brian, Jenny (2x),
Angela, Mani, Maggie, Sandra
ACHIEVE Manual
distributor:
Timekeeper: Mildred (2x), Edgar (12x),
Steven
Presenter: Alfredo
Filer: Jenny (3x), Jailene (7x), Brian
(4x), Sandra (2x), Steven (3x), Angela
(2x), Jasmine, Jessica, Gabino, Areli,
Josefina, Alfredo, Edgar
Tools Distributor: Edgar, Anarely,
Gabino, Alfredo, Alejandra
Models
ANGLE ADDITION
Looking at the Grand Union Flag…
Given: mFLA = 44, mFLG = 88
Prove: mALG = 44
SUPPLEMENTARY ANGLES
Given: 1 and 2 form a linear pair,
m2 = 67
Prove: m1 = 113
COMPLEMENTARY ANGLES
VERTICAL ANGLES
Given: 1 and 3 are complementary, Given: 1 and 2 are vertical angles,
2 and 3 are complementary
m1 = x, m2 = 228 – 3x
Prove: 1  2
Prove: m1 = m2 = 57
m
W
X
V
Y
1
2
4
3
l
Z
BISECTED ANGLES
Given: Ray VX bisects WVY,
Ray VY bisects XVC
Prove: WVX  YVZ
RIGHT ANGLES
Given: l  m
Prove: 1, 2, 3 & 4 are right s
(This is Theorem 2.9!!)
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
DEFINITION OF
CONGRUENCE
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
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
Whenever you change from
 to = or from = to .
IF YOU NEED TO WRITE A PROOF ABOUT ANGLES…LOOK AT THESE:
Postulate 2.11
The  Addition
Postulate
Theorem 2.3
Supplement Theorem
Theorem 2.4
Complement Theorem
Theorem 2.5
The Equalities Theorem
Theorem 2.6
The  Supplements
Theorem
Theorem 2.7
The  Complements
Theorem
If R is in the interior of PQS,
then mPQR + mRQS = mPQS.
THE CONVERSE IS ALSO TRUE!!!!!! Q
If 2 s form a linear pair,
then they are
P Q
supplementary s.
If the non-common sides of
2 adjacent s form a right ,
then they are complementary s. Q
R
S
P
Congruence of s is
Reflexive, Symmetric & Transitive
s supplementary to the
same  or to  s are .
(If m1 + m2 = 180 and
1
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.)
2
3
2
1
3
Theorem 2.8
Vertical s
Theorem
Theorem 2.9
4 Right s
Theorem
Theorem 2.10
Right 
R Congruence
S Theorem
Theorem 2.11
 Adjacent Right
s Theorem
Theorem 2.12
 Supplementary
Right s
Therorem
Theorem 2.13
 Linear Pair
Right s
Therorem
P
R
S
If 2 s are vertical, then they are .
(1  3 and 2  4)
1 2 3
4 4
Perpendicular lines intersect to form
right s.
All right s are .
Perpendicular lines form  adjacent s.
If 2 s are  and supplementary, then
each  is a right .
If 2  s form a linear pair, then they
are right s.