Download Keys GEO SY13-14 Openers 4-3

Geometry Opener(s) 4/3
4/3
It’s National Chocolate Mousse Day, National Find-ARainbow Day, Don’t Go To Work Unless It’s Fun Day,
Corrupt Society Day, Tweed Day, Pony Express Day,
American Circus Day and Hospital Admitting Clerks
Day!!! Happy Birthday Picabo Street, Eddie Murphy,
David Hyde Pierce, Alec Baldwin, Sandra Boynton,
Tony Orlando, Jane Goodall, Doris Day, Marlon
Brando, Henry Luce, Leslie Howard, Dooley Wilson
and Washington Irving!!!
4/3
What to do today:
1. Do the opener.
2. Work on Cognitive Tutor.
3. Do the exit pass.
TODAY’S OPENER
Find x AND the indicated sides.
Agenda
1. Opener (5)
2. Individual Work: Cognitive Tutor (41)
3. Exit Pass (5)
Standard(s)

CCSS-M-G-SRT.5: Use congruence and similarity criteria for
triangles to solve problems and to prove relationships in
geometric figures
Essential Question(s)
 How do I use similarity and congruence to find
numerical relationships among triangle parts?
Objective(s)
 Students will be able to (SWBAT) establish the
congruence or non-congruence of two geometric
figures.
 SWBAT establish the similarity or non-similarity of
two geometric figures.
 SWBAT find missing angle measures using
congruence or similarity.
 SWBAT find missing side measures using
congruence or similarity
The Last Opener
Find x so that l || m.
Exit Pass
The Last Exit Pass
HOMEWORK Period 1
Text ?s, p. 311, #6-8, 12.
HOMEWORK Period 6
Text ?s, p. 311, #6-8, 12.
Extra Credit
Period 1
Period 6
Marisol
Mireya
Valerie
Imelda
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
Theorem 2.10
Right  Congruence Theorem
Theorem 2.11
 Adjacent Right s Theorem
Theorem 4-6
Leg-Leg (LL) Congruence
Theorem 4-8
Leg-Angle (LA)
Congruence
Perpendicular lines
intersect to form 4
right s.
All right s are .
Theorem 3-4
Perpendicular
Transversal Theorem
Postulate 3.2
Slope of  Lines
Perpendicular lines
form  adjacent s.
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.
Postulate 3.2
 and || Lines
Postulate
Theorem 4-7
If the 2 legs of one right  are  to
Hypotenuse-Angle
the corresponding parts of another
(HA) Congruence
right , then both s are .
Postulate 4-4
If the leg and acute  of one right 
Hypotenuse-Leg (HL)
are  to the corresponding parts of
Congruence
another right , then both s are .
If the hypotenuse and acute  of one right
 are  to the corresponding parts of
another right , then both s are .
If the hypotenuse and one leg of one right
 are  to the corresponding parts of
another right , then both s are .
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
If 2 lines are cut by a transversal so
that each pair of CO s is , then the
lines are ||.
If 2 lines are cut by a transversal so
that each pair of AE s is , then the
lines are ||.
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.
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
Notes: Parallel Postulates
1
3-21
2
3
4
5
6
7
CO s
|| lines
Postulate
AE s
|| lines
Postulate
CI s
|| lines
Postulate
AI s
|| lines
Postulate
 lines
|| lines
Postulate
Parallel
Postulate
8
If corresponding s are , then lines are ||.
If alternate exterior s are , then lines are ||.
If consecutive interior s are supplementary, then lines
are ||.
If alternate interior s are , then lines are ||.
If 2 lines are  to the same line, then lines are ||.
You are given a line and a point not on the line. There is
ONLY ONE line through that point that’s || to the given
line.