Download Keys GEO SY14-15 Openers 4-30

Geometry Opener(s) 4/30
TODAY’S OPENER
4/30
It’s Bugs Bunny Day, Dia de Los Ninos, International
Jazz Day, National Adopt a Shelter Pet Day, National
Animal Advocacy Day, National Honesty Day, National
Raisin Day, Spank Out Day and Walpurgis Night!!!
Happy Birthday Carl Gauss, Alice B. Toklas, Eve
Arden, Percy Heath, Cloris Leachman, Annie Dillard,
Jill Clayburgh, Lars von Trier and Kirsten Dunst!!!
What is the
value of x?
a. Make a
conjecture.
b. Provide some
mathematical
evidence
supporting or
negating your conjecture.
c. Write a sentence explaining why your
evidence supports or negates your
conjecture.
Agenda
1. Opener (8)
2. Video: http://www.pbslearningmedia.org/resource/90e224b5-564d-47f0-956658a11370dee9/architecture-trigonometry/ (7)
3. Practice 1/Notes 1: Sohcahtoa PPT completion
(10)
4. Practice 2/CW: Trig ratios investigation – Wksht 74, p. 369 (20)
5. HW Assignment: p. 371-2, #1-2 on each (5)
6. Computer Room Instructions (7)
7. Notes 2: Double Entry Journal till ‘Unit Circle’
https://www.mathsisfun.com/algebra/trigonometry.html# (20)
8. Ind. Work: Chart Completion (10)
9. Calculator: http://www.mathworksheetsgo.com/trigonometry-
THE LAST OPENER
What is the value of x?
a. Make a conjecture.
b. Provide some mathematical
evidence supporting or negating your
conjecture.
c. Write a sentence explaining why
your evidence supports or negates
your conjecture.
calculators/sine-cosine-calculator-online.php
10. CW: 8 problems/8 solutions
https://www.khanacademy.org/math/geometry/right_triangles_topic/ccgeometry-trig/e/trigonometry_0.5 (15?)
11. Exit Pass (5?)
Standard(s)
Essential Question(s)
 CCSS-HSG-SRT.C.6: Understand that by similarity, side ratios in right
triangles are properties of the angles in the triangle, leading to
definitions of trigonometric ratios for acute angles.
 CCSS-HSG-SRT.C.7: Explain and use the relationship between the
sine and cosine of complementary angles.
 CCSS-HSG-SRT.C.8: Use trigonometric ratios and the Pythagorean
Theorem to solve right triangles in applied problems.




How Do I (HDI) use PYT?
HDI use the converse of PYT?
HDI find trig ratios using right triangles?
HDI solve problems using trig ratios?
Objective(s)
 Students will be able to (SWBAT) identify a PYT triple.
 SWBAT calculate the side lengths of a  using PYT.
 SWBAT use the convers of PYT.
 SWBAT identify the opp. side, adj. side and hyp. In a right .
 SWBAT determine the sin, cos and tan ratios for an  in a


right .
SWBAT find values of trig functions for acute angles.
SWBAT find angle measures of right triangles.
Exit Pass (12/11 – 13/14)
The Last Exit Pass
For numbers 1-6 find the coordinates of each
image.
1. Rx-axis (A)
2. Ry-axis (B)
3. Ry = x (C)
4. Rx = 2 (D)
5. Ry= -1 (E)
6. Rx = -3 (F)
HOMEWORK Period 1
Complete Wksht. 7-4, p. 369; Wksht. 7-4, p. 371, #1-2; Wksht. 7-4, p. 372, #1-2; Special Triangles Trig Ratios
HOMEWORK Period 7
No homework tonight.
HOMEWORK Period 2A
Complete Wksht. 7-4, p. 369
HOMEWORK Period 3
Complete Wksht. 7-4, p. 369; Wksht. 7-4, p. 371, #1-2; Wksht. 7-4, p. 372, #1-2; Special Triangles Trig Ratios
HOMEWORK Period 5
No homework tonight.
HOMEWORK Period 8
No homework tonight.
Extra Credit
Period 1 Period 2A Period 3
Alex H. (4x)
Israel A. (4x)
Jocelyn L. (4x)
Amal S. (6x)
Angel V.
Victor C. (4x)
Israel H. (3x)
Stephanie L.
(4x)
Yazmin C.
Melissa A. (2x)
Evelyn A. (2x)
Mirian S.
Alexis S.
Brandon S. (5x)
Gabriel M. (2x)
Rodrigo F. (6x)
Josh P. (5x)
Alejandra G.
(4x)
Jaime A. (5x)
Jacob L. (9x)
Anthony P. (3x)
Enrique G.
Abrahan G.
Alex A. (2x)
Jocelyn J. (2x)
Michelle S.
Carlos O.
Arslan A.
Ronny V. (2x)
Rosie R. (4x)
Javier L. (2x)
Gaby O. (3x)
Claudia M. (2x)
Omar R. (2x)
Ricardo D. (2x)
Josue A.
Alicia R.
Javier D. (2x)
Catalina Z. (2x)
Luis H.
Period 5
Period 7
Period 8
Antonio B. (3x)
Liz L. (2x)
Anadelia G.
Jose C. (5x)
Jose B. (2x)
Jesus H. (3x)
Carlos L. (2x)
Jose D.
Rob C.
Briana T. (3x)
Jose G. (2x)
Eraldy B.
Rogelio G. (2x)
Aurora G.
Jorge L.
Kamil L.
Alfredo F. (2x)
Gustavo C.
Ana R.
Ruby L. (3x)
Danny G.
Adriana H. (3x)
Gabriela G. (5x)
Jackie B.
Julian E. (2x)
Cristian A
Jocelyn C. (4x)
Lily F.
Cynthia R. (3x)
Jessica T.
Jorge C. (2x)
Santi H.
Kevin A. (2x)
Andrea N. (3x)
Ernesto M.
Maria M. (2x)
Fernando V. (2x)
Alejandra P.
Joe C.
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
Perpendicular lines
Theorem 3-4
If a line is  to the 1st of two || lines,
Perpendicular Transversal Theorem
4 Right s Theorem
intersect to form 4 right s.
then it is also  to the 2nd line.
Theorem 2.10
Postulate 3.2
All right s are .
2 non-vertical lines are  if and only if the PRODUCT of their
Right  Congruence Theorem
Slope of  Lines
slopes is -1. (In other words, the 2nd line’s slope is the 1st line’s
slope flipped (reciprocal) with changed sign.)
Theorem 2.11
Perpendicular lines
Postulate 3.2
If 2 lines are  to the same 3rd line, then thhose 2
 Adjacent Right s Theorem
form  adjacent s.
 and || Lines Postulate
lines are || to each other.
Theorem 4-6
Theorem 4-7
If the 2 legs of one right  are  to
If the hypotenuse and acute  of one right
Leg-Leg (LL) Congruence
Hypotenuse-Angle
the corresponding parts of another
 are  to the corresponding parts of
(HA) Congruence
right , then both s are .
another right , then both s are .
Theorem 4-8
Postulate 4-4
If the hypotenuse and one leg of one right
If the leg and acute  of one right  are
Leg-Angle (LA)  to the corresponding parts of another
Hypotenuse-Leg (HL)
 are  to the corresponding parts of
Congruence
Congruence
another right , then both s are .
right , then both s are .
IF YOU NEED TO WRITE A PROOF ABOUT ICCE ANGLES or PARALLEL LINES…LOOK AT THESE:
Postulate 3.1
If 2 || lines are cut by a
Postulate 3.4
If 2 lines are cut by a transversal
Corresponding Angles
transversal, then each pair of CO
Corresponding Angles/|| Lines
so that each pair of CO s is ,
Postulate (CO s Post.)
s is .
Postulate (CO s/|| Lines Post.) then the lines are ||.
Theorem 3.1
If 2 || lines are cut by a
Theorem 3.5
If 2 lines are cut by a transversal so
Alternate Interior Angles
transversal, then each pair
Alternate Exterior Angles/|| Lines
that each pair of AE s is , then the
Theorem (AI s Thm.)
of AI s is .
Theorem (AE s/|| Lines Thm.)
lines are ||.
Theorem 3.2
If 2 || lines are cut by a
Theorem 3.6
If 2 lines are cut by a transversal
Consecutive Interior Angles
transversal, then each pair
Consecutive Interior Angles/|| Lines so that each pair of CI s is
Theorem (CI s Thm.)
of CI s is supplementary.
Theorem (CI s/|| Lines Thm.)
supplementary, the lines are ||.
Theorem 3.3
If 2 || lines are cut by a
Theorem 3.7
If 2 lines are cut by a transversal so
Alternate Exterior Angles transversal, then each pair
Alternate Interior Angles/|| Lines that each pair of AI s is , then the
Theorem (AE s Thm.)
of AE s is .
Theorem (AI s/|| Lines Thm.)
lines are ||.
Postulate 3.2
2 non-vertical lines have the same
Postulate 3.5
If you have 1 line and 1 point NOT on that
Slope of || Lines
slope if and only if they are ||.
|| Postulate
line, ONE and only ONE line goes through
that point that’s || to the 1st line.
Theorem 6.6
Theorem 6.4
A midsegment of a  is || to one
In ACE with ̅̅̅̅̅
𝑩𝑫 || ̅̅̅̅
𝑨𝑬 and
 Midsegment Thm.

Proportionality
Thm.
intersecting
the
other 2 sides in distinct
side of the , and its length is ½
̅̅̅̅
𝑩𝑨 ̅̅̅̅
𝑫𝑬
the length of that side.
points, = .
̅̅̅̅
𝑪𝑩
̅̅̅̅
𝑪𝑫
Postulates and Theorems to IDENTIFY CONGRUENT TRIANGLES: SSS, ASA, SAS or AAS
Postulates and Theorems to IDENTIFY SIMILAR TRIANGLES: AA, SSS or SAS
Linear Equation in
Slope-Intercept
Form
Linear Equation in
Point-Slope Form
y = mx + b
m = slope, b = yintercept
y – y1 = m(x – x1)
m = slope,
(x1, y1) = 1 point on
the 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
Degrees
0°
30°
45°
60°
90°
120°
135°
150°
180°
210°
225°
240°
270°
300°
315°
330°
360°
Special Angle Trig Ratios
Sine
Cosine
Tangent
0
1
0
Some Of Hancock’s Classes Are Hard To Obtain A’s (in)
I know about
PYT…
If you have the measures of 2 sides of a right triangle, you can find the 3 rd
by using:
12
a 2 + b2 = c 2
15
but what happens
if I only know an
angle measure
and 1 side?
Essential
Question(s)
 How do I (HDI)
find the measure
of ‘immeasurable’
distances in a
triangle?
Objective(s)
 Students will be
able to (SWBAT)
name the three
trigonometric
ratios and their
corresponding
fractions.
 SWBAT
determine
numerical
trigonometric
ratios in a
triangle.
 SWBAT find
missing side
lengths in a
triangle using
trigonometric
ratios.
If you have the measure of 1 side and 1 angle of a right triangle, you can
find the other sides by using:
12
SOHCAHTOA
15°
Some Of
Hancock’s
SOH
Classes Are
Hard
CAH
To Obtain
A’s (in)
TOA
Sine
(Sin)
Cosine
(Cos)
Tangent
(Tan)
Opposite
Hypotenuse
𝑶𝑷𝑷
𝑯𝒀𝑷
Adjacent
Hypotenuse
𝑨𝑫𝑱
𝑯𝒀𝑷
Opposite
Adjacent
𝑶𝑷𝑷
𝑨𝑫𝑱
The HYPOTENUSE is the looooooooooongest side of the triangle.
The OPPOSITE side is the side across from the known angle.
The ADJACENT side is the side touching the known angle.
JENNY’S FISHTAIL
Step 1.
Draw an arrow from the known (NON-90°)  to the side
across from it. That’s the OPP (opposite side).
Step 2.
Draw an arrow from that arrowhead around the right 
box. That’s the ADJ (adjacent side).
Step 3.
Draw an arrow from that arrowhead to the longest
side. That’s the HYP (hypotenuse).
Solving Δ Problems with SOHCAHTOA
Let’s recap…
1st, remember we are working with RIGHT triangles!!!
YOU HAVE:
2 sides of a Δ
YOU WANT:
The 3rd side of the Δ
2 sides of a Δ
A non-right  of the
Δ
Another side of the Δ
1 side & 1  of a Δ
1 side & 1  of a Δ
To solve the Δ…find
ALL the missing
parts
YOU USE:
PYT (a + b = c )
SOHCAHTOA
SOHCAHTOA
SOHCAHTOA
then PYT
2
2
2
SOHCAHTOA
So what sort of
problems will I
encounter?
Some Of
Hancock’s
SOH
Classes Are
Hard
CAH
To Obtain
A’s (in)
TOA
A
Sine
(Sin)
Cosine
( os)
Objective(s)
 Students will be
able to (SWBAT)
name the three
trigonometric ratios
and their
corresponding
fractions.
𝑨𝑫𝑱
𝑯𝒀𝑷
Adjacent
Hypotenuse
Tangent
(Tan)
𝑶𝑷𝑷
𝑨𝑫𝑱
Opposite
Adjacent
A
52°
Essential Question(s)
 How do I (HDI) find
the measure of
‘immeasurable’
distances in a
triangle?
𝑶𝑷𝑷
𝑯𝒀𝑷
Opposite
Hypotenuse
A
52°
X°
?
X
8
9
?°
C
6
?°
B
You know an  and a side…
You want to find a side…
1. Draw your arrows from the
 and label OAH.
2. Match OAH with
SOHCAHTOA.
3. Set up your equation.
4. Find decimal for trig ratio.
5. If X is in
denominator…DIVIDE side
by trig decimal.
12
C
X
You know an  and a side…
You want to find a side…
1. Draw your arrows from the
 and label OAH.
2. Match OAH with
SOHCAHTOA.
3. Set up your equation.
4. Find decimal for trig ratio.
5. If X is in
numerator…MULTIPLY
side by trig decimal.
?°
B
C
You know two sides…
You want to find an …
1. Draw your arrows from
the  and label OAH.
2. Match OAH with
SOHCAHTOA.
3. Set up your equation.
4. Find decimal for trig ratio.
5. On your calculator, hit
BLUE BUTTON, hit trig
ratio and type in decimal.
6. Hit ENTER.
B
Practice / Homework
Pits, Hips, Legs and ABCs
Where do you find
pits, hips, legs and
ABCs?
In a…
RIGHT TRIANGLE
Let’s label the parts:
Essential
Question(s)
 How do I (HDI)
use PYT?
or
Objective(s)
 Students will be
able to (SWBAT)
or
name the parts
of a right
triangle.
 SWBAT label
the HYP of a
right triangle.
 SWBAT label
or
the LEGs of a
right triangle.
 SWBAT label
What’s the relationship between the LEGs and the HYP?
the parts of a
right triangle
with ABC.
The Pythagorean Theorem
 SWBAT use
PYT to find a
which is, in formulaic form, …
right triangle’s
missing length.
+
=
Classwork/Homework
7.
8.
http://www.youtube.com/watch?v=5l_0BqSmXwA
http://www.youtube.com/watch?v=pSgUvclFnrQ
https://www.youtube.com/watch?v=_OJ1yVs0aQE
2
3
2
3
12
Imagine a 9th Hopewell Triangle…Triangle I with hypotenuse
of length 15. What would be the length of the shortest leg if
Triangle I and Triangle A are similar? Give your answer
correct to one decimal place.
Show your calculations.