x/ÿj, liÿfjÿ,r [/[1)ÿ/.f'!!) -points that ÿ on the ,jt .... 1 notes... · the length of a...
TRANSCRIPT
Nal'nÿ ÿ/ÿ l/d-. o
MDLDate
Fill in Notes 1,1
Period
Notes 1.1- Points, Lines, & Planes
Quadrant II Quadrant ; Ordered Pair: (x, y)
x=coordinate:
Quadrant III Quadrant IVy-coordfnate:
X/ÿJ, LIÿfJÿ,r [/[1)ÿ/.f'!!) -points that ÿ__ on the ,Jt/!ÿlt, ].,,i!lÿ.,[
Picture:
- points that tÿl(l .ÿII{, l/lil on the ÿIVÿ/I, J /
AI' it takes is for [)ÿIO point/ within the set of points to be/ .
/ ÿ the line to be consideredNfl; K,
Picture: ÿ (ÿ,/
,14ÿf'''11ÿ
E× 1:lie on the line,
Find the coordinates of three points that lie on the graph of y = 3x+5 and one point that doesn't
, d
-5 :< riÿ 1,LI, }
JL5
EX 2: Do points R(4, 8) and S(-3, -7) lie on the line with the equation y = 2x - 1?
Guided Practice:1, Write the ordered pair for points
T
u 1
i
.... ÿ4m', .... L.ÿ., , • 4-° ...... ' ....... ',1 , ÿ'
t'-, d i ÿ: I 13 I Iÿ 1 ,
i
.... ' ...... =} ............... E1'',- ÿ" ". ........ 4ÿ,
1
f
2, Which quadrant is point C located in? point D? ....
3, Does the point(2,5) lie on the graph of y = 3x+67 I\lU
5 = Iÿ f-bÿk-lm '/, kid<"
4, Does the point (-4, -6) lie on the graph of y = 3x + 6?
{!ÿ ,:: ><,+ {ÿ,• -- Irÿ ...... IZ +,/,ÿ,ÿ
Vocabulary:
Undefined Terms - no formal
- an
by a
Written as:
picture:
°A
- collection of
extending directions,
Written as:Picture:
/./
//1
in all directions.
Written as:Picture:
- Points that in the
Picture:
ll it takes is forÿ pointÿÿwithin the set of points to be ÿ xÿ
e plane to be considered /
Defined Terms- terms that can be t/]ÿ,, ,
such as p ÿ)!iÿ or
- part of a lineL
al, _ÿ4/ÿ/ÿiÿ , on the line the endpoints,
and
Picture:
J
Written as:
-part of a line with ÿ endpoint that ,[4Xÿ[//¢ÿ ÿ
T be written as ÿ._@ÿ Theÿÿ
is always the first letter 'ÿ
the notation for a ray, The )he ray is pointing Is always /
.ÿ2nÿ letter written'
Picture:
If point C lies on A--Bbetween A & B, then C-A'and C-B"are opposite rays.
Picture:
at exactly
point,
Picture:
ii " • ........... ÿ...... ..... '-bI
u9
, __- two lines that
0
Line AB is perpendicular to line CD
Picture:
7
,o,.,,o. for PIlÿYbtÿ(ÿ£I
Line AB is parallel to line Cb
- two lines that,
Picture:
intersect, .,Fj fÿ!l tj,ÿl ÿ,
4
Pictur______£: ÿ_ÿ ÿ..ÿ ÿ.,
Line m intersects line k at point F
and a intersect at a
Line k intersects plane A at point B
planes intersect at a
Picture:ÿ,ÿ
° I
Plane M and plane C intersect at line AB
EXAMPLE: FinishJ
the sketch.
line in the plane line not in the plane
III
line intersects at one Dÿint1
NamÿMDL
Date, Period
Fill in Notes 1.2
Notes 1.2 - Segments & Congruence
The length of a segment is ÿhÿI OÿA ÿ ÿ between its IÿAIÿ)
Draw an example of a point that would be between A and C,
The measure of AB is written as
Zt iS the 0!ÿk5 ÿ (ÿ i bÿtween i ÿ and ÿ i' So the measure of a ,,.<ÿ,#ÿ/!,,#ÿJA ifÿ i ÿ
its two endpoints.
Is the distance from A to B different than the distance from B to A?
....
\!lÿ ,, ÿ,, Why or why not?
A postulate is a
The Ruler Postulate:
The numbers on a ruler are a
between twopointsonanumberlineisthe ÿ(!lÿt'I/ ÿI/ÿlÿ/ÿI.,/
of the difference of the coordinates. It can be found using:
41 ¢ ÿ, ÿ AB: zÿ° I
EX 1:x
< I I I-+-f-, ,ÿ-q-,, I I I I I I i I ,ÿ-+-f-ÿ-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
EX____Z2: Find PQ, QR, and PR if P is located at -3, Q is located at 1, and R is located at 6,
Y-=ll--ÿl=1-ÿ1==5 ÿ 9"
Seqment Addition Postulate: If Q is
P and R.
P q
F----PQ I Qÿ
small segment + small se.xjmÿnt = big scgnlÿnt
EX 3: Find LM if L is between N and M, NL = 6x- 5, LM = 2x + 3, and NM = 30,
I'4IJÿ-LIÿ4, ÿ t',t ÿA 5-> xÿ q-
y,]
• 7;,(ÿ 7y, :ÿ,}7ÿo, .... ÿ-ff;771E×___44: Find the measure of MN if N is between M and P, MN = 3× + ÿ7 and MP = 5x,
lÿi,l+klP':Mr> /> 'tO 7.)t i,ltl,.I-O(/d)-i 7.g # "@ Iÿft., ¢1
'I " ......... 1 r'*ti'l : %2'ÿj-p <-at----------- 'ÿ'1,-t.. 70 :J.!)X. " ..... -ÿ-
Coÿ_.ÿrLleflceÿ
Congruent means .__ÿ#i) ÿ-'I} and is written as
Co2D_g_ruent semi_s- segments that have the
S,\
congruency marks
Lengths are equal,
Aÿ -- (.,[.ÿ:>"iÿ equal to"
Segments are congruent,
"is congruent to"
EXÿ: Use the picture below to find AB if K is between A and B, AK = 2x + 10 and KB = 5x + 4,
A KX t- I0
B
,ÿ/n)//ÿ: y//<'>
/ÿ-B- IV- ÷/V- --- 2ÿ .< .......... __ .......... .,ÿ
Name Iv,ÿ Date PeriodMDL(] Fill in Notes 1.3
Notes 1.3 - Midpoint & Distance FormulasI
*Remember: To find *he ÿ from one endpoint to another ona number line use the iJrÿ]l,ÿr ÿl)ÿÿ,ÿ[/ :,
AB: IX,r)(,I
- the midpoint of a segment is the point thatthe segment into ÿ//) _ congruent[ ÿ )segments.
the midpoint M of PQ is the pointbetween P and Q such that PM : MQ 8
P
Number Line Midpoint Formula:To determine the midpoint on a number line add the two endpointsand divide by 2,
EX 1:X A Y B
<t111 l l l l I I 11111-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
What is the coordinate of the midpoint of AB? -ÿ# ÿ ÿ.
illWhat is the coordinate of the midpoint XY?
means to . in
A bÿ V/ÿ/ff, r/Y,\bt?!O- is a point, rp.v, line, llne segment, or plane thata segment at its ÿ, A midpoint or segment
bisector bisects a segment.
AI]ÿx.'l -: f4ÿ!'> ÿ,,Ad., tlH .- H l;.ÿ,
wEX___22: Line RM bisecÿ LK at point b. Find DK if LK is 16 inches.
\ !IL.-ÿ ........... . ......... o°[
EX 3: U is the midpoint of XY. If XY = ;16x - 6 andUY = 4x + 9, find the value of x and XY.
I{ÿ )t-b
7, ÿ,<q bt,, ÿtx+'] \/
Xu, t+//ÿ,/=X'7h<t t <1j, ÿ ty, P} ÿ-ÿ' IIo X-{,:
'77= 7;,:*"7 X
+,
)t\/=
EX 4: If P is the midpoint of Cb and PC = x and PD = 5x - 4,find the value of x and the measure of CD.
c x F ÿX.Ur p
= FP
P6- P ÿ 76= 77, :: D7 ÿ LI Pc., :: I
Pl):: I..... q>::-"-I o
o o ÿ,p=14-1
CoÿCÿoJardinate plane midpoint formula:
The coordinates of the of a segment whoseendpoints are (xl, yl) and (x2, yz) are
EX___55: The endpoints of RS are R(I, -3) and S(4, 2). Find the midpoint.
EX 6; The midpoint of RQ in the graph below is P(4, -1),
Find the( 0)c°°rdinates of R if Q is at (3,-2)rÿiÿll ÿ ÿÿ ÿbÿ ÿ ÿ ÿ1 ÿ ÿ ÿ. Jm{ÿlÿ--]ÿ'ÿ&0 { I;
I2L'T !-":
}ÿi-ÿ--ÿ v-.i-4.-q," 4...,-
-TI , ÿ ' ' T-'z
-ÿ-+-, I ....... >4-,,-1
7:;. ÿ2ÿ T L 5<ÿb(ÿ,bh,ÿcÿcu
EX 7 What happens when you are missing an endpoint?)ÿlfÿ Iÿjÿ t)lÿIIÿIÿ #The midpoint of JK is M(2, 1). One endpoint is J(1, 4). ()Find the coordinates of endpoint K.
,ÿ
t
Co_oral)hate Plane Distance For u_m_uÿla:
The ÿ between 1-wo points (xl, Y0 and (xz, yz) on a coordinateplane can be found using the JÿfSÿTxÿ4 ÿ, ÿÿ.
d Uÿ Z
EX 8: Find the length of RS if R(1, -3) and S(4, 2),
TiLt: : iiiÿ ÿ: i ÿ::i:ÿiÿ"z "]Fÿ,,,;t .ÿ r-lZ.
V ,2 r" 0
...... L.!/ÿs:ÿ
t lEX £: Determine if the two segments A B and L-K are congruent,
AB: A(4, 6), B(7, 2) LK: L(-1,-6), K(4,-6)
LIL
baleFill in Notes 1.4
Period
Notes 1.4 - Measure & Classify Angles
directions along the same line.are rays that share an ÿ/ÿ(I,pÿltÿ,l and extend inÿ!Iÿ /
H
AnlKÿisformedby 2ÿ with aÿendpoint,
B
BA and BC are fheÿ. of the angle. /B is the common endpoint called theÿ
C
Name angles with three letters
C
bP
Angles are measured in
Aÿcan be used to find the measure of an angle.
Addition Postulate:
Ps
If P is the interior of < RST, then
EX 1: Given that m< LKhÿ 145°, find m< LKÿ and m< MKN.
÷ lo)o IN
(4× - 3)°
K M
ivi ÿ- Llÿtÿl-i vvÿ z ÿIF.M :, tvÿ ,ÿ I.VM
X; 7..ÿ;
tv,,,, Ltrÿ : 7/,7,:ÿ)t !0/77to,>
Classffÿg Ang_/ÿ
AJ .ÿ
fÿ,')I-, qO" '70<ÿt' >'l,1 ÿ- f,,77
EX 3: Given that <KLM is a straight angle, find m< KLN and m< NLM,
M
,
--IDo-_ÿ
.,-ÿ rFT.--l- 71
ivv.ÿ,lf-ul --: rjÿ7.:::=..__
EX 4: Given that < EFG is a right angle, find m< EFH and m< HFG,
8%-77Xÿ fi°/
E;
(2× +2)ÿ/
6
Yÿ Zr H F(ÿtÿ ÿq "t" I
iÿz!7:F 1tÿ t(11")ÿ 7 oÿ --#10-ÿ0W - t:O°
d' ,9.1-GD ÿ qO'',,'
,.:( ,y,,-angles that have theÿmeasure,How would you mork theangles 1o show congruency?
m< ABC : m< A FG"iS ÿqUol to"
,,' ., oj:aO, oAÿls a ray that divides an angle into tw ] nglÿ.
/ -Congr, uÿncy markÿ ÿ BC bisects < ABDfor o.glÿÿÿÿ'ÿ < A BC N= < CBD
B ÿ-.-._.zÿ
b_._..J
LOktt-,/ÿ z tXtÿ-I"x-ÿt lo -5X-16
b X A
DateFill in Notes 1.5
Period
Notes 1.5 - Anqle Pair Relationships
**Re_ÿ:
Intersectinÿ - two lines that meet at exactly one point.
Perpendicular ÿ ( _L. ) - two lines that intersect to form 900 angles,
Parallel lines ( // ) - two lines that NEVER intersect.
S
I ,
,ÿ1t!ÿ,. V . Z-cÿ - two angles whose sum is 90o
'*angles can be named with one letter or number. You
IÿDC__ÿ.must use three #etters when clnglÿ ore Odjfltent alld_ there IS nO number.
Must use 3 letters fÿr this pro.lure
H
,K ÿ
U
OR
zÿ
> ,-A 5(., + ÿ4 L e/13I>--/ÿ0°
--.,,
- ?wo angles whose sum is 180°R
OR
Find the measure of the complement and the supplement of <1.m< .!. = 22o
Xÿ_Z: < i and < 2 are complementary angles and <2 and < 3 are supplementary angles.Given the measures of < 1. Find m< 2 and m< 3,
m< 1 = 36o
m<2= ÿL[lÿ-I +msg =qO°
z ;if-, yq"EX_.__!: < A and < B are complementary angles, Find the measure of each
angle if m< A = (Tx + 21)o and m< B = (9x + 5)°,
>4-1,m,9.-- q0
I(0>l¢t4: '16qf-ÿ qd #
.EX 4.: < A and < B are supplementary angles, Find the measure of eachangle if m< A = (7x + 21)0 and m< g : (9x + 5)°,
Iÿ,!t-ÿ qÿ-/,r,s 110
I{ÿX ÿ IÿI' 22 HI"re<A= re<B=
Two adjacent angles are aÿ if theirÿare oppositerays, The angles in a linear palr &e anglea, Remember palrmeans two!!!!
j
JML and < LMK are a linear pair < PQR and < RQT are a linear pair< PQS and < PQR are a linear pair< RQT and <TQS are a linear pair< TQ5 and ÿ PQ5 are a linear pair
< I and < 3 are vertical angles
< 2 and < 4 are vertical angles
***Vertical angles are always_ÿ_.ÿ!!!%1
EX 5: < FGH and < HG3 form a linear pair, Find the measures ofthe angles if m< FGH = (12x ÷ 1)° and m< HGJ = (4x -9)0
4; 11 ,ÿg
............ f
EX 6: ÿ-ÿand ÿ intersect at point E, Find the measure of<AEC and < CEB if m< AEC : (12x - 5)ÿ and mÿ BED = (4x + 19 °*Hint* Draw a picturetl
.. !tÿ, g ÿ,:,, ÿ-1 x.ÿl., i,;1
lÿ/ÿ
There are some things you can conclude from a diagram and some you cannot conclude,
,
C
Can Concludÿ
° L iÿ iv, I4ÿ,ÿ ÿ'ÿ
Cannot Conclude
,dc,
MI>LPeriod
Fill in Notes 1,6
Noÿes 1.6 - Classify Polygons
A !ÿ0!tAf;ÿYbis a closed figure formed by a finite number of-/ \ ÿ ÿ" ,'o '
segments calledÿwhere each s de intersects exactlytwo other sides at their endpoints.
Each endpoint of.a side is aÿof the polygon
- no line that contains a side of thea point in the interior of the polygon.
conwx po lygo n
/ ÿ( l V ÿ / j;ÿ)lÿ'ÿ{)ÿ]ÿ(ÿ ÿ p 0 ÿ y ÿ 0ÿ that is not convex, Also callednonconvex. 0 (]'I)
concave polygon
EX!: Tell whether the figure is a polygon, If it is a polygon, tellwhether it is concave or convex,
I'
\
/
!
' all interior ang
and Iÿlÿ(ÿl@kt/IV coNvEx polygon,0 J
EX2z How would you mark the following polygons to show that they are regular?
Regular Pentagon
0Regular Hexagon
# of :{Ides
3
4
5
6
7
B
9
I0
2
Typÿ of polygon
@ikti, O4%rÿ.
Classifying polygons
EX 3: Classify the polygon by the number of sides, Tell whether theupolygon is equilateral, equiangular, or regular, Explain your reasoning,
i
EX 4: Draw a figure that fitsVrhe description.
A pentagon that is equilateralbut not equiangular. ÿ'-ÿ
A concave heptagon