> ac`5@ 6%bjbj22 RXXQ8@\<y5v%%%4444444$6RA95(#%((5' 35
+
+
+(4
+(4
+z
++3@4.$?lH)$44I50y564~9$*9$494D%{&
+&d['q%%%55$*(Discovering Properties of Parallelograms, Trapezoids, and kites
You will use Cabri Geometry software on the Voyage 200 to explore special properties of parallelograms. Define all terms, fill in the blanks, and answer all questions with complete sentences.
DEFINITIONS:
A parallelogram is _______________________________________________________.
A diagonal is ___________________________________________________________.
THE PARALLELOGRAM:
Construct a parallelogram. Label your parallelogram PARL.
Make sure that opposite sides are parallel.
Measure the angles of PARL.
Compare a pair of opposite angles. Drag a vertex to see if this comparison remains for all parallelograms.
State your observations below.
Name a pair of consecutive angles of PARL. _____________ Find a relationship that exists between consecutive angles of a parallelogram. State a conjecture about the consecutive angles of a parallelogram.
Compare the lengths of the opposite sides of the parallelogram you made.
State a conjecture about the opposite sides of a parallelogram.
The opposite sides of a parallelogram ______________________________________.
Construct the diagonals PR and AL. Label the point of intersection M.
Does AL = PR? ______ Does AM = ML? ______ Does PM = MR? _______
If AM = ML, then M is the _____________ of diagonal AL. If PM = MR, then M is the _____________ of diagonal PR.
Does AL bisect PR? ______ Does PR bisect AL? ______
State a conjecture about the diagonals of a parallelogram. The diagonals of a parallelogram ______________________.
Discovering Properties of Special Parallelograms (cont.)
THE RHOMBUS:
If two parallel lines are intersected by a second pair of parallel lines the same distance apart as the first pair, then the parallelogram formed is a rhombus. In other words a rhombus is an equilateral parallelogram.
Construct a rhombus. (Be careful to construct the rhombus so that when dragging a vertex the quadrilateral will remain a rhombus.)
Draw in both diagonals of the rhombus you created.
Measure the angles formed by the intersection of the two diagonals. What kind of angle is formed?_____________ Test this by dragging a vertex. Do the diagonals of a rhombus always intersect each other this way? _____________
A previous conjecture told us that the diagonals of a parallelogram bisected each other. A rhombus is a parallelogram. Do the diagonals of a rhombus bisect each other? _________
In question 3, you discovered that the diagonals of a rhombus intersect each other in another special way. Make a conjecture combining these two ideas.
The diagonals of a rhombus are ___________________________________________________.
Compare the angles formed by the diagonals and the sides of the rhombus at each vertex. What do you observe? State a conjecture about the pair of angles at each vertex.
The diagonals of a rhombus ______________________________________________________.
THE RECTANGLE:
A rectangle is ______________________________________________________________.
Construct a rectangle.
Make a conjecture comparing the measure of each angle of a rectangle.
Make a conjecture about the diagonals of a rectangle.
THE SQUARE:
A square is ________________________________________________________________.
Before answering the last question below, you may want to construct a square and test to see if all the properties of the rectangle and rhombus are true for the square. There are many ways to construct a square. Test your construction by dragging a vertex. If the quadrilateral remains a square then your construction was correct.
Is a square a parallelogram?, a rectangle?, a rhombus? (EXPLAIN)
The Trapezoid and Kite
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
In a trapezoid, the parallel sides are called bases.
A pair of angles that share a base as a common side are
called a pair of base angles.
THE TRAPEZOID
Construct two parallel segments. Label one segment TR and the other segment PA. Connect T and P with a segment and connect R and A with a segment so that you have trapezoid TRAP.
Measure the consecutive angles between the bases of TRAP. (Remember consecutive angles are not the same as base angles.)
Make a conjecture about the consecutive angles between the bases of a trapezoid.
_________________________________________________________.
Draw in the diagonals of TRAP. Compare the lengths of the diagonals.
Do they bisect each other? ____. Are they perpendicular? ____. Are the diagonals congruent?________
ISOSCELES TRAPEZOID
A trapezoid whose two nonparallel sides are the same length is called an isosceles trapezoid.
One way to construct an isosceles trapezoid follows:
Create three noncollinear points I, S, and O.
Connect point S and O using the segment tool.
Find the perpendicular bisector of segment SO.
Use the reflection menu to reflect I over the perpendicular bisector and call the reflection point T.
Use the polygon tool to connect ISOT.
ISOT is an isosceles trapezoid.
Measure each pair of base angles.
What do you notice about each pair of base angles in an isosceles trapezoid? Complete the following statement.
The base angles of an isosceles trapezoid are __________________.
Construct the diagonals.
Compare the lengths of the two diagonals.
The diagonals of an isosceles trapezoid are _______________.
Do the diagonals bisect each other? _____. Are they perpendicular? ____. Are the diagonals of an isosceles trapezoid congruent? ___________
THE KITE
A kite is a quadrilateral with exactly two pairs of distinct congruent consecutive sides.
One way to construct a kite follows:
Construct a segment KT.
Put a point, I, not on the line containing segment KT.
Reflect I over KT to create point E.
Use the polygon tool to connect KITE.
KITE is a kite.
In a kite, the angles between each pair of congruent
sides are called the vertex angles and the other pair
of angles is called the nonvertex angles. So, in the above construction, angles K and ____ are vertex angles and angles I and ____ are nonvertex angles. Why do you think that mathematicians used the names vertex and nonvertex to identify the angles ?
The diagonals of a kite:
Draw in the diagonals of KITE. How do they intersect?
The diagonals of a kite are _______________.
What else seems to be true about the diagonals? How do the diagonals divide each other? Does either one bisect the other?
The diagonal connecting the vertex angles of a kite is the ____________ ______________ of the other diagonal.
How do the diagonals divide the opposite angles? Does either diagonal bisect one of the angles? ______.
The _____________ angles of a kite are ____________ by a diagonal.
What else seems to be true about kites?
Measure each pair of opposite angles (vertex and nonvertex) of your kite. Are both pairs of opposite angles congruent? _____. One pair? _____. Which pair? __________. Are any pairs of angles complementary? _____. Supplementary? ______.
The ___________ angles of a kite are _____________.
(Sources: Discovering Geometry, Serra (1997), Key Curriculum Press)
Name: __________________________
Created by Dana Long
Ashe County High School
DEj ls %ȿȸȿȪȿȦȠȔȸȪȉxr
h}CJh%y6CJ]hNhCJaJhNhh%yCJaJhNhh%y5CJ\
h%yCJh%yjh%yUmHnHu
h%y5CJh%y5CJ\
h%yCJh(T35CJ]h(T35\hnh(T3
h?CJ
h(T3CJjh(T3UmHnHu+EF ` a
D
l
$
&Fa$$a$^
&F
^
$a$$/%5%89yz
$
s
$a$gd%y$
&F
']a$$a$$
&Fa$6iO
k$
&Fda$gd%ygd%ydgd%y$
&Fda$gd%y
$da$gd%ydgd%yZPR(y
z
gd}h^hgd}$
880^8`0a$gd}gd}gd}gd%y
$da$gd%y
PQ`abcd=LMZԪvjjh}OJQJUjh}5OJQJU\"jh}OJQJUmHnHuh5dOJQJh>h}OJQJ(jh}5OJQJU\mHnHu(jh5d5OJQJU\mHnHujh5dUmHnHuh}OJQJh}5OJQJ\h}5\h}&y`adx>qAghij=>WKLNWgd}gd}gd}WXDt *!+!,!F!~!!&""""A#B#j#[$$$gd}gd}Z^A#B#[$$$$$$$$%%,%.%/%5%6%h/2h(T36]h!e46]h(T3
h%yCJ
h(T3CJh/2OJQJh}5OJQJ\"jh}OJQJUmHnHuh}OJQJh}h}5OJQJ$$$$$$$$$%%%.%/%0%1%2%3%4%5%6%$a$$a$$a$gd}#&P/ =!"v#$%0nO`*ϙ,PNG
IHDRgAMAQRIDATX1hWg4kPR2Km29סM.5!cB!!!dHtCBHR(.s' pJkC%\SOwޓ:x?O= Ll@T
!C4
6zcg7} wpp}uđdJe&vK(FKTuym4R>ʿ!ozUyZ+z>^zE];ڕk3E;?kZif~]jv=vnq=.P']VԀy]EnޜEͣ/r:B
P{e4wq(MJO{<=v$A0w!
?@֥Fwpt xլG!M0nNY=K3Ĳɍa77˸zEGE*]LcNI#wqw*Wby#d*'%Q&tPg223X, Zc~f2xY,s1XLq`B>K=ENf9ML^2O͠ĨdB&ǅUߜ^
ɩ
8`n $Y,MfXLq`p#4>aN$JB0ڑw4ca28ɞ#)grd#0Mgl3>sxV+Ü*~ :S7at$<8U<QG@<XXg3ōxk}Y~
wjkīǭ^)>ݵIENDB`n7_ciht'8PNG
IHDRv;gAMAQIDATX1hQW;G8tDCv*"U^$CE.:E\:H%請
Kж5$gp:w! B7jdBҷI@B}5rKc@P<lFs]~BkZ;=xf1҇q2T,CcW.'d9(˼~wŷ{"V&euef{lmydQ0y~})sǛii5W$.J=WV:ӶmPLIv@]7aBf0muEw!2BuW_2`5ݾL~B_';x:01@U\pq{`Stض'62F6.+cu臾7*Ο?[)}!
u_oz
i
4c\VPb3[jZCU<;ȉ*z:;";BEոW1[jeVbWy_W` .5n:okpQ܊~ԝx;D܍~ezO))JXKIz}M]A_@7Z5JAϹ]
f]/R+o{ߍ zƭ4(Fkwm<0ڹWl]vyϷw3Ҁ=:31{^gS{]~FԞ>?$>Өy=t mCS,IENDB`n!UXUq)9PNG
IHDRv;gAMAQIDATX?UBȐ"7vV!NĐ"^89=DpEDlc!gD8B{+[Hޮ+vo79ϼ(ZCH&!brc]Bl5%oo] P"qʼy?Dx/ˈ?w%躒Q>O[Dj~~XG>z۹uާoX~_d]ѤuzڻViE;e`Il~t}~۫xAY\i%<R,ok_W~Ir/v"`~7CǽRwhBjYw7{Ko4 Q2.{ cߢwbuIG+cN4Tsu'ÆZ f\'Rq+3j̫_z<^g>rܚ);>Vk]9ͦp+ I>)E(u=jAgs'͍>7i[Ds4_7wăqP"z`8Α+ O_a:зyW!J,4IlgouȖҜz觠\ҟ;@LhE*D8a}ǇJOi]dA8qt}n8{Y%%SvSMzXwD"V+^p݊k$ W1̻ R9Q{}C=_Π.ÚZMKwu.?U]WSTq97*Z̹vSPޕl
M~A5SNdQհL}ArnEQP <dI<\,fϺHKJt2%˱8&vE=j(,Wj=/r)e+DuHKTm]y3,k\zeyjRL:{= xrio0wdfʔ*Qʡ[TֽLHN{0.R2(wy(q@z5VfCrqԚ?Fp2>PϿ҈&S?cԚS_rJ$斵OIENDB`nQKG~BtPNG
IHDRv;gAMAQIDATXOUtRC%F0KeaS<]
^UG~_
x4O#DqRwϧ+紻h;7z{[>'}/{*}(w奿=/`DJ]().p8}?z"ptW
y
}!.@1=⚪+OsrXɷ1uxoKS3ia؋+\xA:j'_沎TATe1{Zf['3y>&ډeƻV2Isvj%1QY.$ůUQYbNR_I~'k[.@?QC/.i4,[4c3F1#5pV8hwŹVL@sYdxYw;4݃r'NCG}}E 2}t]}tn1] yYdx9vi0 i1#2\e4K_ψ6fG^.k'$bRmԣ>Ol9D?:ٔ7Z&kM$=#k[sMțȘ&Fk/̋ۂ4L^}=r?~q?>~V&I5IENDB`Dd!<P
33"((Dd!<P
33"((8@8Normal_HmH sH tH @@@ Heading 1$$@&a$56\@\%y Heading 2$<@& 56CJOJQJ\]^JaJDADDefault Paragraph FontViVTable Normal :V44
la(k(No List4@4Header
!4 @4Footer
!>B@> Body Text$a$
CJ\]2>@"2%yTitle$a$56@"`@}Caption
&F5CJ\aJ66<EF`aDl89yz$s6iO
k
Z
PR(y`adx>qAghij=>WKLNWXDt*+,F~&ABj[./01234700000 00 0000 00 00 00 000 00000 00000 00 00000000000 00 0 0
0 000000 0 0 0 000 00000000 0 0 00000000000000000000000000000 0000000000000000000000 000000000000000000000000000000000000@0My00@0@0My000ZM900M900M900M900M9000
0PR7Oy00Oy00Oy00O?Oy00Oy00Oy00
0xEz""PPPSZ6%yW$6%5%H(.ob$O`*ϙ,%<b$_ciht'8??b$)TQy*KCb$UXUq)9)cFb$#ɚIo0Jb$QKG~BtN@l(
v
CFA.@S`TS`T(r
6
G6
G6
G6
BeCDEF<%%in7crxenw3QwlepZLxOqiZGN<<@
#
CFA.@S`TS`T"NB
$
SDv
%
CFA.@S`TS`Tv
&
CFA.@S`TS`Tv
'
CFA.@S`TS`T
(
CFA. S`TS`T"` HB
)@
CDNB
*
SD
NB
+@
SDNB
,
SDTB

c$D
B
S ?Pa6!4tJMt!4t=4tt Z)#Ul(( T%% $V.Vt('$@*"t+t)!t"tVUZt,VA
t&P''&f4H
f4
f4
f4L
f4\I
f4G
f44
f4I
f4\
f4
f4T*
f4
))!7
++ ,,7
=*urn:schemasmicrosoftcom:office:smarttags PlaceType=*urn:schemasmicrosoftcom:office:smarttags PlaceName9*urn:schemasmicrosoftcom:office:smarttagsStateV*urn:schemasmicrosoftcom:office:smarttagsplacehttp://www.5iantlavalamp.com/)EN./7zVXWX/!)lpDR./73333333333333333GsvA O
dx?LNX=./7./7asheasheasheasheasheasheasheasheashexpRz 4.0e5 6\;F?vi hh^h`o(.h^`OJQJo(h^`OJQJo(ohpp^p`OJQJo(h@@^@`OJQJo(h^`OJQJo(oh^`OJQJo(h^`OJQJo(h^`OJQJo(ohPP^P`OJQJo(hh^h`o(.h^`OJQJo(h^`OJQJo(oh^`OJQJo(h^`OJQJo(hLL^L`OJQJo(oh^`OJQJo(h^`OJQJo(h^`OJQJo(oh^`OJQJo(808^8`0o(.^`.pLp^p`L.@@^@`.^`.L^`L.^`.^`.PLP^P`L.hh^h`o(.Rz 64vi0e5F? %0
?/2(T3!e4kXX5d%y=jn}Nh@H6@UnknownGz Times New Roman5Symbol3&z ArialCFComic Sans MS?5 z Courier New;Wingdings"qhBfBfǂzN4N4!24d
3QH(?!e4(Discovering Properties of ParallelogramsBeaver Creek HSxp$Oh+'0 $
@LX
dpx)Discovering Properties of Parallelograms WoiscBeaver Creek HSeavNormalCxpm3mMicrosoft Word 10.0@G@B%@>l@?lN՜.+,0,hp
Ashe County Board of Educatioi4A
)Discovering Properties of ParallelogramsTitle
!"#$%&'()+,./013456789:;<=>?@ABCDEFGHIJKLMNOQRSTUVWYZ[\]^_bRoot Entry FA$?ldData
*1Table2:WordDocumentRSummaryInformation(PDocumentSummaryInformation8XCompObjj
FMicrosoft Word Document
MSWordDocWord.Document.89q