Download Solutions Key 8 - Schilling Farms Middle School

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'EOMETRIC &IGURES
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PROPORTION
ORDERED PAIR
PERCENT
INTEGER
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N N
N B B B D
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D POINT ! S D
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POINT # S D
POINT $ S D
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POINT & S D
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LINES 13 ???
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AB
4HINK AND $ISCUSS
0OSSIBLE ANSWER 9OU CAN NAME A LINE USING ANY
POINTS ON THE LINE AND NAME A PLANE USING ANY
OF ITS POINTS THAT ARE NOT ON THE SAME LINE ! LINE
SEGMENT CAN BE NAMED USING ONLY ITS ENDPOINTS
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4
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SIMPLE
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0OSSIBLE ANSWER
.AME THE ENDPOINT
A RAY FIRST
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BLACKLINE
%XERCISES
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,
0OSSIBLE ANSWER "ECAUSE A LINE HAS ONLY
DIMENSION YOU NEED POINTS NOT ON THE
SAME LINE TO SHOW THE DIMENSIONS OF A PLANE
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
!24 &),%-3-'4%?#?,?4
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#HOOSE ANY TWO POINTS
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j???k j???k
0OSSIBLE ANSWER 13 24
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LINE SEGMENTS 78 79 7: 89 8: AND 9:
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(OLT -C$OUGAL -ATHEMATICS
ON
MOD
COM
GREYSCALE
%XERCISES
.O TWO ENDPOINTS CANNOT BE SHARED BY TWO DIFFERENT
LINE SEGMENTS )F TWO LINE SEGMENTS SHARE TWO
ENDPOINTS THEN THE LINE SEGMENTS ARE THE SAME
RIGHT ANGLE
ACUTE ANGLE
STRAIGHT ANGLE
! LINE IS A STRAIGHT PATH THAT EXTENDS FOREVER IN TWO
DIRECTIONS A RAY IS A STRAIGHT PATH THAT EXTENDS
FOREVER IN ONE DIRECTION FROM AN ENDPOINT AND A
SEGMENT IS A STRAIGHT PATH FROM ONE ENDPOINT TO
ANOTHER )T IS POSSIBLE-4%?#?4%#?
TO ESTIMATE THE LENGTH OF A
!24 &),%
SEGMENT BECAUSE IT DOES NOT EXTEND FOREVER IN
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ANY DIRECTION
33
#2%!4%$
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9ES SINCE
A PLANE EXTENDS
FOREVER IT IS $!4%
POSSIBLE
#3 PLANE
THAT TWO%$)4%$
FACES COULD
AND STILL
"9 BE ON THE SAME
$!4%
M!8" AND M"8# 3INCE !8" AND "8# ARE
NEITHER
M"8# AND M$8% 3INCE "8# AND $8% ARE
COMPLEMENTARY
4%#(
M$8% AND M!8$ $8% AND !8$ ARE
3INCE
SUPPLEMENTARY
M#8$ AND M!8" 3INCE #8$ AND !8" ARE
COMPLEMENTARY
NOT TOUCH
4)-% # )) AND )6 BOTH SETS OF THESE SEGMENTS ARE
CREATED .%43
ONLY ALTERED
CONGRUENT
SIMPLE
MOD
COMPLEX
#HECK STUDENTS WORK 0OSSIBLE ANSWER
.%43
M, V COMPLEX
M# OBTUSE ANGLE
ACUTE ANGLE
M.:/ AND M-:. 3INCE .:/ AND -:. ARE
COMPLEMENTARY
S D RIGHT ANGLE
S D M-:. AND M/:0 3INCE -:. AND /:0 ARE
S D NEITHER
CHANGE
IN
QUANTITY
??????????????? PERCENT
OF CHANGE
M,:. AND M.:0
ORIGINAL QUANTITY
3INCE
,:. AND .:0 ARE
???
???????
! SUPPLEMENTARY
M.:/
AND M,:- 3INCE
.:/ AND ,:- ARE
CHANGE IN QUANTITY
NEITHER
??????????????? PERCENT
OF CHANGE
(*
&%#-#*(
ORIGINAL QUANTITY
M/ '#"$("
???
????????
M*
$&
) $'#
&$'#,
+&$'#,
COMPLEMENTARY
SUPPLEMENTARY &#&(
#"# %
(-)#
X X CHANGE IN QUANTITY
X X ??????????????? PERCENT OF CHANGE
ORIGINAL QUANTITY
SUPPLEMENTARY
?? ?????
X X 4HE HANDS FORM A STRAIGHT ANGLE AT 4HE HANDS FORM A RIGHT ANGLE AT 4HE HANDS FORM AN OBTUSE ANGLE AT 0!'%3 n
A ,INES OF LATITUDE AND LONGITUDE CROSS TO FORM RIGHT
ANGLES
B 7ASHINGTON $# IS ABOUT . 7
4HINK AND $ISCUSS
0OSSIBLE ANSWER ! RIGHT ANGLE MEASURES !N
ACUTE ANGLE IS LESS THAN !N OBTUSE ANGLE IS
GREATER THAN AND A STRAIGHT ANGLE MEASURES 0OSSIBLE ANSWER 3INCE THE SUM OF THE MEASURES OF
COMPLEMENTARY ANGLES IS THE MISSING MEASURE
CAN BE FOUND USING THE EQUATION M0 4HEREFORE M0 #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
! STRAIGHT ANGLE MEASURES SO THE TWO ANGLES
MUST BE SUPPLEMENTARY 4WO ANGLES WHOSE SUM
EQUALS ARE SUPPLEMENTARY
4WO ANGLES ARE SUPPLEMENTARY WHEN THEIR SUM
EQUALS 3INCE OBTUSE ANGLES MEASURE GREATER
THAN THE SUM OF TWO OBTUSE ANGLES IS GREATER
THAN (OLT -C$OUGAL -ATHEMATICS
&IND THE M"!# IN THE FIGURE
M#!$ M$!& M#!$ M#!$ M"!# M#!% M"!# M"!# 0OSSIBLE ANSWER PARALLELˆRAILROAD TRACKS SIDES OF A
LADDER PERPENDICULARˆSIDE AND BOTTOM EDGES OF A
DESK DRAWER SKEWˆA TELEPHONE POLE AND THE EDGE
OF THE CURB OF SIDEWALK
%XERCISES
??
???
j k
j k
*, AND +4HE LINES ARE IN THE SAME PLANE AND DO NOT
INTERSECT
j??k j???k
*, +j???k
j???k
,- AND +.
4HE LINES ARE IN DIFFERENT PLACES AND DO NOT
INTERSECT
j???k
j???k
,- AND +. ARE SKEW
j???k
j???k
,- AND +4HE
LINES APPEAR TO INTERSECT TO FORM RIGHT ANGLES
j???k j???k
,- +-
# &!% AND %!$ ARE COMPLEMENTARY ANGLES
M&!% M%!$ SO THIS PAIR OF ANGLES IS NOT
COMPLEMENTARY
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' M&!% M%!$ 3O M&!$ $ATA SET 4HE MEAN IS !RRANGE THE NUMBERS IN ORDER 4HE MEDIAN IS 4HE NUMBER THAT APPEARS THE MOST IS SO THE
MODE IS GREATEST VALUE LEAST VALUE 4HE RANGE IS AND THE ANGLE ARE OBTUSE ANGLES 3INCE ALL
OF THE OBTUSE ANGLES IN THE FIGURE ARE CONGRUENT
M AND THE ANGLE ARE VERTICAL ANGLES 3INCE ALL
OF THE VERTICAL ANGLES ARE CONGRUENT M AND THE ANGLE ARE CONGRUENT BECAUSE THEY
ARE ALTERNATE EXTERIOR ANGLES
M j???k
j???k
58 AND 9:
4HE LINES ARE IN DIFFERENT PLACES AND DO NOT
INTERSECT
j???k
j???k
58 AND 9: ARE SKEW
j???k
j???k
9: AND 89
4HE
LINES APPEAR TO INTERSECT TO FORM RIGHT ANGLES
j???k j???k
9: 89
j???k
j???k
58 AND 67
4HE
LINES ARE IN THE SAME PLANE AND DO NOT INTERSECT
j???k j???k
58 67
$ATA SET 4HE MEAN IS !RRANGE THE NUMBERS IN ORDER 4HE MEDIAN IS 4HERE ISNT A NUMBER THAT APPEARS MORE THAN ONCE
SO THERE IS NO MODE
GREATEST VALUE LEAST VALUE 4HE RANGE IS $ATA SET 4HE MEAN IS !RRANGE THE NUMBERS IN ORDER $ETERMINE THE AVERAGE OF THE TWO MIDDLE NUMBERS
4HE MEDIAN IS 4HE NUMBER THAT APPEARS THE MOST IS SO THE
MODE IS GREATEST VALUE LEAST VALUE 4HE RANGE IS ??k
4HE FIGURE IS A RAY NAMED *+ 0!'%3 n
4HINK AND $ISCUSS
0OSSIBLE ANSWER
)
AND THE ANGLE ARE ACUTE ANGLES 3INCE ALL
OF THE ACUTE ANGLES IN THE FIGURE ARE CONGRUENT
M AND THE ANGLE ARE CONGRUENT BECAUSE THEY
ARE ALTERNATE INTERIOR ANGLES
M IS AN OBTUSE ANGLE
)N THE FIGURE THE ACUTE AND OBTUSE ANGLES ARE
SUPPLEMENTARY
M ?? ?
M PARALLEL
PERPENDICULAR
SUPPLEMENTARY ADJACENT
ALTERNATE EXTERIOR CONGRUENT
4HE SUM OF THE MEASURES OF TWO ANGLES THAT ARE
COMPLEMENTARY IS DEGREES )F THE ANGLES ARE
CONGRUENT EACH ONE MUST MEASURE DEGREES
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
(OLT -C$OUGAL -ATHEMATICS
COMPLEMENTARY
MX MX 2%!$9 4/ '/ /. 0!'% 3INCE VERTICAL ANGLES ARE CONGRUENT THE ANGLE
0OSSIBLE ANSWER ! " #
OPPOSITE THE GIVEN ANGLE HAS THE SAME MEASURE
#HOOSE ANY TWO POINTS
ONk Aj???
LINE
AS THE GIVEN ANGLE 4HE REMAINING ANGLES ARE
j???k j???
k TO NAME THE LINE
0OSSIBLE ANSWER !' #% "&
BOTH ADJACENT TO THE ANGLES THAT MEASURE 3INCE
ADJACENT ANGLES ARE SUPPLEMENTARY BOTH REMAINING
#HOOSE ANY THREE POINTS ON
A PLANE TO NAME THE
'4%?#?,?4
4%#(
ANGLES MEASURE !24 &),%
PLANE
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0OSSIBLE */"
ANSWER
PLANE "#$
SOMETIMES
JH
#2%!4%$ "9
5SE THE $!4%
ENDPOINTS IN ANY ORDER TO NAME A SEGMENT
NEVER
NEVER
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ALWAYS
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CREATED
0OSSIBLE$!4%
ANSWER "$ "& $%
A /2
? AND 23 ARE PERPENDICULAR
B 04 IS A TRANSVERSAL
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C AND ARE CORRESPONDING ANGLES SIMPLE
)F TWO INTERSECTING LINES FORM CONGRUENTBLACKLINE
ADJACENT
ANGLES THE LINES ARE PERPENDICULAR
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
?
COLOR
$8% AND !8$
M$8% AND M!8$ SO $8% AND !8$ ARE
SUPPLEMENTARY
!8" AND #8$
M!8" AND M#8$ SO !8" AND #8$ ARE
COMPLEMENTARY
$8% AND !8"
M$8% AND M!8" SO $8% AND !8" ARE NEITHER
"8# AND $8%
M"8# AND M$8% SO "8# AND $8% ARE
COMPLEMENTARY
M2 M2 & AND THE ARE VERTICAL ANGLES WHICH MAKES
THEM CONGRUENT 3O M 3INCE R AND S ARE
PARALLEL AND BECOME CORRESPONDING ANGLES
BY THE TRANSVERSAL LINE 4HEREFORE AND ARE
CONGRUENT MAKING M SUPPLEMENTARY
MX MX ? ?
STRAIGHT
$ M ?? ?
M COMPLEMENTARY
MX MX ? ?
ACUTE
$RAW A TRANSVERSAL THAT INTERSECTS TWO OR MORE LINES
IN THE PARKING LOT AND MEASURE A PAIR OF ANGLES
THAT SHOULD BE CONGRUENT SUCH AS CORRESPONDING
ANGLES )F YOU CAN ASSUME THAT YOUR MEASUREMENTS
ARE CORRECT AND THE ANGLES ARE CONGRUENT THEN YOU
CAN CONCLUDE THAT THE LINES IN THE PARKING LOT ARE
LIKELY TO BE PARALLEL
S D ? ?
GREYSCALE
OBTUSE
4HE NONADJACENT SIDES OF A PAIR OF ADJACENT ANGLES
ARE FORMED BY ONE OF THE TWO STRAIGHT LINES ! LINE IS
A STRAIGHT ANGLE WHICH BY DEFINITION HAS A MEASURE
OF 4HEREFORE THE MEASURES OF THE ADJACENT
ANGLES MUST ADD UP TO THE THAT IS THEY ARE
SUPPLEMENTARY
?
!&CHECKMARK
AND #$ "& AND #%
PLACE
!" AND
%$ "#
AND &%
RIGHT
MOD
COMPLEX
V COMPLEX
# $RAW A $IAGRAM OR -AKE A -ODEL
S D 4)-%
.AME THE
ENDPOINT
A RAY???FIRST
???kOF ???
k
k
0OSSIBLE ONLY
ANSWER
"$ #%
$%
ALTERED
.%43
.%43
M& M& 4HE LINES ARE
IN DIFFERENT
PLANES AND DO NOT
j??k
j???k
INTERSECT +, AND -. ARE SKEW
4HE LINES ARE
IN THE
SAME PLANE AND DO NOT
j??k
j???k
INTERSECT *, AND -. ARE PARALLEL
4HE
LINES??APPEAR TO INTERSECT TO FORM RIGHT ANGLES
j??k
j k
+, AND *, ARE PERPENDICULAR
4HE LINES ARE
IN DIFFERENT
PLANES AND DO NOT
j??k
j???k
INTERSECT )* AND -. ARE SKEW
AND THE ANGLE ARE VERTICAL ANGLES 3INCE
VERTICAL ANGLES ARE CONGRUENT M (OLT -C$OUGAL -ATHEMATICS
IS AN OBTUSE ANGLE )N THE FIGURE THE ACUTE AND
OBTUSE ANGLES ARE SUPPLEMENTARY
M M M ??
?
M AND ARE OBTUSE ANGLES 3INCE ALL OF THE
OBTUSE ANGLES IN THE FIGURE ARE CONGRUENT
M AND THE ANGLE ARE ACUTE ANGLES 3INCE ALL OF
THE ACUTE ANGLES IN THE FIGURE ARE CONGRUENT
M AND ARE VERTICAL ANGLES 3INCE VERTICAL
ANGLES ARE CONGRUENT M AND ARE OBTUSE AND VERTICAL ANGLES 3INCE
ALL OF THE VERTICAL OBTUSE ANGLES IN THE FIGURE ARE
CONGRUENT M 0!'%3 n
4HINK AND $ISCUSS
0OSSIBLE ANSWER ! DIAMETER IS A CHORD BECAUSE
BOTH OF ITS ENDPOINTS LIE ON THE CIRCLE ! RADIUS HAS
ONLY ON ENDPOINT ON THE CIRCLE
0OSSIBLE ANSWER
OF OF IS !LL ACUTE ANGLES MADE BY A TRANSVERSAL
AND TWO PARALLEL LINES ARE CONGRUENT
0OSSIBLE ANSWER ! CIRCLE GRAPH SHOWS A POPULATION
DISTRIBUTION OF MALES AND FEMALES 7HAT IS THE
MEASURE OF THE CENTRAL ANGLE OF THE SECTOR THAT
SHOWS THE POPULATION DISTRIBUTION OF FEMALES
0OSSIBLE ANSWER ! CENTRAL ANGLE IS AN ANGLE FORMED
BY TWO RADII ! SECTOR IS THE REGION INSIDE THE CIRCLE
FORMED BY A CENTRAL ANGLE AND AN ARC
4HE CLOCK IS A CIRCLE DIVIDED INTO EQUAL PARTS
%ACH FIVEMINUTE SECTION EQUALS 4HERE ARE
FIVEMINUTE SECTIONS BETWEEN THE HANDS
4HE ANGLE MEASURE BETWEEN THE HANDS IS '&( AND *&) ARE NOT SUPPLEMENTARY SO $
?
?
)F (& IS PERPENDICULAR TO DIAMETER ') THE ANGLES
FORMED BY THEIR INTERSECTION ARE MAKING
(&) OF %STIMATING OF OF %STIMATING OF OF %STIMATING OF %XERCISES
2ADII
? ? ? ?
/1 /2 /3 /4
$IAMETER
?
24
#HORDS
? ? ? ?
24 23 34 41
OF 4HE CENTRAL ANGLE OF THE SECTOR MEASURES 2ADII
? ? ? ? ?
#! #" #$ #% #&
OF %STIMATING OF ,ETTERS % & ( - . AND : APPEAR TO HAVE PARALLEL
LINES
,ETTERS % & ( , AND 4 APPEAR TO HAVE
PERPENDICULAR LINES
0!'%3 n
4HINK AND $ISCUSS
0OSSIBLE ANSWER !LTHOUGH A CIRCLE IS A CLOSED
FIGURE IT IS NOT FORMED BY THREE OR MORE LINE
SEGMENTS SO IT CANNOT BE A POLYGON
$IAMETERS
? ?
!% "&
0OSSIBLE ANSWER 4HE FIGURE IS NOT CLOSED THE FIGURE
HAS MORE THAN TWO DIMENSIONS ONE OF THE LINE
SEGMENTS CROSSES ANOTHER LINE SEGMENT
#HORDS
? ? ? ? ?
'" "& $% &% !%
OF 4HE CENTRAL ANGLE OF THE SECTOR MEASURES %XERCISES
4HE FIGURE IS NOT A POLYGON .OT ALL OF THE SIDES OF
THE FIGURE ARE LINE SEGMENTS
!DD THE RADII OF THE CIRCLES
4HE DISTANCE BETWEEN THE CENTERS IS CENTIMETERS
4HE FIGURE IS A POLYGON )T IS A CLOSED FIGURE WITH
SIDES
4HE FIGURE IS NOT A POLYGON )T IS NOT A CLOSED FIGURE
4HE MEASURE OF THE CENTRAL ANGLE OF EACH SECTOR
IS #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
SIDES ANGLES OCTAGON
(OLT -C$OUGAL -ATHEMATICS
SIDES ANGLES QUADRILATERAL
x
SIDES ANGLES NONAGON
N
2ULE
Y
4HE FIGURE IS A REGULAR QUADRILATERAL OR SQUARE
4HE FIGURE IS A QUADRILATERAL RECTANGLE )T IS NOT
A REGULAR POLYGON BECAUSE NOT ALL THE SIDES ARE
CONGRUENT
4HE FIGURE IS A TRIANGLE )T IS NOT A REGULAR POLYGON
BECAUSE NOT ALL OF THE SIDES AND NOT ALL OF THE
ANGLES ARE CONGRUENT
4HE FUNCTION Y N DESCRIBES THIS SEQUENCE
x
4HE FIGURE IS NOT A POLYGON 4HERE ARE LINE SEGMENTS
IN THE FIGURE THAT CROSS
N
2ULE
Y
4HE FIGURE IS NOT A POLYGON .OT ALL OF THE SIDES OF
THE FIGURE ARE LINE SEGMENTS
4HE FIGURE IS A POLYGON )T IS A CLOSED FIGURE WITH
SIDES
SIDES ANGLES PENTAGON
4HE FUNCTION Y N DESCRIBES THIS SEQUENCE
SIDES ANGLES TRIANGLE
IS WHAT PERCENT OF N ???
????
N N IS OF SIDES ANGLES HEPTAGON
4HE FIGURE IS A REGULAR OCTAGON
4HE FIGURE IS A PENTAGON )T IS NOT A REGULAR POLYGON
BECAUSE NOT ALL OF THE SIDES AND NOT ALL OF THE
ANGLES ARE CONGRUENT
7HAT IS OF OF IS 4HE FIGURE IS A HEXAGON )T IS NOT A REGULAR POLYGON
BECAUSE NOT ALL OF THE SIDES ARE CONGRUENT
A 0OSSIBLE ANSWER HEXAGONS PENTAGONS
RHOMBUSES TRAPEZOIDS PARALLELOGRAMS
B REGULAR TRIANGLES AND HEXAGONS
IS OF WHAT NUMBER
N
N
IS OF 4HE SMALLER SHAPES ARE QUADRILATERALS
IS WHAT PERCENT OF N ???
????
N N IS OF )T HAS SIDES AND IT IS CALLED A GON
-AKE SURE THAT STUDENTS DRAW AND IDENTIFY ONE
OF EACH TYPE OF POLYGON TRIANGLE QUADRILATERAL
PENTAGON HEXAGON HEPTAGON OCTAGON NONAGON
AND DECAGON
! )T IS A POLYGON
4HE FIGURE IS A CLOSED FIGURE WITH SIDES .OT ALL OF
ITS SIDES ARE EQUAL
0!'%3 n
0OSSIBLE ANSWER
4HINK AND $ISCUSS
0OSSIBLE ANSWER
4HE FIGURE IS NOT A POLYGON BECAUSE IT IS NOT FORMED
BY LINE SEGMENTS
x
2ULE
Y
)SOSCELES ACUTE
0OSSIBLE ANSWER
N
4HE FUNCTION Y N DESCRIBES THIS SEQUENCE
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
)SOSCELES OBTUSE
3CALENE RIGHT TRIANGLE
(OLT -C$OUGAL -ATHEMATICS
0OSSIBLE ANSWER 3INCE AT LEAST TWO OF THE
SIDES OF AN EQUILATERAL TRIANGLE ARE CONGRUENT
IT IS ALSO ISOSCELES ! TRIANGLE WITH EXACTLY TWO
CONGRUENT SIDES IS AN ISOSCELES TRIANGLE BUT IT IS
NOT EQUILATERAL BECAUSE ALL OF ITS SIDES ARE NOT
CONGRUENT
%XERCISES
TWO CONGRUENT SIDES ONE RIGHT ANGLE
4HIS IS AN ISOSCELES RIGHT TRIANGLE
NO CONGRUENT SIDES ONE OBTUSE ANGLE
4HIS IS A SCALENE OBTUSE TRIANGLE
TWO CONGRUENT SIDES THREE ACUTE ANGLES
4HIS IS AN ISOSCELES ACUTE TRIANGLE
4HE FIGURE HAS ISOSCELES RIGHT ISOSCELES ACUTE
AND SCALENE OBTUSE TRIANGLES
NO CONGRUENT SIDES ONE RIGHT ANGLE
4HIS IS A SCALENE RIGHT TRIANGLE
TWO CONGRUENT SIDES ONE OBTUSE ANGLE
4HIS IS AN ISOSCELES OBTUSE TRIANGLE
THREE CONGRUENT SIDES THREE ACUTE ANGLES
4HIS IS AN EQUILATERAL ACUTE TRIANGLE
4HE FIGURE HAS ISOSCELES RIGHT ISOSCELES ACUTE
AND SCALENE RIGHT TRIANGLES
NO CONGRUENT SIDES
4HIS IS A SCALENE TRIANGLE
THREE CONGRUENT SIDES
4HIS IS AN EQUILATERAL TRIANGLE
TWO CONGRUENT SIDES
4HIS IS AN ISOSCELES TRIANGLE
$ 3OLVE A 3IMPLER 0ROBLEM
3MALL TRIANGLES -EDIUM TRIANGLES MADE UP OF SMALL TRIANGLES ,ARGE TRIANGLE MADE UP OF ALL THE SMALL TRIANGLES .O THE ANGLES IN AN EQUILATERAL TRIANGLE ARE ALL
ACUTE AND A TRIANGLE MUST HAVE ONE OBTUSE ANGLE TO
BE CLASSIFIED AS AN OBTUSE TRIANGLE
5SING THE FACT THAT ALL THE RADII OF A GIVEN CIRCLE ARE
CONGRUENT FIND THE MISSING RADII AND THEREBY THE
MISSING SIDE LENGTHS
N!"# IS EQUILATERAL N"#$ IS ISOSCELES N"$%
IS SCALENE
" 4HE ONE ANGLE MAKES THIS A RIGHT TRIANGLE NOT
ACUTE
( ISOSCELES OBTUSE TRIANGLE
4WO CONGRUENT SIDES AND ONE OBTUSE ANGLE MAKES
THIS AN ISOSCELES OBTUSE TRIANGLE
?? ???
#ONVERT THE MEASUREMENTS TO DECIMALS FOR EASY
COMPARISON
?? ???
&ROM LEAST TO GREATEST ?? ???
ONE OBTUSE ANGLE
4HIS IS AN OBTUSE TRIANGLE
4HIS FIGURE IS A HEPTAGON )T IS NOT REGULAR BECAUSE
NOT ALL THE SIDES AND NOT ALL THE ANGLES ARE
CONGRUENT
ONE RIGHT ANGLE
4HIS IS A RIGHT TRIANGLE
4HIS FIGURE IS A QUADRILATERAL )T IS REGULAR BECAUSE
ALL THE SIDES AND ALL THE ANGLES ARE CONGRUENT
THREE ACUTE ANGLES
4HIS IS AN ACUTE TRIANGLE
4HIS FIGURE IS AN OCTAGON )T IS NOT REGULAR BECAUSE
NOT ALL THE SIDES AND NOT ALL THE ANGLES ARE
CONGRUENT
?
!# IS INCHES 4HE TRIANGLE IS ISOSCELES
%ACH TRIANGLE HAS TWO CONGRUENT SIDES AND ONE
RIGHT ANGLE 4HEY ARE ISOSCELES RIGHT TRIANGLES
0!'%3 n
4HINK AND $ISCUSS
0OSSIBLE ANSWER )F THE ANGLES OF A RHOMBUS ARE
RIGHT ANGLES THEN THE RHOMBUS IS ALSO A SQUARE )F
THE DIAGONALS OF A RHOMBUS ARE CONGRUENT THEN THE
RHOMBUS IS ALSO A SQUARE
TWO CONGRUENT SIDES THREE ACUTE ANGLES
4HIS IS AN ISOSCELES ACUTE TRIANGLE
0OSSIBLE ANSWER
1UADRILATERALS
NO CONGRUENT SIDES ONE OBTUSE ANGLE
4HIS IS A SCALENE OBTUSE TRIANGLE
2HOMBUSES
0ARALLELOGRAMS
NO CONGRUENT SIDES ONE RIGHT ANGLE
#! 4RAPEZOIDS
4HIS IS A SCALENE RIGHT TRIANGLE
3QUARES
%ACH FACE IS A SCALENE TRIANGLE
2ECTANGLES
%ACH FACE OF THE PYRAMID IS AN ISOSCELES TRIANGLE
*,
('%/%,*
%IGHT TRIANGLES ARE CREATED %ACH TRIANGLE HAS TWO
)%!$&*$
CONGRUENT SIDES AND THREE ACUTE ANGLES
SO THE
TRIANGLES ARE ALL ISOSCELES ACUTE TRIANGLES
+"&)%
&(
(&)%.
-(&)%.
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
%$%"'
*/+%
(%(*
!24 &),%'4%?#?,?4
#534/-%2
#2%!4%$ "9
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(OLT -C$OUGAL -ATHEMATICS
(27
JH
JH
%XERCISES
! PARALLELOGRAM RECTANGLE RHOMBUS AND SQUARE ALL
HAVE OPPOSITE SIDES THAT ARE CONGRUENT
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES SO IT IS
A PARALLELOGRAM 4HIS FIGURE IS BEST DESCRIBED AS A
PARALLELOGRAM
$RAW LINES PARALLEL TO EACH SIDE 4HEIR INTERSECTION IS
THE FOURTH VERTEX
4RUE 3QUARES HAVE FOUR CONGRUENT SIDES
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES SO IT IS
A PARALLELOGRAM )T HAS FOUR RIGHT ANGLES SO IT IS
ALSO A RECTANGLE 4HIS FIGURE IS BEST DESCRIBED AS A
RECTANGLE
4RUE /PPOSITE SIDES OF A RECTANGLE ARE PARALLEL
4RUE 3QUARES HAVE FOUR RIGHT ANGLES
&ALSE ! RHOMBUS MAY OR MAY NOT HAVE FOUR RIGHT
ANGLES IF IT DOES NOT IT IS NOT A RECTANGLE
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES SO IT IS A
PARALLELOGRAM )T HAS FOUR CONGRUENT SIDES SO IT IS
ALSO A RHOMBUS 4HIS FIGURE IS BEST DESCRIBED AS A
RHOMBUS
&ALSE ! SQUARE HAS TWO SETS OF PARALLEL SIDES BUT A
TRAPEZOID HAS ONLY ONE SET
4RUE 2ECTANGLES WITH FOUR CONGRUENT SIDES ARE
SQUARES
TRIANGLES HEXAGON AND TRAPEZOIDS
4HE QUADRILATERAL IS A TRAPEZOID
! PARALLELOGRAM HAS TWO PAIRS OF OPPOSITES THAT
ARE PARALLEL BUT A TRAPEZOID HAS EXACTLY ONE PAIR OF
OPPOSITE SIDES THAT IS PARALLEL )T IS NOT !24
POSSIBLE
TO'4%?#?,?4
&),%
(27
DRAW THE FIGURE
#534/-%2
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES
SO IT IS A"9
JH
#2%!4%$
PARALLELOGRAM )T HAS FOUR CONGRUENT SIDES SO IT IS
%$)4%$AS"9
ALSO A RHOMBUS 4HIS FIGURE IS BEST DESCRIBED
A RHOMBUS
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES
SO IT IS .%43
CREATED
A PARALLELOGRAM 4HIS FIGURE IS BEST DESCRIBED AS
2%6)3)/.
A PARALLELOGRAM
SIMPLE
4HE FIGURE HAS EXACTLY ONE PAIR OF PARALLEL SIDES SO
BLACKLINE
IT IS A TRAPEZOID 4HIS FIGURE IS BEST DESCRIBED
AS
A TRAPEZOID
#
$
4%#(
*/"
.5-"%2
$!4%
!
$!4%
4)-% (OW MANY SQUARES ARE IN THE DESIGN
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES SO IT IS
A PARALLELOGRAM )T HAS FOUR CONGRUENT SIDES SO IT
IS ALSO A RHOMBUS )T HAS FOUR RIGHT ANGLES SO IT IS
ALSO A RECTANGLE AND A SQUARE 4HIS FIGURE IS BEST
DESCRIBED AS A SQUARE
3OME SWIMMING POOLS ARE RECTANGLES ! BASEBALL
DIAMOND IS A RHOMBUS BUT THE ANGLE AT EACH BASE
IS SO IT IS ALSO A SQUARE
0OSSIBLE ANSWER
-4%?#?4%#?
!24OF&),%
4HE COORDINATES
THE FOURTH VERTEX OF THE
(27
PARALLELOGRAM#534/-%2
ARE */" .5-"%2
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES SO IT IS
A PARALLELOGRAM )T HAS FOUR RIGHT ANGLES SO IT IS
ALSO A RECTANGLE 4HIS FIGURE IS BEST DESCRIBED AS
A RECTANGLE
Y
#2%!4%$
"9
%$)4%$
"9
CREATED
! PARALLELOGRAM RECTANGLE RHOMBUS AND SQUARE ALL
HAVE TWO PAIRS OF OPPOSITE PARALLEL SIDES
$!4%
JH
$!4%
.%43
2%6)3)/.
#3
SIMPLE
ONLY ALTERED
X
MOD
BLACKLINE
COMPLEX
GREYSCALE
! RHOMBUS AND A SQUARE BOTH HAVE FOUR CONGRUENT
SIDES
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
4)-% ! SQUARE AND A RECTANGLE BOTH HAVE FOUR RIGHT
ANGLES
X
"
ONLY ALTERED .%43
4HE FOLLOWING POLYGONS
COULD BE MADE
PLACE CHECKMARK
PARALLELOGRAM RHOMBUS SQUARE RECTANGLE
MODTRAPEZOID
COMPLEX
RIGHT TRIANGLEV COMPLEX
COLOR ANSWER
GREYSCALE
!NSWERS WILL VARY 0OSSIBLE
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES SO IT IS A
PARALLELOGRAM )T HAS FOUR CONGRUENT SIDES SO IT IS
ALSO A RHOMBUS 4HIS FIGURE IS BEST DESCRIBED AS
A RHOMBUS
Y
(OLT -C$OUGAL -ATHEMATICS
.%
PLACE
V C
COLOR
# 3OME TRAPEZOIDS ARE RECTANGLES
!LL RECTANGLES HAVE TWO PAIRS OF CONGRUENT SIDES
WHILE TRAPEZOIDS ONLY NEED ONE PAIR OF CONGRUENT
SIDES
%XERCISES
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS 4HIS FIGURE IS A PARALLELOGRAM RECTANGLE AND
SQUARE )T IS BEST DESCRIBED AS A SQUARE
!
Y
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS "
X
$
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS #
X X ??
?
X
4HE MEASURE OF THE UNKNOWN ANGLE IS 3TEMS ,EAVES
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS 3NOWFALL IN #OLORADO 3PRING
3NOWFALL
IN
&REQUENCY
#OMULATIVE
&REQUENCY
X X ?? ?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS 4HERE ARE TRIANGLES
4HE SUM OF THE ANGLE MEASURES OF A HEXAGON
IS 4WO ACUTE ANGLES MAKE THIS AN ACUTE TRIANGLE
4HERE ARE TRIANGLES
4HE SUM OF THE ANGLE MEASURES OF A PENTAGON
IS /NE RIGHT ANGLE MAKES THIS A RIGHT TRIANGLE
/NE OBTUSE ANGLE MAKES THIS AN OBTUSE TRIANGLE
/NE OBTUSE ANGLE MAKES THIS AN OBTUSE TRIANGLE
4HERE ARE TRIANGLES
4HE SUM OF THE ANGLE MEASURES OF A QUADRILATERAL
IS 0!'%3 n
4HINK AND $ISCUSS
0OSSIBLE ANSWER 3UBTRACT THE SUM OF THE MEASURES
OF THE TWO ANGLES FROM 0OSSIBLE ANSWER 4HE OCTAGON HAS MORE TRIANGLES
WHEN THE DIAGONALS ARE DRAWN 3INCE THERE ARE
MORE TRIANGLES THERE ARE MORE DEGREES 4HE
OCTAGON HAS WHILE THE PENTAGON HAS ONLY
0OSSIBLE ANSWER 4HE SIZE OF EACH ANGLE IN A
REGULAR POLYGON INCREASES AS THE NUMBER OF THE
SIDES INCREASE &OR EXAMPLE EACH ANGLE OF AN
EQUILATERAL TRIANGLE IS AND EACH ANGLE OF A
SQUARE IS #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS X X ??
?
X
4HE MEASURE OF THE UNKNOWN ANGLE IS (OLT -C$OUGAL -ATHEMATICS
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS X X ?? ?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS 4HERE ARE TRIANGLES
4HE SUM OF THE ANGLE MEASURES OF A NONAGON
IS 4HERE ARE TRIANGLES
4HE SUM OF THE ANGLE MEASURES OF AN OCTAGON
IS 4HERE ARE TRIANGLES
4HE SUM OF THE ANGLE MEASURES OF A QUADRILATERAL
IS X X ??
?
X 4HE MEASURE OF THE OTHER ACUTE ANGLE IS X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS /NE
OBTUSE ANGLE MAKES THIS AN OBTUSE TRIANGLE
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS /NE
OBTUSE ANGLE MAKES THIS AN OBTUSE TRIANGLE
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS 4HREE
ACUTE ANGLES MAKE THIS AN ACUTE TRIANGLE
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS /NE
RIGHT ANGLE MAKES THIS A RIGHT TRIANGLE
4HERE ARE TRIANGLES
4HE 0ENTAGON HAS CONGRUENT ANGLES SO DIVIDE BY %ACH ANGLE MADE BY THE 0ENTAGONS OUTER WALLS
IS 3INCE IT IS A RIGHT TRIANGLE IT HAS ONE ANGLE
3INCE IT IS ISOSCELES THE TWO ACUTE ANGLES ARE
CONGRUENT 4HEY HAVE A SUM OF SO EACH
MEASURES ! POLYGON HAS FEWER INTERIOR TRIANGLES THAN ITS
NUMBER OF SIDES 3UBTRACT FROM THE NUMBER OF
SIDES S TO FIND THE NUMBER OF TRIANGLES THE POLYGON
CAN BE DIVIDED INTO -ULTIPLY S BY TO FIND
THE SUM OF THE ANGLE MEASURES S THE SUM OF THE INTERIOR ANGLE MEASURES IN A POLYGON
WITH S SIDES
0OSSIBLE ANSWER 3IX TRIANGLES CAN BE DRAWN INSIDE
AN OCTAGON NOT SEVEN
$IVIDE THE QUADRILATERAL INTO TWO TRIANGLES AND THEN
FIND THE SUM OF THE ANGLES MEASURES OF EACH TRIANGLE
X X X
X X ????
???
X X S D 4HE ANGLES AT THE TUGBOAT AND CARGO SHIP ARE AND " X X ??
?
X
4HE MEASURE OF THE UNKNOWN ANGLE IS ?
???
?X? ???
?
P X P X
P
?N?
T
?? ???
???
T N T N ! PARALLELOGRAM RECTANGLE RHOMBUS AND SQUARE ALL
HAVE TWO PAIRS OF OPPOSITE CONGRUENT SIDES
! SQUARE AND A RHOMBUS BOTH HAVE FOUR CONGRUENT
SIDES
(OLT -C$OUGAL -ATHEMATICS
2%!$9 4/ '/ /. 0!'% ? ? ?
2ADII "# "$ "!
?
$IAMETER #$
? ?
#HORDS #$ %&
4HE MEASURE OF THE CENTRAL ANGLE OF EACH SECTOR
IS 4HE FIGURE IS A REGULAR POLYGON )T IS A PENTAGON
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS 0!'%3 n
4HINK AND $ISCUSS
0OSSIBLE ANSWER )F THE SIDES AND ANGLES OF THE
ISOSCELES TRIANGLE ARE CONGRUENT TO THE SIDES AND
ANGLES OF THE RIGHT TRIANGLE THEN AN ISOSCELES
TRIANGLE CAN BE CONGRUENT TO A RIGHT TRIANGLE
4HE FIGURE IS AN OCTAGON "UT IT IS NOT A REGULAR
OCTAGON BECAUSE NOT ALL SIDES AND NOT ALL ANGLES
ARE CONGRUENT
4HE FIGURE IS A TRIANGLE "UT IT IS NOT A REGULAR
TRIANGLE BECAUSE NOT ALL SIDES AND NOT ALL ANGLES
ARE CONGRUENT
4HE FIGURE IS A REGULAR POLYGON )T IS A HEXAGON
TWO CONGRUENT SIDES ONE RIGHT ANGLE
4HIS IS AN ISOSCELES RIGHT TRIANGLE
NO CONGRUENT SIDES ONE OBTUSE ANGLE
4HIS IS A SCALENE OBTUSE TRIANGLE
TWO CONGRUENT SIDES ONE OBTUSE ANGLE
4HIS IS AN ISOSCELES OBTUSE TRIANGLE
0OSSIBLE ANSWER 4HE CORRESPONDING ANGLE
MEASURES OF CONGRUENT FIGURES ARE ALWAYS EQUAL
SO THE FIGURES ARE SIMILAR
%XERCISES
4HE TRIANGLES ON THE GAME BOARD ARE CONGRUENT
AND THE HOLES ON THE GAME BOARD ARE ALSO
CONGRUENT 4HE PAIRS OF DRAGONS AND WORDS
h#HINESE #HECKERSv MAY ALSO BE CONSIDERED TO BE
CONGRUENT
NO CONGRUENT SIDES ONE RIGHT ANGLE
4HIS IS A SCALENE RIGHT TRIANGLE
4HE FIGURE HAS EXACTLY ONE PAIR OF PARALLEL SIDES
SO IT IS A TRAPEZOID 4HIS FIGURE IS BEST DESCRIBED AS
A TRAPEZOID
.ONE 4HE TRIANGLES IN THE KITES DESIGN ARE NOT
CONGRUENT
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES SO IT IS
A PARALLELOGRAM )T HAS FOUR CONGRUENT SIDES SO
IT IS ALSO A RHOMBUS )T HAS FOUR RIGHT ANGLES SO IT
IS ALSO A RECTANGLE AND SQUARE 4HIS FIGURE IS BEST
DESCRIBED AS A SQUARE
0OSSIBLE ANSWER 4HERE ARE TWO SHAPES IN THE
MIDDLE THAT DO NOT HAVE A CONGRUENT SHAPE THE
KITE IN THE UPPER MIDDLE AND THE TRIANGLE AT THE
BOTTOM IN THE MIDDLE /THERWISE EACH SHAPE ON THE
LEFT SIDE HAS A CONGRUENT SHAPE ON THE RIGHT SIDE
TWO TRIANGLES TWO TRAPEZOIDS TWO PARALLELOGRAMS
TWO OTHER TRIANGLES TWO PENTAGONS TWO OTHER
PENTAGONS TWO RECTANGLES TWO OTHER PENTAGONS
AND TWO KITES
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES SO IT IS A
PARALLELOGRAM )T HAS FOUR CONGRUENT SIDES SO IT IS
ALSO A RHOMBUS 4HIS FIGURE IS BEST DESCRIBED AS
A RHOMBUS
!# MM
$& MM
!" MM
$% MM
#" MM
&% MM
"Y THE 3IDEBY3IDE 2ULE N!"# IS CONGRUENT TO
N$%& OR N!"# z N$%& )F YOU ROTATE IT IT WILL FIT
EXACTLY OVER THE OTHER
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES SO IT IS
A PARALLELOGRAM 4HIS FIGURE IS BEST DESCRIBED AS
A PARALLELOGRAM
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS 4HE CORRESPONDING SIDES ARE NOT CONGRUENT
THEREFORE THE TRIANGLES ARE NOT CONGRUENT
4HE CORRESPONDING ANGLES IN CONGRUENT POLYGONS
ARE CONGRUENT 4HE UNKNOWN ANGLE MEASURE IS X X ?? ?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
4HE CORRESPONDING SIDES OF CONGRUENT POLYGONS ARE
CONGRUENT 4HE UNKNOWN SIDE MEASURE IS .ONE 4HE GEARS ARE NOT CONGRUENT
4HE TRIANGLES IN THE KITES DESIGN ARE CONGRUENT
SQUARES RECTANGLES TAN BROWN YELLOW
SMALL YELLOW LARGE
4HE CORRESPONDING SIDES ARE NOT CONGRUENT
THEREFORE THE TRIANGLES ARE NOT CONGRUENT
(OLT -C$OUGAL -ATHEMATICS
*+ M
!" M
+, M
"# M
*, M
!# M
"Y THE 3IDEBY3IDE 2ULE N+*, IS CONGRUENT TO
N"!# OR N+*, z N"!# )F YOU ROTATE IT IT WILL FIT
EXACTLY OVER THE OTHER
X
4HE CORRESPONDING ANGLES IN CONGRUENT POLYGONS
ARE CONGRUENT 4HE UNKNOWN ANGLE MEASURE IS 4HE CORRESPONDING SIDES IN CONGRUENT POLYGONS ARE
CONGRUENT 4HE UNKNOWN SIDE MEASURE IS CM
4HE CORRESPONDING ANGLES OF CONGRUENT POLYGONS
ARE CONGRUENT 4HE UNKNOWN ANGLE MEASURE IS
4HE CORRESPONDING SIDES OF CONGRUENT POLYGONS
ARE ALSO CONGRUENT 4HE UNKNOWN SIDE MEASURE
IS IN
X
X
X
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
.O 4HE THREE ANGLES IN EACH TRIANGLE ARE CONGRUENT
BUT THE SIDES IN ONE TRIANGLE CAN BE OF DIFFERENT
LENGTH THAN THOSE IN THE OTHER TRIANGLE
0OSSIBLE ANSWER #OMPARE THE MEASURES OF
THE THREE SIDES )F THE CORRESPONDING SIDES ARE
CONGRUENT THE TRIANGLES ARE CONGRUENT
Y
# 4HEY ARRIVE AT THE SAME TIME
5SE ,OGICAL 2EASONING
!NJIS VERTICAL DISTANCES WHEN COMBINED ARE
EQUAL TO !RTS VERTICAL DISTANCE !NJIS HORIZONTAL
DISTANCES WHEN COMBINED ARE EQUAL TO !RTS
HORIZONTAL DISTANCE 4HEY ARE WALKING THE SAME
DISTANCE
' MM
4HE CORRESPONDING
?
?SIDES OF
?CONGRUENT TRIANGLES ARE
CONGRUENT !" z $% AND $% MM 4HEREFORE
!" MM
Y
4HE SQUARES APPEAR TO BE CONGRUENT AND THE
TRIANGLES APPEAR TO BE CONGRUENT
# THE LIGHTNING BOLTS
4HE LIGHTNING BOLTS ARE THE SAME SIZE AND SAME
SHAPE
4HE CORRESPONDING
SIDES OF
CONGRUENT TRIANGLES ARE
?
?
?
CONGRUENT !" z $% AND $% M 4HEREFORE
!" M AND THE DISTANCE BETWEEN THE TREES
IS M
THE LENGTH OF EACH SIDE AND THE MEASURE OF EACH
ANGLE IN EACH PENTAGON
Y
THE LENGTH OF ADJACENT SIDES IN EACH RECTANGLE
THE LENGTH OF ONE SIDE IN EACH SQUARE
THE LENGTHS OF ALL OF THE SIDES
Y
X X ?? ?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS /NE
OBTUSE ANGLE MAKES THIS AN OBTUSE TRIANGLE
X X ?? ?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS /NE
OBTUSE ANGLE MAKES THIS AN OBTUSE TRIANGLE
X X ?? ?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS 4HREE
ACUTE ANGLES MAKE THIS AN ACUTE TRIANGLE
X X ?? ?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS /NE
RIGHT ANGLE MAKES THIS A RIGHT TRIANGLE
(OLT -C$OUGAL -ATHEMATICS
0!'%3 n
, - . 4HINK AND $ISCUSS
0OSSIBLE ANSWER )F A SKATER OUTLINES THE SHAPE OF A
LOWERCASE D ON THE ICE HE WOULD BE ABLE TO PERFORM
A TRANSLATION AND A ROTATION AT THE SAME TIME
a
%XERCISES
ROTATION
a
TRANSLATION
! " # :
X
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a
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PLACE CHECKMARK
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MOD
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('%/%,*
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COLOR
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!LL RIGHTS RESERVED
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TRANSLATION
!`
9
REFLECTION
Y
(OLT -C$OUGAL -ATHEMATICS
!+ 0,&
)'*& /
4HE CORRESPONDING
SIDES OF CONGRUENT TRIANGLES ARE
?
?
CONGRUENT -/ z 02 AND -/ IN 4HEREFORE
02 IN
! " # $ 0!'%3 n
4HINK AND $ISCUSS
0OSSIBLE ANSWER
, - . 0OSSIBLE ANSWER 9ES AN
EQUILATERAL TRIANGLE HAS
ROTATIONAL SYMMETRY )T SHOWS ROTATIONAL SYMMETRY
#! THREE TIMES WITHIN A ROTATION
%XERCISES
*,
('%/%,*
+"&)%
&(
(&)%.
0OSSIBLE ANSWER 4HE TWO PEOPLE ARE THE SAME SIZE
%$%"'
*/+%
AND SHAPE BUT THEY ARE HOLDING DIFFERENT OBJECTS
AND THEIR SKIRTS ARE SLIGHTLY DIFFERENT
0OSSIBLE ANSWER 4HE STICK FIGURES
ARE A REFLECTION OF
EACH OTHER BUT THEY ARE ALSO A
ROTATION OF EACH
OTHER /NE IS ALSO A TRANSLATED
IMAGE OF THE OTHER
+"&)%
-(&)%.
(%(*
#! *,
)%!$&*$
('%/%,*
&(
%$%"'
(&)%.
*/+%
)%!$&*$
-(&)%.
(%(*
! S D
4HERE IS LINE OF SYMMETRY
0OINT 8 IS ORIGINALLY LOCATED AT S D -OVING IT
4HERE ARE NO LINES OF SYMMETRY
UNITS DOWN AND UNITS TO THE RIGHT WILL PUT IT AT
S D
! " # #! 4HERE IS LINE OF SYMMETRY
$RAW LINES
FROM THE CENTER OF THE FIGURE OUT THROUGH
IDENTICAL
PLACES IN THE FIGURE #OUNT THE NUMBER
*,
('%/%,*
OF LINES DRAWN 4HE HEXAGON WILL SHOW ROTATIONAL
SYMMETRY
)%!$&*$
TIMES WITHIN A ROTATION
+"&)%
&(
-(&)%.
(&)%.
$RAW LINES FROM
THE CENTER OF THE FIGURE OUT THROUGH
%$%"'
*/+% IDENTICAL PLACES
(%(*IN THE FIGURE #OUNT THE NUMBER
OF LINES DRAWN 4HE FIGURE WILL SHOW ROTATIONAL
SYMMETRY TIMES WITHIN A ROTATION
4HE MEDIAN OF THE DATA IS 4HE RANGE OF THE DATA IS 4HE CORRESPONDING
SIDES OF CONGRUENT TRIANGLES ARE
?
?
CONGRUENT !# z &$ AND !% M 4HEREFORE
&$ M
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
(OLT -C$OUGAL -ATHEMATICS
$RAW LINES FROM THE CENTER OF THE FIGURE OUT THROUGH
IDENTICAL PLACES IN THE FIGURE #OUNT THE NUMBER
OF LINES DRAWN 4HE HEXAGON WILL SHOW ROTATIONAL
SYMMETRY TIMES WITHIN A ROTATION
)N ORDER TO HAVE ROTATIONAL SYMMETRY A FIGURE MUST
MATCH ITSELF AT LEAST ONCE BEFORE ROTATING COMPLETELY
AROUND OTHERWISE ALL FIGURES WOULD HAVE ROTATIONAL
SYMMETRY
0OSSIBLE ANSWERS "%$ AND -/ $ 4HE FIGURE WILL SHOW ROTATIONAL SYMMETRY TIMES
WITHIN A ROTATION
$RAW LINES FROM THE CENTER OF THE FIGURE OUT THROUGH
IDENTICAL PLACES IN THE FIGURE #OUNT THE NUMBER
OF LINES DRAWN 4HE FIGURE WILL SHOW ROTATIONAL
SYMMETRY TIMES WITHIN A ROTATION
CM ??????
CM
?????
X
M
X X 4HE LENGTH OF THE ACTUAL BRIDGE IS M
4HERE ARE NO LINES OF SYMMETRY
4HERE IS LINE OF SYMMETRY
7HEN TRIANGLE *+, IS TRANSLATED UNITS TO THE RIGHT
AND UNITS DOWN THE VERTICES BECOME * S D
+ S D AND , S D
7HEN TRIANGLE *+, IS REFLECTED ACROSS THE YAXIS THE
VERTICES BECOME
* S D + S D AND , S D
2%!$9 4/ '/ /. 0!'% %ACH OF THE CORRESPONDING SIDES OF THESE TWO
TRIANGLES IS CONGRUENT SO THE TRIANGLES ARE
CONGRUENT
4HERE ARE LINES OF SYMMETRY
4HE CORRESPONDING SIDES OF THESE TWO TRIANGLES ARE
NOT CONGRUENT SO THE TRIANGLES ARE NOT CONGRUENT
$RAW LINES FROM THE CENTER OF THE FIGURE OUT THROUGH
IDENTICAL PLACES IN THE FIGURE #OUNT THE NUMBER
OF LINES DRAWN 4HE FIGURE WILL SHOW ROTATIONAL
SYMMETRY TIMES WITHIN A ROTATION
4HE CORRESPONDING
?SIDES OF CONGRUENT TRIANGLES ARE
?
CONGRUENT 9: z #$ AND 9: M 4HEREFORE
#$ M
$RAW LINES FROM THE CENTER OF THE FIGURE OUT THROUGH
IDENTICAL PLACES IN THE FIGURE #OUNT THE NUMBER
OF LINES DRAWN 4HE OCTAGON WILL SHOW ROTATIONAL
SYMMETRY TIMES WITHIN A ROTATION
Y
2
$RAW LINES FROM THE CENTER OF THE FIGURE OUT THROUGH
IDENTICAL PLACES IN THE FIGURE #OUNT THE NUMBER
OF LINES DRAWN 4HE HEXAGON WILL SHOW ROTATIONAL
SYMMETRY TIMES WITHIN A ROTATION
2@
3@ 4HE SNOWFLAKE HAS LINES OF SYMMETRY 4HE
SNOWFLAKE WILL SHOW ROTATIONAL SYMMETRY TIMES
WITHIN ONE FULL ROTATION
4
! REGULAR NONAGON WILL SHOW ROTATIONAL SYMMETRY
TIMES WITHIN ONE FULL ROTATION BECAUSE LINES
FROM THE CENTER CAN BE DRAWN TO EACH VERTEX
X
3
4 @ 2 S D 3 S D AND 4 S D
4HE DESIGN HAS VERTICAL AND HORIZONTAL LINES OF
SYMMETRY ALONG THE FOLDS 4HE DESIGN SHOWS
ROTATIONAL SYMMETRY TIMES IN ONE ROTATION
)T WILL SHOW ROTATIONAL SYMMETRY TIMES IN ONE FULL
TURN IF YOU CONSIDER ONLY SHAPE #ONSIDERING BOTH
SHAPE AND COLOR IT HAS NO ROTATIONAL SYMMETRY
0OSSIBLE ANSWER 7HAT IS THE SMALLEST ANGLE OF
ROTATIONAL SYMMETRY FOR THE SQUARE
! " # $ #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
(OLT -C$OUGAL
-ATHEMATICS
#! EQUILATERAL ACUTE TRIANGLE
SCALENE RIGHT TRIANGLE
PARALLELOGRAM RHOMBUS RHOMBUS
PARALLELOGRAM RECTANGLE RECTANGLE
X X ??
?
X X " X * + , ??
?
)+ X '&$.$+)
($ #%)#
POLYGONS
#!4HE
CORRESPONDING ANGLES
IN CONGRUENT
ARE CONGRUENT
4HE
UNKNOWN
ANGLE
MEASURE
IS
*!%($
%'
'%($,'%($
4HE CORRESPONDING SIDES IN CONGRUENT POLYGONS ARE
$#$!&
).*$
'$')
CONGRUENT
4HE UNKNOWN SIDE MEASURE IS CM
*,
('%/%,*
+"&)%
%$%"'
4HIS FIGURE HAS LINES OF SYMMETRY
&(
)%!$&*$
(&)%.
-(&)%.
(%(* */+%
4HE FIGURE WILL SHOW ROTATIONAL SYMMETRY TIMES
WITHIN A ROTATION
345$9 '5)$% 2%6)%7 0!'%3 n
ACUTE OR ISOSCELES
PARALLEL LINES
CHORD
j???k
$&
???k ???k ???k
%$ &$ $&
$ % &
PLANE $%&
? ? ?
$% $& %&
ACUTE
STRAIGHT
SKEW
PARALLEL
" # $ 4HERE IS ONE VERTICAL LINE OF SYMMETRY THROUGH THE
CENTER OF THE FLAG
#(!04%2 4%34 0!'% 0OSSIBLE ANSWERS ! " # $ %
j???k j???k j???k
0OSSIBLE ANSWER !" #$ #"
AND THE ANGLE ARE CONGRUENT BECAUSE THEY
ARE VERTICAL ANGLES
M 0OSSIBLE ANSWER PLANE !"#
? ? ? ? ?
0OSSIBLE ANSWER !" #$ %# %" #"
???k ???k ???k ???k ???k ???k
0OSSIBLE ANSWER !" "! "# #" #$ $#
AND THE ANGLE ARE SUPPLEMENTARY
M M M !"# AND THE ANGLE ARE ACUTE ANGLES 3INCE
ALL OF THE ACUTE ANGLES IN THE FIGURE ARE CONGRUENT
M!"# M"#% BECAUSE IT IS A STRAIGHT ANGLE
AND ARE CONGRUENT BECAUSE THEY ARE
ALTERNATE INTERIOR ANGLES &ROM QUESTION M M $#% IS AN OBTUSE ANGLE )N THE FIGURE THE ACUTE
AND OBTUSE ANGLES ARE SUPPLEMENTARY
M$#% ?? ?
M$#% AND THE ANGLE ARE CONGRUENT BECAUSE THEY
ARE ALTERNATE EXTERIOR ANGLES
M 4HE LINES ARE
IN THE???
SAME PLANE AND DO NOT
j k
j???k
INTERSECT -. AND 0/ ARE PARALLEL
? ? ?
(& &) &'
?
')
? ? ? ?
() ') '* *)
4HE LINES ARE
IN DIFFERENT
PLANES AND DO NOT
j???k
j???k
INTERSECT ,- AND 0/ ARE SKEW
9ES ALL SIDES ARE CONGRUENT AND ALL ANGLES ARE
CONGRUENT
4HE
LINES???
APPEAR TO INTERSECT TO FORM RIGHT ANGLES
j???k
j k
./ AND -. ARE PERPENDICULAR
.O ALL SIDES ARE NOT CONGRUENT
2ADII !% %# "%
? ? ?
? ?
#HORDS !$ !#
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
(OLT -C$OUGAL -ATHEMATICS
?
$IAMETER !#
X X ??
?
X
4HE MEASURE OF THE UNKNOWN ANGLE IS .O ALL THE INTERIOR ANGLES ARE NOT CONGRUENT
.O ONE SIDE IS CURVED
9ES !LL SIDES ARE CONGRUENT AND ALL ANGLES ARE
CONGRUENT
4HREE CONGRUENT SIDES AND THREE ACUTE ANGLES
MAKE THIS AN EQUILATERAL ACUTE TRIANGLE
X X ??
?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS .O CONGRUENT SIDES AND ONE RIGHT ANGLE MAKE THIS A
SCALENE RIGHT TRIANGLE
4HE CORRESPONDING SIDES OF CONGRUENT POLYGONS ARE
CONGRUENT 4HE MISSING MEASURE IS IN
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES SO
IT IS A PARALLELOGRAM )T IS BEST DESCRIBED AS A
PARALLELOGRAM
4WO CONGRUENT SIDES AND THREE ACUTE ANGLES MAKE
THIS AN ISOSCELES ACUTE TRIANGLE
X
4HE FIGURE HAS TWO PAIRS OF PARALLEL SIDES SO IT IS A
PARALLELOGRAM )T HAS FOUR CONGRUENT SIDES SO IT IS
ALSO A RHOMBUS )T HAS FOUR RIGHT ANGLES SO IT IS ALSO
A RECTANGLE AND SQUARE )T IS BEST DESCRIBED AS A
SQUARE
4HE FIGURE HAS EXACTLY ONE PAIR OF PARALLEL SIDES SO
IT IS A TRAPEZOID )T IS BEST DESCRIBED AS A TRAPEZOID
X X ?? ?
X 4HE MEASURE OF THE UNKNOWN ANGLE IS #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
Y
! " # 4HERE IS VERTICAL LINE THROUGH THE CENTER OF
THE FLAG
4HERE ARE NO LINES OF SYMMETRY
(OLT -C$OUGAL -ATHEMATICS