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#(!04%2 n -ÕÌÃÊiÞ 'EOMETRIC &IGURES !2% 9/5 2%!$9 0!'% 5SE THE ENDPOINTS? IN ANY ? TO NAME A SEGMENT ? ORDER 0OSSIBLE ANSWER 15 25 35 ? ? PROPORTION ORDERED PAIR PERCENT INTEGER "! ? /NE TICK MARK ? AND "# !% ? 4WO TICK MARKS ? AND #% !$ AND #$ 4HREE TICK MARKS $ % & OF OF OF OF OF OF N N N N N N N N N T T T K K K j???k j???k 0OSSIBLE ANSWER $% %& N N N B B B D ? ? D D POINT ! S D POINT " S D POINT # S D POINT $ S D POINT % S D POINT & S D 0OSSIBLE ANSWER PLANE $%& ???k ???k ???k 0OSSIBLE ANSWER $% &$ %& ? ? ? 0OSSIBLE ANSWER $% %& $& ? ? !& ? AND "# ? !% ? AND "$ ? !" AND %$ ? ? &% AND #$ j???k 0LANE???!"# CONTAINS POINTS ! " AND # LINES !" ? ? ? j k AND "# LINE SEGMENTS !" !# AND "# AND ???k ???k ???k ???k ???k RAYS !" "! "# #" AND #! 0LANE !#$ CONTAINS ? ? ? POINTS !??? # AND $ LINE SEGMENTS !# !$ AND #$ k AND RAY #! ? ? ? ? ? 2 1 3 5 4 0OSSIBLE ANSWER PLANE 123 POINTS ??? 1 2 3 4 AND 5 j k j???k LINES 13 ??? 24 ??? k k ???k ???k AND RAYS 51 54 53 52 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 3 4 ! " # $ % ' ( ) * + - . & 2 / 0 CREATED .%43 2%6)3)/. SIMPLE 1 B 0OSSIBLE ANSWER? ? ? ? ? ? ? ? !" z '( #) z %* (+ z ,. -0 z .1 0OSSIBLE ANSWER .AME THE ENDPOINT A RAY FIRST ???OF k ???k ???k 0OSSIBLE ANSWER 51 54 53 % # " ! #HOOSE ANY THREE POINTS ON A PLANE TO NAME THE PLANE 0OSSIBLE ANSWER PLANE 123 $ $! 4)- (+ ^ ,. 0OSSIBLE ANSWER 1 2 3 */ JH $! %$)4%$ "9 BLACKLINE %XERCISES (27 #2%!4%$ "9 , 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 #534/-%2 0!'%3 n #HOOSE ANY TWO POINTS ON A LINE TO NAME THE LINE j???k j???k 0OSSIBLE ANSWER 13 24 ? LINE SEGMENTS 78 79 7: 89 8: AND 9: -4%?#?!$$?2 (OLT -C$OUGAL -ATHEMATICS ON MOD COM GREYSCALE %XERCISES .O TWO ENDPOINTS CANNOT BE SHARED BY TWO DIFFERENT LINE SEGMENTS )F TWO LINE SEGMENTS SHARE TWO ENDPOINTS THEN THE LINE SEGMENTS ARE THE SAME RIGHT ANGLE ACUTE ANGLE STRAIGHT ANGLE ! LINE IS A STRAIGHT PATH THAT EXTENDS FOREVER IN TWO DIRECTIONS A RAY IS A STRAIGHT PATH THAT EXTENDS FOREVER IN ONE DIRECTION FROM AN ENDPOINT AND A SEGMENT IS A STRAIGHT PATH FROM ONE ENDPOINT TO ANOTHER )T IS POSSIBLE-4%?#?4%#? TO ESTIMATE THE LENGTH OF A !24 &),% SEGMENT BECAUSE IT DOES NOT EXTEND FOREVER IN (27 #534/-%2 */" .5-"%2 ANY DIRECTION 33 #2%!4%$ "9 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 PARALLELRAILROAD TRACKS SIDES OF A LADDER PERPENDICULARSIDE AND BOTTOM EDGES OF A DESK DRAWER SKEWA 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 ) ' 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%!$9 4/ '/ /. 0!'% 3INCE VERTICAL ANGLES ARE CONGRUENT THE ANGLE 0OSSIBLE ANSWER ! " # OPPOSITE THE GIVEN ANGLE HAS THE SAME MEASURE #HOOSE ANY TWO POINTS ONk Aj??? LINE AS THE GIVEN ANGLE 4HE REMAINING ANGLES ARE j???k j??? k TO NAME THE LINE 0OSSIBLE ANSWER !' #% "& BOTH ADJACENT TO THE ANGLES THAT MEASURE 3INCE ADJACENT ANGLES ARE SUPPLEMENTARY BOTH REMAINING #HOOSE ANY THREE POINTS ON A PLANE TO NAME THE '4%?#?,?4 4%#( ANGLES MEASURE !24 &),% PLANE (27 #534/-%2 .5-"%2 0OSSIBLE */" ANSWER PLANE "#$ SOMETIMES JH #2%!4%$ "9 5SE THE $!4% ENDPOINTS IN ANY ORDER TO NAME A SEGMENT NEVER NEVER ? ? ? ? %$)4%$ "9 ALWAYS ? CREATED 0OSSIBLE$!4% ANSWER "$ "& $% A /2 ? AND 23 ARE PERPENDICULAR B 04 IS A TRANSVERSAL 2%6)3)/. C AND ARE CORRESPONDING ANGLES SIMPLE )F TWO INTERSECTING LINES FORM CONGRUENTBLACKLINE ADJACENT ANGLES THE LINES ARE PERPENDICULAR #OPYRIGHT © BY (OLT -C$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 %$)4%$ "9 (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 8 9 : / X # " #! $ % & #! ' a a * , 8` + - X 8 9 / : :` 9` ,#'*& )(&0&- + ( ' %/ %, * & ' ( * ) % ! $ & *@ 4%#( a a !24 &),%-4%?#,?! + "& ) % & ( ( & ) % . - ( & ) % (27 #534/-%2 */" .5-"%2 % $ %"'- * / + / % , . ( %( * JH #2%!4%$ "9 $!4% &@ JH %$)4%$ "9 $!4% (@ Y * '@ ( X ' & 4)-% #! CREATED .%43 ONLY ALTERED .%43 PLACE CHECKMARK 2%6)3)/. SIMPLE MOD COMPLEX V COMPLEX *, ('%/%,* + , - . BLACKLINE GREYSCALE COLOR Y)%!$&*$ 8 9 : -` +"&)% &( (&)%. -(&)%. Y :` : %$%"' 8` / */+% X 8 9` #OPYRIGHT © BY (OLT -C$OUGAL !LL RIGHTS RESERVED . , / / .` (%(* ,` X /` 9 ') &%&#( $" $" Y #` ! "` TRANSLATION !` 9 REFLECTION Y (OLT -C$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