e zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) ab = dc = 8 cm, ad = 4 cm and bc = 4.4...
TRANSCRIPT
![Page 1: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/1.jpg)
7
NCERT Textual Exercises and Assignm
entsExErcisE 3.1
1. Givenherearesomefigures.
(i) (ii)
(iii) (iv)
(v) (vi)
(vii) (viii)
Classifyeachofthemonthebasisofthefollowing. (a) Simplecurve (b) Simpleclosedcurve (c) Polygon (d) Convexpolygon (e) Concavepolygon 2. Howmanydiagonalsdoeseachofthefollowinghave? (a) Aconvexquadrilateral (b) Aregularhexagon (c) Atriangle 3. Whatisthesumofthemeasuresoftheanglesofaconvexquadrilateral?Will
thispropertyholdifthequadrilateralisnotconvex?(Makeanon-convexquadrilateralandtry!)
4. Examinethetable.(Eachfigureisdividedintotrianglesandthesumoftheanglesdeducedfromthat.) Figure
Side 3 4 5 6Angle sum
180° 2 × 180° = (4 – 2) × 180°
3 × 180° = (5 – 2) × 180°
4 × 180° = (6 – 2) × 180°
Maths VIII – Understanding Quadrilateral
![Page 2: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/2.jpg)
8
NCERT Textual Exercises and Assignm
ents Whatcanyousayabouttheanglesumofaconvexpolygonwithnumberof
sides? (a) 7 (b) 8 (c) 10 (d) n 5. Whatisaregularpolygon? Statethenameofaregularpolygonof (i) 3 sides (ii) 4 sides (iii) 6 sides 6. Findtheanglemeasurexinthefollowingfigures.
(a) (b)
(c) (d)
7.
(a) Findx + y + z (b) Findx + y + z + w
Maths VIII – Understanding Quadrilateral
![Page 3: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/3.jpg)
9
NCERT Textual Exercises and Assignm
entsTest Yourself - UQ1
1. Findthevalueoftheunknownangleineachofthefollowingfigures:
(i) (ii)
(iii)
2. Whatisthesumofallexterioranglesofaconvexquadrilateral? 3. Whatissumofallinterioranglesofapentagon? 4. InaquadrilateralABCD,∠A = 150° and ∠B=∠C = ∠D,find∠B,∠C and
∠D. 5. Themeasuresofthreeanglesofaquadrilateralare39°,141°and13°.Find
thefourthangle. (a) 117° (b) 167° (c) 108° (d) 137°
ExErcisE 3.2 1. Findxinthefollowingfigures.
(a) (b)
2. Findthemeasureofeachexteriorangleofaregularpolygonof (i) 9 sides (ii) 15 sides
Maths VIII – Understanding Quadrilateral
![Page 4: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/4.jpg)
10
NCERT Textual Exercises and Assignm
ents 3. Howmanysidesdoesaregularpolygonhaveifthemeasureofanexterior
angleis24°? 4. Howmanysidesdoesaregularpolygonhaveifeachofitsinteriorangles
is165°? 5. (a) Isitpossibletohavearegularpolygonwithmeasureofeachexterior
angleas22°? (b) Canitbeaninteriorangleofaregularpolygon?Why? 6. (a) Whatistheminimuminterioranglepossibleforaregularpolygon?
Why? (b) Whatisthemaximumexterioranglepossibleforaregularpolygon?
Test Yourself - UQ2 1. Aquadrilateralhasallfouranglesofthesamemeasure,whatisthemeasure
ofeachangle? 2. Twoanglesofaquadrilateralareofmeasure50°andtheothertwoangles
areequal.Whatisthemeasureofeachofthesetwoangles? 3. ABCDisaquadrilateral.AOandBOaretheanglebisectorsofangleAand
BwhichmeetatO.If∠C=70°,∠D=50°,find∠AOB. 4. Thevalueofyinthegivenfigureis:
(a) 89° (b) 91° (c) 101° (d) 79° 5. Findthemeasureof∠ADC.
Maths VIII – Understanding Quadrilateral
![Page 5: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/5.jpg)
11
NCERT Textual Exercises and Assignm
ents (a) 37° (b) 47° (c) 133° (d) 43° 6. Whatisthevalueofa?
(a) 90° (b) 10° (c) 180° (d) 2° 7. Findthevaluesofx,y and zinthediagram.Givereasonswherevernecessary.
8. Inthegivendiagram,thevalueofx is
(a) 170° (b) 190° (c) 100° (d) 90° 9. Iftheanglesofaquadrilateralarex°,(2x+13)°,(3x+10)°,(x–6)°,findx.
Maths VIII – Understanding Quadrilateral
![Page 6: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/6.jpg)
12
NCERT Textual Exercises and Assignm
entsExErcisE 3.3
1. Given a parallelogramABCD.Complete each statement alongwith thedefinitionorpropertyused.
(i) AD = ...... (ii) ∠DCB=...... (iii) OC=...... (iv) m∠DAB+m∠CDA = ...... 2. Considerthefollowingparallelograms.Findthevaluesoftheunknownsx,
y,z.
(i) (ii)
(iii) (iv)
(v)
3. CanaquadrilateralABCDbeaparallelogramif (i) ∠D + ∠B=180°? (ii) AB=DC=8cm,AD=4cmandBC=4.4cm? (iii) ∠A = 70° and ∠C=65°? 4. Drawaroughfigureofaquadrilateralthatisnotaparallelogrambuthas
exactlytwooppositeanglesofequalmeasure.
Maths VIII – Understanding Quadrilateral
![Page 7: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/7.jpg)
13
NCERT Textual Exercises and Assignm
ents 5. Themeasuresof twoadjacent anglesof aparallelogramare in the ratio
3:2.Findthemeasureofeachoftheanglesoftheparallelogram. 6. Twoadjacentanglesofaparallelogramhaveequalmeasure.Findthemeasure
ofeachoftheanglesoftheparallelogram.
7. TheadjacentfigureHOPEisaparallelogram.Findtheanglemeasuresx,y and z.Statethepropertiesyouusetofindthem.
8. ThefollowingfiguresGUNSandRUNSareparallelograms.Findx and y. (Lengthsareincm)
(i) (ii)
9.
IntheabovefigurebothRISKandCLUEareparallelograms.Findthevalueof x.
10. Explainhowthisfigureisatrapezium.Whichofitstwosidesareparallel?(Fig3.32)
Maths VIII – Understanding Quadrilateral
![Page 8: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/8.jpg)
14
NCERT Textual Exercises and Assignm
ents 11. Findm∠CinFig3.33ifAB DC|| .
12. Findthemeasureof∠P and ∠S if SP RQ|| inFig3.34.(Ifyoufindm∠R,istheremorethanonemethodtofindm∠P?)
Test Yourself - UQ3 1. Theanglesofaquadrilateralareintheratio2:3:5:8.Findthemeasure
ofeachofthefourangles. 2. ABCDisaparallelogram:If∠A=70°,calculate∠B,∠C and ∠D.
3. Theperimeterofaparallelogramis150cm.Oneofitssidesisgreaterthantheotherby25cm.Findthelengthsofallthesidesoftheparallelogram.
4. Theratiooftwosidesofaparallelogramis3:5,anditsperimeteris48cm.Findthesidesoftheparallelogram.
5. DiagonalsofaparallelogramABCDintersectat).XYcontainsO,andX,Yarepointsonoppositesidesoftheparallelogram.Givereasonsforeachofthefollowingstatements:
Maths VIII – Understanding Quadrilateral
![Page 9: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/9.jpg)
15
NCERT Textual Exercises and Assignm
ents (i) OB=OD (ii) ∠OBY=∠ODX (iii) ∠BOY=∠DOX (iv) DBOY≅ DDOX Now,stateifXYisbisectedatO. 6. ABCDisaparallelogram.CEbisects∠CandAFbisects∠A.Ineachofthe
following,ifthestatementistrue,giveareasonforthesame.
(i) ∠A = ∠C (ii) ∠FAB=1/2∠A (iii) ∠DCE=1/2∠C (iv) ∠FAB=∠DCE (v) ∠DCE=∠CEB (vi) ∠CEB=∠FAB (vii) CE||AF (viii) AE||FC 7. In a DABC,D,E,Farerespectively,themid-pointsofBC,CAandAB.If
thelengthsofsideAB,BCandCAare17cm,18cmand19cmrespectively,findtheperimeterofDDEF.
ExErcisE 3.4 1. StatewhetherTrueorFalse. (a) Allrectanglesaresquares (b) Allrhombusesareparallelograms (c) Allsquaresarerhombusesandalsorectangles (d) Allsquaresarenotparallelograms. (e) Allkitesarerhombuses. (f) Allrhombusesarekites.
Maths VIII – Understanding Quadrilateral
![Page 10: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/10.jpg)
16
NCERT Textual Exercises and Assignm
ents (g) Allparallelogramsaretrapeziums. (h) Allsquaresaretrapeziums. 2. Identifyallthequadrilateralsthathave. (a) foursidesofequallength (b) fourrightangles 3. Explainhowasquareis. (i) aquadrilateral (ii) aparallelogram (iii) arhombus (iv) arectangle 4. Namethequadrilateralswhosediagonals. (i) bisecteachother (ii) areperpendicularbisectorsofeachother (iii) are equal 5. Explainwhyarectangleisaconvexquadrilateral. 6. ABCisaright-angledtriangleandOisthemidpointofthesideoppositeto
therightangle.ExplainwhyOisequidistantfromA,BandC.(Thedottedlinesaredrawnadditionallytohelpyou).
Test Yourself - UQ4 1. ABCDisarhombuswith∠ABC=126°.Determine∠ACD.
2. ABCDisasquare.Determine∠DCA.
Maths VIII – Understanding Quadrilateral
![Page 11: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/11.jpg)
17
NCERT Textual Exercises and Assignm
ents 3. Thediagonalsofarhombusare6cmand8cm.Findthelengthofasideof
therhombus.
4. ABCDisarhombuswith∠ABC=56°.Determine∠CAD. 5. ABCDisatrapeziumandABEDisasquare.IfBE=EC,find:(a)∠BAE
(b) ∠ABC(c)WhatshapeisthefigureABCE? 6. ABCDisakiteand∠A = ∠C. If ∠CAD=70°,∠CBD=65°,find:(a)
∠BCD(b)∠ADC.
Maths VIII – Understanding Quadrilateral
![Page 12: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/12.jpg)
22
NCERT Textual Exercises and Assignments
Exercise – 3.1 1. (a) Simplecurve
(1) (2) (5) (6) (7) (b) Simpleclosedcurve
(1) (2) (5) (6) (7) (c) Polygons
(1) (2) (4) (d) Convexpolygons
(1) (e) Concavepolygon
(1) (4) 2. (a) Aconvexquadrilateralhastwodiagonals. Here,ACandBDaretwodiagonals.
D C
A B (b) Aregularhexagonhas9diagonals. Here,diagonalsareAD,AE,BD,BE,FC,FB,AC,ECandFD.
Maths VIII – Understanding Quadrilateral
![Page 13: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/13.jpg)
23
E D
C
BA
F
(c) Atrianglehasnodiagonal. 3. LetABCDisaconvexquadrilateral,thenwedrawadiagonalACwhichdividesthequadrilateral
intwotriangles. ∠A+B+∠C + ∠D = ∠1 + ∠6 + ∠5 + ∠4 + ∠3 + ∠2 = (∠1 + ∠2 + ∠3) + (∠4 + ∠5 + ∠6) =180°+180°[ByAnglesumpropertyoftriangle] Hence,thesumofmeasuresofthetrianglesofaconvexquadrilateralis360°
16
5
2 34
CD
B
A Yes,ifquadrilateralisnotconvexthen,thispropertywillalsobeapplied. LetABCDisanon-convexquadrilateralandjoinBD,whichalsodividesthequadrilateralin
twotriangles. Usinganglesumpropertyoftriangle, InDABD,
A1
D
C6
4
3B 2
5
∠1 + ∠2 + ∠3 = 180° ...(i) ∠4 + ∠5 + ∠6 = 180° ...(ii) Addingequation(i)and(ii), ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360° ⇒ ∠1 + ∠2 + (∠3 + ∠4) + ∠5 + ∠6 = 360° ⇒ ∠A + ∠B+∠C + ∠D = 360° Henceproved.4. (a) Whenn =7,then Angle sum of a polygon = (n – 2) × 180° = 5 × 180° = 900° (b) Whenn=8,then Angle sum of a polygon = (n – 2) × 180° = (8 – 2) × 180° = 6 × 180
=1080°
Maths VIII – Understanding Quadrilateral
![Page 14: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/14.jpg)
24
(c) Whenn=10,then Angle sum of a polygon = (n – 2) × 180° = (10 – 2) ×180° = 8 × 180°= 1440° (d) Whenn = n, then Angle sum of a polygon = (n – 2) × 180 5. Aregularpolygon:Apolygonhavingallsidesofequallengthandtheinterioranglesofequal
sizeisknownasregularpolygon. (i) 3 sides Polygonhavingthreesidesiscalledatriangle. (ii) 4 sides Polygonhavingfoursidesiscalledaquadrilateral. (iii) 6 sides Polygonhavingsixsidesiscalledahexagon. 6. (a) Usinganglesumpropertyofaquadrilateral, 50° + 130° + 120° + x = 360° ⇒ 300° + x = 360° ⇒ x = 360° – 300 ⇒ x = 60° (b) Usinganglesumpropertyofaquadrilateral, 90° + 60° + 70° + x = 360° ⇒ 220° + x = 360° ⇒ x = 360° – 220 ⇒ x = 140° (c) Firstbaseinteriorangle=180°–70°=110° Secondbaseinteriorangle=180°–60°=120° \ Angle sum of a polygon = (n – 2) × 180° = (5 – 2) × 180° = 3 × 180° = 540° \ 30° + x + 110° + 120° + x = 540° ⇒ 260° + 2x = 540° ⇒ 2x = 280° ⇒ x = 140° (d) Angle sum of a polygon = (n – 2) × 180° = (5 – 2) × 180° = 3 × 180° = 540° \ x + x + x +x + x = 540° ⇒ 5x = 540° ⇒ x = 108° Henceeachinteriorangleis108°.
Maths VIII – Understanding Quadrilateral
![Page 15: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/15.jpg)
25
7. (a) Sincesumoflinearpairangleis180° \ 90° + x = 180 ⇒ x = 180° – 90° = 90° and z + 30° = 180° ⇒ z = 180° – 30° = 150° Also y=90°+30°=120° [Exteriorangleproperty] \ x + y + x = 90° + 120° + 150° = 360° (b) Usinganglesumpropertyofaquadrilateral, 60° + 80° + 120° + n = 360° ⇒ 260° + n = 360° ⇒ n = 360° – 260° ⇒ n = 100° Sincesumoflinearpairanglesis180° \ w + 100 = 180° ....(i) x + 120° = 180° ....(ii) y + 80° = 180° ...(iii) z+60°=180° ...(vi) Addingeq.(i),(ii),(iii)and(iv) ⇒ x + y + z + w + 100° + 120° + 80° + 60° = 180° + 180° + 180° ⇒ x + y + z + w + 360° = 720° ⇒ x + y + z + w = 720° – 360° ⇒ x + y + z + w = 360°
Test Yourself - UQ1 1. (i) 80° (ii) 40° (iii) 233° 2. 360º 3. Sum=540º 4. 70°,70°,70° 5. 167°
Maths VIII – Understanding Quadrilateral
![Page 16: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/16.jpg)
26
Exercise – 3.2ExErcisE 3.2 ExErcisE – 3.1 1. (a) Here,125°+m=180° [Linearpair] ⇒ m = 180° – 125° = 55°
and 125° + n=180° [Linearpair] ⇒ n = 180° – 125° = 55° Exterioranglex°=sumofoppositeinteriorangles \ x° = 55° + 55° = 110 (b) Sumofanglesofapentagon =(n – 2) × 180° = (5 × 2) × 180° = 3 × 180° = 540° Bylinearpairsofangles, ∠1 + 90° = 180° ...(i) ∠2 + 60° = 180° ...(ii) ∠3 + 90° = 180° ...(iii) ∠4+70°=180° ....(iv) ∠5 + x=180° ...(v) Addingequation(i),(ii),(iii),(iv)and(v) x + (∠1 + ∠2 + ∠3 + ∠4 + ∠5) + 310° = 900 ⇒ x + 540° + 310° = 900 ⇒ x + 850° = 900° ⇒ x = 900° – 850° = 50° 2. (i) Sum of angles of a regular polygon = (n – 2) × 180° = (9 – 2) × 180° = 7 × 180° = 1260°
Eachinteriorangle = = ° = °Sum of interior angles Number of sides
12609
140
(ii) Sumofexterioranglesofaregularpolygon=360°
Eachinteriorangle = = ° = °Sum of interior angles Number of sides
36015
24
3. Letnumberofsidesben. Sumofexterioranglesofaregularpolygon=360°
Maths VIII – Understanding Quadrilateral
![Page 17: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/17.jpg)
27
Number of sides = = °°
=Sum of exterior angles Each interior angles
36024
15
Hence,theregularpolygonhas15sides. 4. Letnumberofsidesben. Exteriorangle=180°–165°=15°
Number of sides = = °°
=Sum of exterior angles Each interior angles
36015
24
Hence,theregularpolygonhas24sides. 5. (a) No.(since22isnotadivisorof360°) (b) No,(Becauseeachexteriorangleis180°–22°=158°,whichisnotadivisorof360°) 6. (a) Theequilateral trianglebeingaregularpolygonof3 sideshas the leastmeasureofan
interiorangleof60°. Sumofalltheanglesofatriangle=180° \ x + x + x = 180° ⇒ 3x= 180° ⇒ x = 60° (b) By(a),wecanobservethatthegreatestexteriorangleis180°–60°=120°.
Test Yourself - UQ2 1. 90° 2. 130° 3. 60° 4. 91° 5. 133° 6. 2° 7. 88°,68°,92° 8. 190° 9. x + 2x + 13 + 3x + 10 + x–1=360º 7x+17=360º 7x=360º–17 7x=343º
x = 343
7 x=49º
Maths VIII – Understanding Quadrilateral
![Page 18: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/18.jpg)
28
Exercise – 3.3ExErcisE 3.2 ExErcisE – 3.1 1. (i) AD=BC[Sinceoppositesidesofaparallelogramareequal] (ii) ∠DCB=∠DAB[Sinceoppositeanglesofaparallelogramareequal] (iii) OC=OA[sincediagonalsofaparallelogrambisecteachother] (iv)m∠DAB+m∠CDA=180°[Adjacentanglesinaparallelogramaresupplementary] 2. (i) ∠B+∠C=180°[Adjacentanglesinaparallelogramaresupplementary]
⇒ 100° + x = 180° ⇒ x = 180° – 100° = 80° and z = x =80°[Sinceoppositeanglesofaparallelogramareequal]
Also y=100°[Sinceoppositeanglesofaparallelogramareequal] (ii) x+50°=180°[Adjacentanglesina||gmaresupplementary] ⇒ x = 180° – 50° = 130° ⇒ z = x =130°[Correspondingangles] (iii) x=90°[Verticallyoppositeangle] ⇒ y + x +30°=180°[Anglesumpropertyofatriangle] ⇒ y + 90° + 30° = 180° ⇒ y + 120 = 180° ⇒ y = 180° – 120° = 60° ⇒ z = y=60°[Alternateangles] (iv) z=80°[Correspondingangles]
⇒ x+80°=180°[Adjacentanglesina||gmaresupplementary] ⇒ x = 180° – 80° = 100° and y=80°[Oppositeanglesareequalina||gm]
Maths VIII – Understanding Quadrilateral
![Page 19: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/19.jpg)
29
(v) y=112°[Oppositeanglesareequalina||gm]
⇒ y=112°[Oppositeanglesareequalina||gm] ⇒ 40° + y + x=180°[Anglesumpropertyofatriangle] ⇒ 40° + 112° + x = 180° ⇒ 152° + x = 180° ⇒ x = 180° – 152° = 28° and z = x =28°[alternateangles] 3. (i) ∠D + ∠B=180° Itcanbe,buthere,itneedsnottobe.
(ii) No,inthiscasebecauseonepairofoppositesidesareequalandanotherpairofoppositesides are unequal.
So,itisnotaparallellogram.
(iii) No. ∠A ≠ ∠C. Sinceoppositeanglesareequalinparallelogramandhereoppositeanglesarenotequalin
quadrilateralABCD. Thereforeitisnotaparallelogram.
4. ABCDisaquadrilateralinwhichangles∠A = ∠C = 110°. Therefore,itcouldbeakite.
Maths VIII – Understanding Quadrilateral
![Page 20: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/20.jpg)
30
5. Lettwoadjacentanglesbe3x and 2x. Sincetheadjacentanglesinaparallelogramaresupplementary. \ 3x + 2x = 180° ⇒ 5x = 180°
⇒ x = ° = °1805
36
\ One angle = 3x = 3 × 36° = 108° And Anotherangle =2x = 2 × 36° = 72° 6. Leteachadjacentaanglebex. Sincetheadjacentanglesinaparallelogramaresupplementary. \ x + x = 180° ⇒ 2x = 180°
⇒ x = ° = °1802
90
Hence,eachadjacentangleis90°. 7. Here ∠HOP=180°–70°=110°[Angleoflinearpair] and ∠E=∠HOP[Oppositeanglesofa||gmareequal] ⇒ x = 110° ∠PHE=∠HPO[Alternateangles] \ y = 40°
Now ∠EHO=∠O=70°[Correspondingangles] ⇒ 40° + z = 70° ⇒ z = 70° – 40° = 30°
Maths VIII – Understanding Quadrilateral
![Page 21: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/21.jpg)
31
8. (i) InparallelogramGUNS, GS=UN [Oppositesidesofparallelogramareequal]
⇒ 3x = 18 x = =183
6 cm
Also GU=SN [Oppositesidesofparallelogramareequal] ⇒ 3y – 1 = 26 ⇒ 3y = 26 + 1
⇒ 3y = 27 ⇒ y = =273
9 cm Hence,x=6cmandy=9cm. (ii) InparallelogramRUNS, y+7=20 [Diagonalsof||gmbisectseachother] ⇒ y =20–7=13cm And x + y = 16 ⇒ x + 13 = 16 ⇒ x = 16 – 3 ⇒ x=3cm Hence,x=3cmandy=13cm. 9. InparallelogramRISK, ∠RIS=∠K=120° [Oppositeanglesofa||gmareequal] ∠m+120°=180° [Linearpair] ⇒ ∠m = 180° – 120° = 60° And ∠ECl=∠L=70° [Correspondingangles]
⇒ m + n + ∠ECl=180° [Anglesumpropertyofatriangle] ⇒ 60° + n + 70° = 180° ⇒ 130° + n = 180° ⇒ n = 180° – 130° = 50° Also x = n =50° [Verticallyoppositeangles] 10. Here,∠M+∠L = 100° + 80° = 180°
[Sumofinterioroppositeangleis180°]
Maths VIII – Understanding Quadrilateral
![Page 22: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/22.jpg)
32
\ NMandKLareparallel Hence,KLMNisatrapezium. 11. Here,∠B+∠C = 180°
AB||DC
\ 120 + m∠C = 180° ⇒ m∠C = 180° – 120° = 60° 12. Here,∠P + ∠Q=180° [Sumofco-interioranglesis180°] ⇒ ∠P + 130° = 180° ⇒ ∠P = 180° – 130° ⇒ ∠P = 50° \ ∠R=90° \ ∠S + 90° = 180° ⇒ ∠S= 180° – 90° ⇒ ∠S = 90° Yes,onemoremethodistheretofind∠P. ∠S + ∠R+∠Q + ∠P=360° [Anglesumpropertyofquadrilateral] ⇒ 90° + 90° + 130° + ∠P = 360° ⇒ 310° + ∠P = 360° ⇒ ∠P = 360° – 310° ⇒ ∠P = 50°.
Test Yourself - UQ3 1. 40°,60°,100°,160° 2. 110°,70°,110° 3. 25cm,50cm,25cm,50cm 4. 9cm,15cm,9cmand15cm 5.
Maths VIII – Understanding Quadrilateral
![Page 23: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/23.jpg)
33
(i) Diagonalsofaparallelogrambisecteachother. (ii) Alternateinterioranglerareequal. (iii) Verticallyoppositeangles. (iv) A.A.S.congrencycriteria. 7. 27cm
Exercise – 3.4ExErcisE 3.2 ExErcisE – 3.1 1. (a) False.Since,allsidesofsquaresareequal. (b) True.Since,inrhombus,oppositeanglesareequalanddiagonalsintersectatmid-point. (c) True.Since,squareshavethesamepropertyofrhombusbutnotarectangle. (d) False.Since,allsquareshavethesamepropertyofparallelogram. (e) False.Since,allkitesdonothaveequalsides. (f) True.Since,allrhombuseshaveequalsidesanddiagonalsbisecteachother. (g) True.Since,trapeziumhasonlytwoparallelsides. (h) True.Since,allsquareshavealsotwoparalleltines. 2. (a) Rhombusandsquarehavesidesofequallength. (b) Squareandrectanglehavefourrightangles. 3. (i) AsquareIsaquadrilateral,ifithasfourunequallengthsofsides. (ii) Asquareisaparallelogram,sinceitcontainsbothpairsofoppositesidesequal. (iii)Asquareisalreadyarhombus.Since,ithasfourequalsidesanddiagonalsbisectat90to
eachother (iv)Asquareisaparallelogram,sincehavingeachadjacentanglearightangleandopposite
sides are equal. 4. (i) Ifdiagonalsofaquadrilateralbisecteachotherthenitisarhombus,parallelogram,rectangle
or square. (ii) Ifdiagonalsofaquadrilateralareperpendicularbisectorofeachother,thenitisarhombus
or square. (iii)Ifdiagonalsareequal,thenItisasquareorrectangle 5. ArectangleisaconvexquadrilateralsinceitsvertexareraisedandbothofItsdiagonalsliein
itsinterior. 6. Since,tworighttrianglesmakearectanglewhere0isequidistantpointfromA,B,CandD
because0 is themid-pointof the twodiagonalsofarectangle.SinceACandBDareequaldiagonalsandintersectatmid-pointSo,0istheequidistantfromA.B,CandD
Test Yourself - UQ4 1. 27° 2. 45°
Maths VIII – Understanding Quadrilateral
![Page 24: E Zdd øãç ½ ø Ù ®Ý Ý Ä ÝÝ®¦Äà ÄãÝ · (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°? 4. Draw a rough figure of a quadrilateral](https://reader031.vdocument.in/reader031/viewer/2022022120/5e5f934e4a3bef154234b3e7/html5/thumbnails/24.jpg)
34
3. 5cm 4. 34 5. 45 6. 95°,40°
Maths VIII – Understanding Quadrilateral