Exercise 10D
Q1 | Ex-10D | Class 8 | SChand New learning Composite Maths | Quadrilaterals | myhelper
Question 1
Find the measures of the numbered angles in each rectangle.
(a)
∠ABD=58°
∵AB⊥BC
DC⊥BC
AB=DC , AD=BC
AB||DC , AD||DC
∴∠ABC=90°
∴∠DBC=∠4=90°-58°=32°
∠DCB=∠3=90°=∠BAD
∴∠BAD=∠5=90°
∠ABC=∠ADC=90°
∴∠BDA=∠1=∠DBC=32°
Again, ∠ABD=∠BDC=∠2=58°
∴∠1=32° ,∠3=90°
∠2=58° ,∠4=32°
∠5=90°
(b)
∵AD=DC=BC=AB
∠ADC=∠ABC=∠BAD=∠DCB=90°
∴∠1=∠ABC=90°
AC and BD bisect each other at O
∴∠ABD=∠DBC$=\frac{90}{2}$=45°
∴∠4=∠BDC=45°
∠BCA=∠5$=\frac{90}{2}$=45°
∴∠3=∠DAC=∠BCA=45°
∴∠CAB=∠2=45°
∴∠1=90° ,∠2=45° ,∠3=45° ,∠4=45°,∠5=45°
Q2 | Ex-10D | Class 8 | SChand New learning Composite Maths | Quadrilaterals | myhelper
Question 2
In rectangle PQRS , S R=20 cm and PR=35 cm. Find:
(a) PQ
(b) SK
PQRS is rectangle
SR=20 , PR=35
(a) PQ=20=SR [∵SR=PS]
(b) ∵PR=SQ=35 [∵SR and PQ bisect each other at O]
∴$SQ=\frac{1}{2}$SK$=\frac{35}{2}$=17.5cm
Q3 | Ex-10D | Class 8 | SChand New learning Composite Maths | Quadrilaterals | myhelper
Question 3
Find the value of each variable in these rectangles. Give brief reasons.
Sol :
(a)
∵ABCD is a rectangle
∴AD=BC=x=7
AB=DC=y=4
∠a=90°=∠c [∵AD⊥DC]
∠b=90°=∠d [BC⊥DC]
(b)
AB=DC
AD=BC
∠BCD=90°
∴∠BCA=55°
∴∠ACD=90°-55°=35°
∴AD||BC and AB||DC
∴∠ACD=∠CAB=35°=y
∴x=∠DAC=∠BCA=55°
∴x=55°
and y=35°
(c)
∵ABCD is rectangle
∴AB=DC ∴AB⊥AD
AD=BC DC⊥AD
∴∠ADC=5x=90°
∴$x=\frac{90}{5}$=18°
∴∠ADC=18°×5=90°
Now , 5y=15
∴$y=\frac{15}{5}=3$
∴AD=5×3=15
x=18 and y=3
(d)
AC and BD are bisect each other at "O"point
∴AC=BD
AD=BC [AD||BC]
AB=DC [AB||DC]
∴a=b=c=7unit
Q4 | Ex-10D | Class 8 | SChand New learning Composite Maths | Quadrilaterals | myhelper
Question 4
Find the value of each variable in these rhombuses. Give brief reasons.
Sol :
(a)
ABCD is rhombus
∴AB||DC
AD||BC
∴AB=DC=AD=BC=x=9
AC and BD are intersection at the point "O"
∴AC=BD
∴AC⊥OD
AC⊥OB
∴∠BOC=a=∠AOB=90°
∴a=90° , x=9 unit ,y=9 unit
(b)
∵ABCD is rhombus
∴AD||BC
AB||DC
∴AD=BC , AB=DC
∴∠a=∠c=x
∠A=∠d=105°
In quadrilateral ABCD
∠A+∠a+∠c+∠d=360°
105°+x+x+105°=360°
210°+2x=360°
2x=360°-210°
$x=\dfrac{150^{\circ}}{2}$
x=75°
(c)
ABCD is rhombus
∵AC and BO are intersecting at point "O"
∴AO=OC=x=4
∴BO=OD=y=6
(d)
∠a=∠b=∠c=d
ΔABC
$2a+c+\frac{a}{2}=180^{\circ}$
$2a+\frac{a}{2}+\frac{a}{2}=180^{\circ}$
[∵AB=BC
∴∠BAC=∠BCA
∴$\frac{a}{2}=C$]
$2a+\frac{2a}{2}=180^{\circ}$
3a=180°
$a=\frac{180}{3}$=60°
Q5 | Ex-10D | Class 8 | SChand New learning Composite Maths | Quadrilaterals | myhelper
Question 5
In the rhombus ABCD, find
(a) BC
∵ABCD is a rhombus
∴AB=BC=AD=DC=4a+5=15a-6
∴We know
4a+5=15a-6
11a=11
a=1
∴BC=4a+5=4×1+5=9
(b) ∠BCO
AC and BD are intersecting at point "O"
∴∠BOC=90°
∴11b+2=90°
11b=90°-2°=88°
$b=\frac{88}{11}$=8
∴∠ACD=4b-10
=(4×8-10)
=32-10=22
∠ACD=∠BCO (Diagonals of rhombus bisect interior angles)
Q6 | Ex-10D | Class 8 | SChand New learning Composite Maths | Quadrilaterals | myhelper
Question 6
Find the measures of the numbered angles in each rhombus.
(a)
Sol :
∠5=28° (Diagonals of rhombus bisect interior angle)
∠2=∠5=28° (alternate interior angle)
∠3=28° (alternate interior angle)
In triangle
∠1+∠2+28°=180° (Angle sum property of triangle)
∠1+28°+28°=180°
∠1+28°+28°=180°
∠1+56°=180°
∠1=180°-56°=124°
∠1=∠4 (Opposite angle of rhombus are equal)
(b)
Sol :
∠A=50°, ∠B=∠2+∠3, ∠C=∠4, ∠D=∠1+∠5
∠4=50°...(i) (Opposite angle of rhombus are equal)
∠B=∠D...(ii) (Opposite angle of rhombus are equal)
In quadrilateral ABCD,
∠A+∠B+∠C+∠D=360° (Angle sum property of quadrilateral)
50°+∠B+∠B+50°=360° from (i) and (ii)
2∠B=360°-100°
2∠B=260°
$\angle B=\dfrac{260^{\circ}}{2}$
B=130°
∠2=∠3 and ∠1=∠5 (Diagonals bisect interior angle)
∠B=∠2+∠3
∠B=∠2+∠2
∠B=130°
∠2+∠2=130°
2∠2=130°
$\angle 2=\dfrac{130^{\circ}}{2}$
∠2=65°
∴∠1=∠2=∠3=∠5=65°
(c)
Sol :
∠3=28° (alternate interior angle)
∠4=90° (Diagonals of rhombus is perpendicular bisectors)
In triangle AOB ,
∠OAB+∠AOB+∠ABO=180°
∠3+∠4+∠ABO=180°
28°+90°+∠ABO=180°
118°+∠ABO=180°
∠ABO=62°
∠ABO=∠OBD=62° (Diagonals of rhombus bisect interior angle)
∠2=∠5=ABO=∠1=62° (alternate interior angle)
Q7 | Ex-10D | Class 8 | SChand New learning Composite Maths | Quadrilaterals | myhelper
Question 7
Find the value of each variable in these squares. Give brief reasons.
(a)
ABCD is a square
∴AD=DC=BC=AB=x=y=6
∴∠BAD=∠ADC=∠DCB=∠CBA=a=b=c=d=90°
∴x=6 ,y=6 also, a=b=c=d=90°
(b)
AC and BD are intersect each other
AC=BD
∴∠BOC=90°
∴2p=90°
∴p=45°
Now ,ΔAOD
OD=OA [∴∠OAD=∠ODA=q]
∠AOD=90°
We know that
∠AOD+∠DOA+∠ODA=180°
90°+2q=180°
2q=180°-90°
2q=90°
$q=\frac{90}{2}$=45°
(c)
ABCD is square
AD=DC=BC=AB
AB and DC are intersecting at a point "O"
∴AO=OC=BO=OD=x=y=z=4
∴x=y=z=4
(d)
ABCD is square
∴AB=BC=DC=AD
∴∠ABC=∠BCD=∠CDA=∠DAB=90°
∴2x=90°
∴x=45°
∠CDA=90°
∠BDC$=\frac{1}{2}$∠CDA$=\frac{90}{2}=45$°=5y
∴5y=45°
y=9
Q8 | Ex-10D | Class 8 | SChand New learning Composite Maths | Quadrilaterals | myhelper
Question 8
One of the diagonals of a rhombus is equal to one of its sides. Find the angles of the rhombus.
Sol :
ABCD is rhombus
∴AC and BD are two diagonals
∴ Let , AC=x
AB=BC=CD=AD
ATQ ,
AB=x
∴ABC is equilateral triangle
ΔABC
AB=BC=AD
∠BAC=∠BCA=∠ABC=60°
∠B=60° , ∠A=60°×2=120° , ∠D=60° ,∠C=60°×2=120°
Q9 | Ex-10D | Class 8 | SChand New learning Composite Maths | Quadrilaterals | myhelper
Question 9
ABCD is a rectangle. E is he midpoint of AB. Prove that △DEC is an isosceles triangle.
Sol :
E is mid point of AB
∴AE=EB
DC=CE
ABCD is a rhombus
AD=BC
ΔDEC➝
DE=CE
∴ΔDEC is isosceles triangle
Q10 | Ex-10D | Class 8 | SChand New learning Composite Maths | Quadrilaterals | myhelper
Question 10
PQRS is a rhombus with diagonals PR which is extended to a point T. Prove that TQ = TS.
Sol :
PQRS is a rhombus
∴PQ=SR
PS=QR
∴In ΔTRS
SR2+TR2=TS2
In ΔTRS
RQ2+TR2=TQ2
SR2+TR2=TQ2
∴We can write
TS2=TQ2
or TS=TQ
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