ICSE Board: Suggested Books     ICSE Board:  Foundation Mathematics
Class 8: Reference Books               Class 8: NTSE Preparation
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Q.1 In the adjourning figure, name:281

  1. The point are M, N, P, Q , X, Y
  2. Five line segments are \overline{XM},\overline{MP},\overline{YN},\overline{NQ},\overline{MN},\overline{PQ}
  3. Four Rays are \overrightarrow{PB}, \overrightarrow{QD}, \overrightarrow{XA}, \overrightarrow{YC} 
  4. Four lines are \overleftrightarrow{AB}, \overleftrightarrow{CD}, \overleftrightarrow{EF}, \overleftrightarrow{HG} 
  5. Four Collinear points are A, X, M, P \ or\  C, Y, N, Q \ or\  X, M, P, B

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Q.2 In the adjoining figure, name :282

  1. Two points of intersecting lines are
    •  \overleftrightarrow{EF}, \overleftrightarrow{GH}  intersecting at R
    •  \overleftrightarrow{CD}, \overleftrightarrow{GH}  intersecting at Q.
  2. Three concurrent lines and their point of concurrence.
    • \overleftrightarrow{AB}, \overleftrightarrow{EF}, \overleftrightarrow{GH} and point of concurrence is R
  3. Three rays are, \overrightarrow{QH}, \overrightarrow{PB},\overrightarrow{PD} some other rays are \overrightarrow{RA}, \overrightarrow{RE},\overrightarrow{RG} , etc.
  4. Two line segments are \overline{QR}, \overline{PQ},\overline{RP}

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Q.3 State whether the following statements are true or false :

  1. A ray has no end Point: False
  2. A line AB is the same as line BA: True
  3. A ray AB is the same as BA: False
  4. A line has a definite length: False
  5. Two planes always meet in a line: True
  6. A plane has length and breadth but no thickness: True
  7. Two distinct points always determine a unique line: True
  8. Two lines may intersect in two points: False
  9. Two intersecting lines cannot be both parallel to the same line: True

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Q.4 Two adjacent angles on a straight line are x^{\circ} \ and\  (2x - 21)^{\circ}   Find (i) the value of x ii) the measure of each angle283

\ \ i)\ \ \angle AOB+\angle COB=180^{\circ}

 2x-21+x=180^{\circ}

 3x=201

 x=67^{\circ}

\ \ i)\ \  Hence\  \angle COB=67^{\circ} \ and\  \angle AOB=113^{\circ}

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Q.5. Two adjacent angles on a straight line are (3x - 2)^{\circ}  \ and\  4(x + 7)^{\circ} – Find :(i) the value of x ii) the measure of each angle

\ \ i)\ 3x-2+4(x+7)=180^{\circ}

 3x+4x+26=180^{\circ}

 7x=154

 or x=22

ii) The measure of angles

Angle 1 = 3\times 22-2=64^{\circ}

Angle 2 = 4\times (22+7)=116^{\circ}

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Q.6 Two adjacent angles on a straight line are in the ratio 3. Find the measure of each angle:

i) The ratio of angles = 3\colon 2

ii) Therefore the angles are 3x and 2x

 3x+2x=180^{\circ}

 5x=180^{\circ}

 x=36^{\circ}

The two angles are 108^{\circ}  \ and\  72^{\circ}

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Q.7 In the adjoining figure, AOB is a straight line. Find the value x . Hence ,find, \angle AOC \ and\  \angle BOD 284

 \angle AOC+\angle COD+\angle DOB=180^{\circ}

 3x-5+55+x+20=180^{\circ}

 4x=110^{\circ}

 \Rightarrow x=27.5^{\circ}

Therefore \angle AOC = 3\times 22-5=77.5^{\circ}

And \angle BOD = 22+20=47.5^{\circ}

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Q.8 In the adjoining figure, AOB is a straight line. If x \colon y \colon  z = 6 \colon 5 \colon  4 , find the values of x, \ y \ and\  z .285

 x\colon y\colon z=6\colon 5\colon 4

Therefore

 6a+5a+4a=180^{\circ}

Or a=12 \ therefore \ x=72^{\circ} , y=60^{\circ}  \ \&\  z=48^{\circ}

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Q.9 In the adjoining figure, what value of x make AOB a straight line?

For \overleftrightarrow{AB}   to be a straight line286

 3x+5+2x-25=180^{\circ}

 5x-20=180^{\circ}

 5x=200^{\circ}

 x=40^{\circ}

 \angle AOC=3\times 40+5=125^{\circ}

 \angle BOC=2\times 40-25=55^{\circ}

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Q 10. In the adjoining figure, find the value of x .287

  \angle DOA+\angle AOB+\angle BOC+\angle COD=360^{\circ}

 x+65+90+120=360

  x=360^{\circ} -275^{\circ}

  x=85^{\circ}

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Q 11. In each of the following figures, two lines   AB \ and\  CD  intersect at a point  O . Find the value of    x, \ y \ and \ z

288

\ \ i)\  \angle AOD=\angle COB (vertically opposite angle)

  \therefore y=75^{\circ}

  \angle AOD+\angle DOB=180^{\circ}

  75+z=180^{\circ}

Or  z=105^{\circ}

  \angle AOD+\angle AOC=180^{\circ}

  75+x=180

  x=105^{\circ}

We could have also used

 \angle DOB=\angle AOC

 x=105^{\circ}

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\ \ ii)\  \angle COB=\angle AOB (vertically Opposite angles)

 \Rightarrow y=125^{\circ}

 125+z=180^{\circ}  (Angles on a straight line are supplementary)

  z=55^{\circ}

 \angle BOD=\angle COA (Vertically opposite angles)

 x=55^{\circ}  

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\ \ iii)\  \angle AOC=\angle DOB (Vertically opposite angles)

 \Rightarrow y=30^{\circ}  

 \angle COB+\angle BOD=180 (Angles on a straight line)

 \Rightarrow z=150^{\circ}  

 \angle COB=\angle AOD (Vertically opposite angles)

 \Rightarrow x=150^{\circ}  

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\ \ iv)\  \angle AOC+\angle COB+\angle BOD+\angle DOA=360^{\circ}

3x-20+x+z+y=360^{\circ} ... i)

 x=y \ \&\ (Vertically opposite angles)

 3x-20=z \ \&\ (Vertically opposite angles)

  z+y=180^{\circ} ... ii) 
 x+z=180^{\circ} ... iii) 

Substituting (ii) in (i)

  3x-20+x+180=360
  4x=180+20
  x=50^{\circ}  
\angle AOD=50^{\circ}   \ \&\  \angle AOC=130^{\circ}  

 \ From\  iii)  z= 180-50 = 130^{\circ}

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Q.12 Prove that the bisectors of two adjacent supplementary angles include a right angle.

Let   \angle AOC=180-x
  \angle BOC=x

Bisector of   \angle BOC=  \frac{x}{2}= \angle DOC

Bisector of   \angle AOC=\frac{1}{2} (180-x)=90-\frac{x}{2}=\angle COE

Therefore
  \angle COE+\angle DOC=90-\frac{x}{2}+\frac{x}{2}=90^{\circ}  

Hence
 \angle DOE=90^{\circ}  =Right \ angle

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Q.13 Find the measure of an angle which is (i) equal to its complement (ii) equal to its supplement.

i) If the  \angle 1=x, its \ complement \angle 2=90-x

If  x=90-x 

  \Rightarrow x=45^{\circ}  

ii) If the  \angle 1=x, its \ supplement = 180-x

  x=180-x

  \Rightarrow x=90^{\circ}  

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Q.14 Find the angle which is   84^{\circ}   more than its complement.

Let the angle   =x  

 Complement   = 90 - x  

 Given    x = (90-x) + 34  

   2x=124  

 or    x=62^{\circ}  

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Q.15 Find the angle which is   16^{\circ}    less than its complement.

Let the angle    = x  

Complement   =90-x  

Given

   x+16=90-x  

   2x=74  

   x=37^{\circ}    

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Q.16 Find the angle which is 26^{\circ}   more than its supplement.

Let the angle   = x  

 Supplement   = 180-x  

 Given

   x=(180-x)+26  

   2x=206  

 Or   x=103^{\circ}    

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Q.17 Find the angle which is   32^{\circ}     less than its supplement.

Let the angle   = x    

 Supplement   = 180-x    

 Given,

   x+32=180-x    

   2x=148^{\circ}      

   or x=74^{\circ}      

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Q.18 Find the angle which is four times its complement.

Let the angle   =x    

Complement   = 90-x    

Given

   x=4(90-x)    

   5x=360    

   \Rightarrow x=72^{\circ}    

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Q.19 Find the angle which is five times its supplement.

Let the angle   =x    

Complement   = 180-x    

Given

   x=5(180-x)    

   6x=5\times 180    

Or   x=150^{\circ}    

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Q.20 Find the angle whose supplement is four times its complement.

Let the angle   =x    

Complement   = 90-x    

Supplement   = 180-x    

Given,

   180-x=4(90-x)    

   3x=180    

   x=60^{\circ}    

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Q.21 Find the angle whose complement is one third of its supplement.

Let the angle   =x    

Complement   = 90-x    

Supplement   =180-x    

Given

   90-x=\frac{1}{3} (180-x)    

   270-3x=180-x    

   2x=90    

   \Rightarrow  x=45^{\circ}    

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Q.22 Two complementary angles are in the ratio   7\colon 11     Find the angles.

Let the angle   =x    

Complement   =90-x    

Given

  \frac{x}{(90-x)}=\frac{7}{11}    

or    18x=630 \ or\   x=35^{\circ}    

The complement   = 35^{\circ}    

Hence the angles are   35^{\circ}  \ and\  55^{\circ}    

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Q.23 Two supplementary angles are in the ratio 7.  Find the angles.

Let the angle   = x    

Supplement   = 180-x    

Given

  \frac{x}{(180-x)}=\frac{7}{8}    

  8x=1260-7x    

  x=84^{\circ}   \ supplement=96^{\circ}      

Hence the angles are   84^{\circ}  \ and\  96^{\circ}    

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Q.24 Find the measure of an angle, if seven times its complement is   10{\circ}     less than three times its supplement.

Let the angle   =x    

Complement   =90-x    

Supplement   = 180-x    

Given

  7(90-x)+10=3(180-x)    

  630-7x+10=540-3x    

  4x=100    

  x=25^{\circ}    

 

ICSE Board: Suggested Books     ICSE Board:  Foundation Mathematics
Class 8: Reference Books               Class 8: NTSE Preparation
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