Question 1: In the adjourning figure, name:

i)six points

ii) five line segments

iii) four rays

iv) four lines

v) four collinear points

i)  The point are $M, N, P, Q , X, Y$

ii)  Five line segments are $\overline{XM},\overline{MP},\overline{YN},\overline{NQ},\overline{MN},\overline{PQ}$

iii)  Four Rays are $\overrightarrow{PB}, \overrightarrow{QD}, \overrightarrow{XA}, \overrightarrow{YC}$

iv)  Four lines are $\overleftrightarrow{AB}, \overleftrightarrow{CD}, \overleftrightarrow{EF}, \overleftrightarrow{HG}$

v)  Four Collinear points are $A, X, M, P \ or\ C, Y, N, Q \ or\ X, M, P, B$

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Question 2: In the adjoining figure, name :

i) two pairs of intersecting lines and their corresponding points of intersection

ii) three concurrent lines and their points of concurrence

iii) three rays

iv) two line segments

i)  Two points of intersecting lines are

$\overleftrightarrow{EF}, \overleftrightarrow{GH}$  intersecting at R

$\overleftrightarrow{CD}, \overleftrightarrow{GH}$  intersecting at Q.

ii)  Three concurrent lines and their point of concurrence.

$\overleftrightarrow{AB}, \overleftrightarrow{EF}, \overleftrightarrow{GH}$ and point of concurrence is R

iii)  Three rays are, $\overrightarrow{QH}, \overrightarrow{PB},\overrightarrow{PD}$ some other rays are $\overrightarrow{RA}, \overrightarrow{RE},\overrightarrow{RG}$, etc.

iv)  Two line segments are $\overline{QR}, \overline{PQ},\overline{RP}$

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Question 3: State whether the following statements are true or false :

i)  A ray has no end Point: $False$

ii)  A line AB is the same as line BA: $True$

iii)  A ray AB is the same as BA: $False$

iv)  A line has a definite length: $False$

v)  Two planes always meet in a line: $True$

vi)  A plane has length and breadth but no thickness: $True$

vii)  Two distinct points always determine a unique line: $True$

viii)  Two lines may intersect in two points: $False$

ix)  Two intersecting lines cannot be both parallel to the same line: $True$

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Question 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 angle

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

$2x-21+x=180^{\circ}$

$3x=201$

$x=67^{\circ}$

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

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Question 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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Question 6: Two adjacent angles on a straight line are in the ratio $3:2$. Find the measure of each angle:

The ratio of angles $= 3\colon 2$

Therefore the angles are $3x$ and $2x$

$3x+2x=180^{\circ}$

$\Rightarrow 5x=180^{\circ}$

$\Rightarrow x=36^{\circ}$

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

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

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

$\Rightarrow 3x-5+55+x+20=180^{\circ}$

$\Rightarrow 4x=110^{\circ}$

$\Rightarrow x=27.5^{\circ}$

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

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

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Question 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 $z$.

$x\colon y\colon z=6\colon 5\colon 4$

Therefore

$6a+5a+4a=180^{\circ}$

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

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Question 9: In the adjoining figure, what value of $x$ make AOB a straight line?

For $\overleftrightarrow{AB}$ to be a straight line

$3x+5+2x-25=180^{\circ}$

$\Rightarrow 5x-20=180^{\circ}$

$\Rightarrow 5x=200^{\circ}$

$\Rightarrow x=40^{\circ}$

$\angle AOC=3\times 40+5=125^{\circ}$

$\angle BOC=2\times 40-25=55^{\circ}$

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Question 10: In the adjoining figure, find the value of $x$.

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

$x+65+90+120=360$

$\Rightarrow x=360^{\circ} -275^{\circ}$

$\Rightarrow x=85^{\circ}$

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

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

$\therefore y=75^{\circ}$

$\angle AOD+\angle DOB=180^{\circ}$

$\Rightarrow 75+z=180^{\circ}$

$\Rightarrow z=105^{\circ}$

$\angle AOD+\angle AOC=180^{\circ}$

$\Rightarrow 75+x=180$

$\Rightarrow x=105^{\circ}$

We could have also used

$\angle DOB=\angle AOC$

$\Rightarrow x=105^{\circ}$

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

$\Rightarrow y=125^{\circ}$

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

$\Rightarrow z=55^{\circ}$

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

$\Rightarrow x=55^{\circ}$

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)

$\Rightarrow x=150^{\circ}$

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

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

$x=y \$   (Vertically opposite angles)

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

$z+y=180^{\circ} \ldots \ldots ii)$

$x+z=180^{\circ} \ldots \ldots iii)$

Substituting (ii) in (i)

$3x-20+x+180=360$

$\Rightarrow 4x=180+20$

$\Rightarrow x=50^{\circ}$

$\angle AOD=50^{\circ} \ \&\ \angle AOC=130^{\circ}$

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

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

Let $\angle AOC=180-x$

$\Rightarrow \angle BOC=x$

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

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

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

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

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

i)  If the $\angle 1=x$, it’s 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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Question 14: Find the angle which is $84^{\circ}$more than its complement.

Let the angle $=x$

Complement $= 90 - x$

Given $x = (90-x) + 34$

$\Rightarrow 2x=124$

$\Rightarrow x=62^{\circ}$

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

Let the angle $= x$

Complement $=90-x$

Given

$x+16=90-x$

$\Rightarrow 2x=74$

$\Rightarrow x=37^{\circ}$

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

Let the angle $= x$

Supplement $= 180-x$

Given

$x=(180-x)+26$

$\Rightarrow 2x=206$

$\Rightarrow x=103^{\circ}$

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

Let the angle $= x$

Supplement $= 180-x$

Given,

$x+32=180-x$

$\Rightarrow 2x=148^{\circ}$

$\Rightarrow x=74^{\circ}$

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Question 18: Find the angle which is four times its complement.

Let the angle $=x$

Complement $= 90-x$

Given

$x=4(90-x)$

$\Rightarrow 5x=360$

$\Rightarrow x=72^{\circ}$

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Question 19: Find the angle which is five times its supplement.

Let the angle $=x$

Complement $= 180-x$

Given

$x=5(180-x)$

$\Rightarrow 6x=5\times 180$

$\Rightarrow x=150^{\circ}$

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

Let the angle $=x$

Complement $= 90-x$

Supplement $= 180-x$

Given,

$180-x=4(90-x)$

$\Rightarrow 3x=180$

$\Rightarrow x=60^{\circ}$

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

Let the angle $=x$

Therefore Complement $= 90-x$

Therefore Supplement $=180-x$

Given

$\displaystyle 90-x= \frac{1}{3} (180-x)$

$\Rightarrow 270-3x=180-x$

$\Rightarrow 2x=90$

$\Rightarrow x=45^{\circ}$

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Question 22: Two complementary angles are in the ratio $7\colon 11$ Find the angles.

Let the angle $=x$

Complement $=90-x$

Given

$\displaystyle \frac{x}{(90-x)} = \frac{7}{11}$

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

The complement $= 35^{\circ}$

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

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Question 23: Two supplementary angles are in the ratio 7. Find the angles.

Let the angle $= x$

Supplement $= 180-x$

Given

$\displaystyle \frac{x}{(180-x)} = \frac{7}{8}$

$\Rightarrow 8x=1260-7x$

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

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

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Question 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

$7(90-x)+10=3(180-x)$

$\Rightarrow 630-7x+10=540-3x$

$\Rightarrow 4x=100$

$\Rightarrow x=25^{\circ}$