\displaystyle \textbf{Question 1: }\text{In the adjoining figure, name:}
\displaystyle \text{(i) six points}
\displaystyle \text{(ii) five line segments} 281
\displaystyle \text{(iii) four rays}
\displaystyle \text{(iv) four lines}
\displaystyle \text{(v) four collinear points}
\displaystyle \text{Answer:}
\displaystyle \text{(i) Six points are }M,\ N,\ P,\ Q,\ X,\ Y.

\displaystyle \text{(ii) Five line segments are } \overline{XM},\ \overline{MP},\ \overline{YN},\ \overline{NQ},\ \overline{MN}.

\displaystyle \text{(iii) Four rays are } \overrightarrow{PB},\ \overrightarrow{QD},\ \overrightarrow{XA},\ \overrightarrow{YC}.

\displaystyle \text{(iv) Four lines are } \overleftrightarrow{AB},\ \overleftrightarrow{CD},\ \overleftrightarrow{EF},\ \overleftrightarrow{GH}.

\displaystyle \text{(v) Four collinear points are }A,\ X,\ M,\ P.

\displaystyle \\

\displaystyle \textbf{Question 2: }\text{In the adjoining figure, name:} 282
\displaystyle \text{(i) two pairs of intersecting lines and their corresponding points}
\displaystyle \text{of intersection}
\displaystyle \text{(ii) three concurrent lines and their point of concurrence}
\displaystyle \text{(iii) three rays}
\displaystyle \text{(iv) two line segments}
\displaystyle \text{Answer:}
\displaystyle \text{(i) Two pairs of intersecting lines and their points of intersection are:}
\displaystyle \overleftrightarrow{EF}\text{ and }\overleftrightarrow{GH} \text{ intersect at }R.
\displaystyle \overleftrightarrow{CD}\text{ and }\overleftrightarrow{GH} \text{ intersect at }Q.

\displaystyle \text{(ii) Three concurrent lines are } \overleftrightarrow{AB},\ \overleftrightarrow{EF}\text{ and } \overleftrightarrow{GH}.
\displaystyle \text{Their point of concurrence is }R.

\displaystyle \text{(iii) Three rays are } \overrightarrow{QH},\ \overrightarrow{PB},\ \overrightarrow{PD}.
\displaystyle \text{Other possible rays include } \overrightarrow{RA},\ \overrightarrow{RE},\ \overrightarrow{RG}.

\displaystyle \text{(iv) Two line segments are } \overline{QR}\text{ and }\overline{PQ}.

\displaystyle \\

\displaystyle \textbf{Question 3: }\text{State whether the following statements are true or false:}
\displaystyle \text{(i) A ray has no endpoint.}
\displaystyle \text{Answer: False}
\displaystyle \text{A ray has exactly one endpoint and extends indefinitely in one direction.}

\displaystyle \text{(ii) A line }AB\text{ is the same as line }BA.
\displaystyle \text{Answer: True}
\displaystyle \overleftrightarrow{AB}=\overleftrightarrow{BA}.

\displaystyle \text{(iii) Ray }AB\text{ is the same as ray }BA.
\displaystyle \text{Answer: False}
\displaystyle \overrightarrow{AB}\ne\overrightarrow{BA}\text{ in general, since they have different endpoints.}

\displaystyle \text{(iv) A line has a definite length.}
\displaystyle \text{Answer: False}
\displaystyle \text{A line extends indefinitely in both directions and has no definite length.}

\displaystyle \text{(v) Two planes always meet in a line.}
\displaystyle \text{Answer: False}
\displaystyle \text{Two distinct planes may be parallel and therefore may have no point in common.}
\displaystyle \text{If two distinct planes intersect, their intersection is a line.}

\displaystyle \text{(vi) A plane has length and breadth but no thickness.}
\displaystyle \text{Answer: True}
\displaystyle \text{A plane is a two-dimensional surface extending indefinitely in all directions.}

\displaystyle \text{(vii) Two distinct points always determine a unique line.}
\displaystyle \text{Answer: True}
\displaystyle \text{Exactly one straight line can be drawn through any two distinct points.}

\displaystyle \text{(viii) Two distinct lines may intersect at two points.}
\displaystyle \text{Answer: False}
\displaystyle \text{Two distinct lines can have at most one point of intersection.}

\displaystyle \text{(ix) Two intersecting lines cannot both be parallel to the same line.}
\displaystyle \text{Answer: True}
\displaystyle \text{Two distinct lines parallel to the same line are parallel to each other}
\displaystyle \text{and therefore cannot intersect.}

\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Two adjacent angles on a straight line are }x^\circ
\displaystyle \text{and }(2x-21)^\circ.\text{ Find:} 283
\displaystyle \text{(i) the value of }x
\displaystyle \text{(ii) the measure of each angle.}
\displaystyle \text{Answer:}
\displaystyle \text{Since }\angle AOB\text{ and }\angle BOC\text{ form a linear pair,}
\displaystyle \angle AOB+\angle BOC=180^\circ.
\displaystyle \text{(i) }(2x-21)+x=180
\displaystyle 3x-21=180
\displaystyle 3x=201
\displaystyle \therefore x=67.

\displaystyle \text{(ii) }\angle BOC=x^\circ=67^\circ.
\displaystyle \angle AOB=(2x-21)^\circ
\displaystyle =[2(67)-21]^\circ=113^\circ.
\displaystyle \therefore \text{The two angles are }67^\circ\text{ and }113^\circ.
\displaystyle \text{Check: }67^\circ+113^\circ=180^\circ.

\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Two adjacent angles on a straight line are }
\displaystyle (3x-2)^\circ\text{ and }4(x+7)^\circ.\text{ Find:}
\displaystyle \text{(i) the value of }x
\displaystyle \text{(ii) the measure of each angle.}
\displaystyle \text{Answer:}
\displaystyle \text{Since adjacent angles on a straight line form a linear pair,}
\displaystyle \text{their sum is }180^\circ.
\displaystyle \text{(i) }(3x-2)+4(x+7)=180
\displaystyle 3x-2+4x+28=180
\displaystyle 7x+26=180
\displaystyle 7x=154
\displaystyle \therefore x=22.

\displaystyle \text{(ii) First angle}=(3x-2)^\circ
\displaystyle =[3(22)-2]^\circ=64^\circ.
\displaystyle \text{Second angle}=4(x+7)^\circ
\displaystyle =[4(22+7)]^\circ=116^\circ.
\displaystyle \therefore \text{The two angles are }64^\circ\text{ and }116^\circ.
\displaystyle \text{Check: }64^\circ+116^\circ=180^\circ.

\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Two adjacent angles on a straight line are in the ratio}
\displaystyle 3:2.\text{ Find the measure of each angle.}
\displaystyle \text{Answer:}
\displaystyle \text{The ratio of the two angles is }3:2.
\displaystyle \text{Let the two angles be }3x^\circ\text{ and }2x^\circ.
\displaystyle \text{Since they form a linear pair, their sum is }180^\circ.
\displaystyle 3x+2x=180
\displaystyle 5x=180
\displaystyle \therefore x=36.
\displaystyle \text{First angle}=3x^\circ=3(36)^\circ=108^\circ.
\displaystyle \text{Second angle}=2x^\circ=2(36)^\circ=72^\circ.
\displaystyle \therefore \text{The two angles are }108^\circ\text{ and }72^\circ.
\displaystyle \text{Check: }108^\circ+72^\circ=180^\circ.

\displaystyle \\

\displaystyle \textbf{Question 7: }\text{In the adjoining figure, }AOB\text{ is a straight line.}
\displaystyle \text{Find the value of }x.\text{ Hence, find }\angle AOC\text{ and }\angle BOD. 284
\displaystyle \text{Answer:}
\displaystyle \text{Since }AOB\text{ is a straight line,}
\displaystyle \angle AOC+\angle COD+\angle DOB=180^\circ.
\displaystyle (3x-5)+55+(x+20)=180
\displaystyle 4x+70=180
\displaystyle 4x=110
\displaystyle \therefore x=27.5.

\displaystyle \angle AOC=(3x-5)^\circ
\displaystyle =[3(27.5)-5]^\circ=77.5^\circ.
\displaystyle \angle BOD=(x+20)^\circ
\displaystyle =(27.5+20)^\circ=47.5^\circ.
\displaystyle \therefore \angle AOC=77.5^\circ\text{ and }\angle BOD=47.5^\circ.
\displaystyle \text{Check: }77.5^\circ+55^\circ+47.5^\circ=180^\circ.

\displaystyle \\

\displaystyle \textbf{Question 8: }\text{In the adjoining figure, }AOB\text{ is a straight line.}
\displaystyle \text{If }x:y:z=6:5:4,\text{ find the values of }x,\ y\text{ and }z. 285
\displaystyle \text{Answer:}
\displaystyle \text{Given, }x:y:z=6:5:4.
\displaystyle \text{Let }x=6a,\quad y=5a,\quad z=4a.
\displaystyle \text{Since }AOB\text{ is a straight line,}
\displaystyle x+y+z=180^\circ.
\displaystyle 6a+5a+4a=180
\displaystyle 15a=180
\displaystyle \therefore a=12.
\displaystyle x=6a=6(12)=72^\circ,
\displaystyle y=5a=5(12)=60^\circ,
\displaystyle z=4a=4(12)=48^\circ.
\displaystyle \therefore x=72^\circ,\quad y=60^\circ,\quad z=48^\circ.
\displaystyle \text{Check: }72^\circ+60^\circ+48^\circ=180^\circ.

\displaystyle \\

\displaystyle \textbf{Question 9: }\text{In the adjoining figure, what value of }x
\displaystyle \text{makes }AOB\text{ a straight line?} 286
\displaystyle \text{Answer:}
\displaystyle \text{For }AOB\text{ to be a straight line,}
\displaystyle \angle AOC+\angle COB=180^\circ.
\displaystyle (3x+5)+(2x-25)=180
\displaystyle 5x-20=180
\displaystyle 5x=200
\displaystyle \therefore x=40.
\displaystyle \text{Hence, }\angle AOC=(3x+5)^\circ
\displaystyle =[3(40)+5]^\circ=125^\circ,
\displaystyle \text{and }\angle COB=(2x-25)^\circ
\displaystyle =[2(40)-25]^\circ=55^\circ.
\displaystyle \text{Check: }125^\circ+55^\circ=180^\circ.

\displaystyle \\

\displaystyle \textbf{Question 10: }\text{In the adjoining figure, find the value of }x.
\displaystyle \text{Answer:} 287
\displaystyle \text{Angles at a point add up to }360^\circ.
\displaystyle \therefore \angle AOB+\angle BOC+\angle COD+\angle DOA=360^\circ.
\displaystyle 65+90+120+x=360
\displaystyle x+275=360
\displaystyle x=360-275
\displaystyle \therefore x=85^\circ.
\displaystyle \text{Check: }65^\circ+90^\circ+120^\circ+85^\circ=360^\circ.

\displaystyle \\

\displaystyle \textbf{Question 11: }\text{In each of the following figures, two lines }AB
\displaystyle \text{and }CD\text{ intersect at a point }O.\text{ Find the values of }x,\ y
\displaystyle \text{and }z. 288
\displaystyle \text{Answer:}
\displaystyle \text{(i) }\angle AOD=\angle COB \qquad\text{[Vertically opposite angles]}
\displaystyle \therefore y=75^\circ.
\displaystyle \angle AOD+\angle DOB=180^\circ \qquad\text{[Linear pair]}
\displaystyle 75+z=180
\displaystyle \therefore z=105^\circ.
\displaystyle \angle AOD+\angle AOC=180^\circ \qquad\text{[Linear pair]}
\displaystyle 75+x=180
\displaystyle \therefore x=105^\circ.
\displaystyle \therefore x=105^\circ,\quad y=75^\circ,\quad z=105^\circ.

\displaystyle \text{(ii) }\angle COB=\angle AOD \qquad\text{[Vertically opposite angles]}
\displaystyle \therefore y=125^\circ.
\displaystyle \angle COB+\angle BOD=180^\circ \qquad\text{[Linear pair]}
\displaystyle 125+z=180
\displaystyle \therefore z=55^\circ.
\displaystyle \angle BOD=\angle COA \qquad\text{[Vertically opposite angles]}
\displaystyle \therefore x=55^\circ.
\displaystyle \therefore x=55^\circ,\quad y=125^\circ,\quad z=55^\circ.

\displaystyle \text{(iii) }\angle AOC=\angle BOD \qquad\text{[Vertically opposite angles]}
\displaystyle \therefore y=30^\circ.
\displaystyle \angle AOC+\angle COB=180^\circ \qquad\text{[Linear pair]}
\displaystyle 30+z=180
\displaystyle \therefore z=150^\circ.
\displaystyle \angle COB=\angle AOD \qquad\text{[Vertically opposite angles]}
\displaystyle \therefore x=150^\circ.
\displaystyle \therefore x=150^\circ,\quad y=30^\circ,\quad z=150^\circ.

\displaystyle \text{(iv) }\angle AOD=\angle BOC \qquad\text{[Vertically opposite angles]}
\displaystyle \therefore x=y.
\displaystyle \angle AOC=\angle DOB \qquad\text{[Vertically opposite angles]}
\displaystyle \therefore z=3x-20.
\displaystyle \angle AOD+\angle DOB=180^\circ \qquad\text{[Linear pair]}
\displaystyle x+z=180
\displaystyle x+(3x-20)=180
\displaystyle 4x-20=180
\displaystyle 4x=200
\displaystyle \therefore x=50^\circ.
\displaystyle \therefore y=x=50^\circ.
\displaystyle z=3x-20=3(50)-20=130^\circ.
\displaystyle \therefore x=50^\circ,\quad y=50^\circ,\quad z=130^\circ.

\displaystyle \\

\displaystyle \textbf{Question 12: }\text{Prove that the bisectors of two adjacent}
\displaystyle \text{supplementary angles include a right angle.}
\displaystyle \text{Answer:}
\displaystyle \text{Let }\angle BOC=x^\circ.
\displaystyle \text{Since }\angle AOC\text{ and }\angle BOC\text{ are supplementary,}
\displaystyle \angle AOC=(180-x)^\circ.
\displaystyle \text{Let }OD\text{ bisect }\angle BOC.
\displaystyle \therefore \angle DOC=\frac{x}{2}^\circ.
\displaystyle \text{Let }OE\text{ bisect }\angle AOC.
\displaystyle \therefore \angle COE=\frac{1}{2}(180-x)^\circ
\displaystyle =\left(90-\frac{x}{2}\right)^\circ.
\displaystyle \angle DOE=\angle DOC+\angle COE
\displaystyle =\frac{x}{2}^\circ+\left(90-\frac{x}{2}\right)^\circ
\displaystyle =90^\circ.
\displaystyle \therefore \angle DOE\text{ is a right angle.}
\displaystyle \text{Hence, the bisectors of two adjacent supplementary angles are perpendicular.}

\displaystyle \\

\displaystyle \textbf{Question 13: }\text{Find the measure of an angle which is:}
\displaystyle \text{(i) equal to its complement}
\displaystyle \text{(ii) equal to its supplement.}
\displaystyle \text{Answer:}
\displaystyle \text{(i) Let the angle be }x^\circ.
\displaystyle \text{Its complement is }(90-x)^\circ.
\displaystyle \text{Since the angle is equal to its complement,}
\displaystyle x=90-x
\displaystyle 2x=90
\displaystyle \therefore x=45^\circ.
\displaystyle \therefore \text{The required angle is }45^\circ.

\displaystyle \text{(ii) Let the angle be }x^\circ.
\displaystyle \text{Its supplement is }(180-x)^\circ.
\displaystyle \text{Since the angle is equal to its supplement,}
\displaystyle x=180-x
\displaystyle 2x=180
\displaystyle \therefore x=90^\circ.
\displaystyle \therefore \text{The required angle is }90^\circ.

\displaystyle \\

\displaystyle \textbf{Question 14: }\text{Find the angle which is }84^\circ
\displaystyle \text{more than its complement.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the angle be }x^\circ.
\displaystyle \text{Its complement is }(90-x)^\circ.
\displaystyle \text{Since the angle is }84^\circ\text{ more than its complement,}
\displaystyle x=(90-x)+84
\displaystyle x=174-x
\displaystyle 2x=174
\displaystyle \therefore x=87^\circ.
\displaystyle \text{Its complement is }90^\circ-87^\circ=3^\circ.
\displaystyle \text{Check: }87^\circ-3^\circ=84^\circ.
\displaystyle \therefore \text{The required angle is }87^\circ.

\displaystyle \\

\displaystyle \textbf{Question 15: }\text{Find the angle which is }16^\circ
\displaystyle \text{less than its complement.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the angle be }x^\circ.
\displaystyle \text{Its complement is }(90-x)^\circ.
\displaystyle \text{Since the angle is }16^\circ\text{ less than its complement,}
\displaystyle x+16=90-x
\displaystyle 2x=74
\displaystyle \therefore x=37^\circ.
\displaystyle \text{Its complement is }90^\circ-37^\circ=53^\circ.
\displaystyle \text{Check: }53^\circ-37^\circ=16^\circ.
\displaystyle \therefore \text{The required angle is }37^\circ.

\displaystyle \\

\displaystyle \textbf{Question 16: }\text{Find the angle which is }26^\circ
\displaystyle \text{more than its supplement.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the angle be }x^\circ.
\displaystyle \text{Its supplement is }(180-x)^\circ.
\displaystyle \text{Since the angle is }26^\circ\text{ more than its supplement,}
\displaystyle x=(180-x)+26
\displaystyle 2x=206
\displaystyle \therefore x=103^\circ.
\displaystyle \text{Its supplement is }180^\circ-103^\circ=77^\circ.
\displaystyle \text{Check: }103^\circ-77^\circ=26^\circ.
\displaystyle \therefore \text{The required angle is }103^\circ.

\displaystyle \\

\displaystyle \textbf{Question 17: }\text{Find the angle which is }32^\circ
\displaystyle \text{less than its supplement.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the angle be }x^\circ.
\displaystyle \text{Its supplement is }(180-x)^\circ.
\displaystyle \text{Since the angle is }32^\circ\text{ less than its supplement,}
\displaystyle x+32=180-x
\displaystyle 2x=148
\displaystyle \therefore x=74^\circ.
\displaystyle \text{Its supplement is }180^\circ-74^\circ=106^\circ.
\displaystyle \text{Check: }106^\circ-74^\circ=32^\circ.
\displaystyle \therefore \text{The required angle is }74^\circ.

\displaystyle \\

\displaystyle \textbf{Question 18: }\text{Find the angle which is four times its complement.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the angle be }x^\circ.
\displaystyle \text{Its complement is }(90-x)^\circ.
\displaystyle \text{Since the angle is four times its complement,}
\displaystyle x=4(90-x)
\displaystyle x=360-4x
\displaystyle 5x=360
\displaystyle \therefore x=72^\circ.
\displaystyle \text{Its complement is }90^\circ-72^\circ=18^\circ.
\displaystyle \text{Check: }4\times18^\circ=72^\circ.
\displaystyle \therefore \text{The required angle is }72^\circ.

\displaystyle \\

\displaystyle \textbf{Question 19: }\text{Find the angle which is five times its supplement.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the angle be }x^\circ.
\displaystyle \text{Its supplement is }(180-x)^\circ.
\displaystyle \text{Since the angle is five times its supplement,}
\displaystyle x=5(180-x)
\displaystyle x=900-5x
\displaystyle 6x=900
\displaystyle \therefore x=150^\circ.
\displaystyle \text{Its supplement is }180^\circ-150^\circ=30^\circ.
\displaystyle \text{Check: }5\times30^\circ=150^\circ.
\displaystyle \therefore \text{The required angle is }150^\circ.

\displaystyle \\

\displaystyle \textbf{Question 20: }\text{Find the angle whose supplement is four times}
\displaystyle \text{its complement.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the angle be }x^\circ.
\displaystyle \text{Its complement is }(90-x)^\circ.
\displaystyle \text{Its supplement is }(180-x)^\circ.
\displaystyle \text{Since its supplement is four times its complement,}
\displaystyle 180-x=4(90-x)
\displaystyle 180-x=360-4x
\displaystyle 3x=180
\displaystyle \therefore x=60^\circ.
\displaystyle \text{Its complement is }90^\circ-60^\circ=30^\circ.
\displaystyle \text{Its supplement is }180^\circ-60^\circ=120^\circ.
\displaystyle \text{Check: }4\times30^\circ=120^\circ.
\displaystyle \therefore \text{The required angle is }60^\circ.

\displaystyle \\

\displaystyle \textbf{Question 21: }\text{Find the angle whose complement is one-third}
\displaystyle \text{of its supplement.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the angle be }x^\circ.
\displaystyle \text{Its complement is }(90-x)^\circ.
\displaystyle \text{Its supplement is }(180-x)^\circ.
\displaystyle \text{Since its complement is one-third of its supplement,}
\displaystyle 90-x=\frac{1}{3}(180-x)
\displaystyle 270-3x=180-x
\displaystyle 90=2x
\displaystyle \therefore x=45^\circ.
\displaystyle \text{Its complement is }45^\circ\text{ and its supplement is }135^\circ.
\displaystyle \text{Check: }\frac{1}{3}\times135^\circ=45^\circ.
\displaystyle \therefore \text{The required angle is }45^\circ.

\displaystyle \\

\displaystyle \textbf{Question 22: }\text{Two complementary angles are in the ratio }7:11.
\displaystyle \text{Find the angles.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the two angles be }7x^\circ\text{ and }11x^\circ.
\displaystyle \text{Since the angles are complementary, their sum is }90^\circ.
\displaystyle 7x+11x=90
\displaystyle 18x=90
\displaystyle \therefore x=5.
\displaystyle \text{First angle}=7x^\circ=7(5)^\circ=35^\circ.
\displaystyle \text{Second angle}=11x^\circ=11(5)^\circ=55^\circ.
\displaystyle \therefore \text{The two angles are }35^\circ\text{ and }55^\circ.
\displaystyle \text{Check: }35^\circ+55^\circ=90^\circ.

\displaystyle \\

\displaystyle \textbf{Question 23: }\text{Two supplementary angles are in the ratio }7:8.
\displaystyle \text{Find the angles.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the two angles be }7x^\circ\text{ and }8x^\circ.
\displaystyle \text{Since the angles are supplementary, their sum is }180^\circ.
\displaystyle 7x+8x=180
\displaystyle 15x=180
\displaystyle \therefore x=12.
\displaystyle \text{First angle}=7x^\circ=7(12)^\circ=84^\circ.
\displaystyle \text{Second angle}=8x^\circ=8(12)^\circ=96^\circ.
\displaystyle \therefore \text{The two angles are }84^\circ\text{ and }96^\circ.
\displaystyle \text{Check: }84^\circ+96^\circ=180^\circ.

\displaystyle \\

\displaystyle \textbf{Question 24: }\text{Find the measure of an angle, if seven times its}
\displaystyle \text{complement is }10^\circ\text{ less than three times its supplement.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the angle be }x^\circ.
\displaystyle \text{Its complement is }(90-x)^\circ.
\displaystyle \text{Its supplement is }(180-x)^\circ.
\displaystyle \text{Since seven times its complement is }10^\circ\text{ less than three}
\displaystyle \text{times its supplement,}
\displaystyle 7(90-x)=3(180-x)-10
\displaystyle 630-7x=540-3x-10
\displaystyle 630-7x=530-3x
\displaystyle 100=4x
\displaystyle \therefore x=25^\circ.
\displaystyle \text{Its complement is }65^\circ\text{ and its supplement is }155^\circ.
\displaystyle \text{Check: }7(65^\circ)=455^\circ\text{ and }3(155^\circ)=465^\circ.
\displaystyle \text{Thus, }455^\circ\text{ is }10^\circ\text{ less than }465^\circ.
\displaystyle \therefore \text{The required angle is }25^\circ.

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.