Geometrical Constructions (Exercise 19.5) RD Sharma Solutions | Mathematics (Maths) Class 6 PDF Download

 Page 1


 
 
 
 
 
 
                                                                           
1. Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC. 
Solution: 
 
Construct an angle ?BAC and draw a ray OP. 
Taking A as centre and suitable radius, construct an arc which intersects AB and AC at points X and Y. 
Taking O as centre and same radius, construct an arc which intersects the arc OP at the point M. 
Now measure XY with the help of compass. 
Taking M as centre and XY as radius construct an arc which intersects the arc which is drawn from O and name it 
as point N. 
Now join the points O and N and extend it to the point Q. 
Here, ?POQ is the required angle. 
 
 
2. Draw an obtuse angle. Bisect it. Measure each of the angles so obtained. 
Solution: 
 
We know that obtuse angles are those which are greater than 90
o
 and less than 180
o
. 
Construct an obtuse angle ?BAC. 
Taking A as centre with appropriate radius construct an arc which intersects AB and AC at the points P and Q. 
Taking P as centre and radius which is more than half of PQ construct an arc. 
Taking Q as centre and same radius construct another arc which intersects the previous arc at the point R. 
Now join A and R and extend it to the point X. 
So the ray AX is the required bisector of ?BAC. 
By measuring ?BAR and ?CAR we get ?BAR = ?CAR = 65
o
. 
 
 
3. Using your protractor, draw an angle of measure 108
o
. With this angle as given, drawn an angle of 54
o
. 
Solution: 
 
Page 2


 
 
 
 
 
 
                                                                           
1. Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC. 
Solution: 
 
Construct an angle ?BAC and draw a ray OP. 
Taking A as centre and suitable radius, construct an arc which intersects AB and AC at points X and Y. 
Taking O as centre and same radius, construct an arc which intersects the arc OP at the point M. 
Now measure XY with the help of compass. 
Taking M as centre and XY as radius construct an arc which intersects the arc which is drawn from O and name it 
as point N. 
Now join the points O and N and extend it to the point Q. 
Here, ?POQ is the required angle. 
 
 
2. Draw an obtuse angle. Bisect it. Measure each of the angles so obtained. 
Solution: 
 
We know that obtuse angles are those which are greater than 90
o
 and less than 180
o
. 
Construct an obtuse angle ?BAC. 
Taking A as centre with appropriate radius construct an arc which intersects AB and AC at the points P and Q. 
Taking P as centre and radius which is more than half of PQ construct an arc. 
Taking Q as centre and same radius construct another arc which intersects the previous arc at the point R. 
Now join A and R and extend it to the point X. 
So the ray AX is the required bisector of ?BAC. 
By measuring ?BAR and ?CAR we get ?BAR = ?CAR = 65
o
. 
 
 
3. Using your protractor, draw an angle of measure 108
o
. With this angle as given, drawn an angle of 54
o
. 
Solution: 
 
 
 
 
 
 
 
Construct a ray OA. 
Using protractor, draw an angle ?AOB of 108
o 
where 108/2 = 54
o
 
Hence, 54
o
 is half of 108
o
. 
In order to get angle 54
o
, we must bisect the angle of 108
o
. 
Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. 
Taking P as centre and radius which is more than half of PQ construct an arc. 
Taking Q as centre and same radius construct another arc which intersects the previous arc and name it as point R. 
Now join the points O and R and extend it to the point X. 
Here, ?AOX is the required angle of 54
o
. 
 
 
4. Using protractor, draw a right angle. Bisect it to get an angle of measure 45
o
. 
Solution: 
 
Construct a ray OA. 
Using a protractor construct ?AOB of 90
o
. 
Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. 
Taking P as centre and radius which is more than half of PQ, construct an arc. 
Taking Q as centre and same radius, construct another arc which intersects the previous arc and name it as point 
R. 
Now join the points O and R and extend it to the point X. 
Here, ?AOX is the required angle of 45
o
 where ?AOB = 90
o
 and ?AOX = 45
o
. 
 
 
5. Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are 
perpendicular to each other. 
Page 3


 
 
 
 
 
 
                                                                           
1. Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC. 
Solution: 
 
Construct an angle ?BAC and draw a ray OP. 
Taking A as centre and suitable radius, construct an arc which intersects AB and AC at points X and Y. 
Taking O as centre and same radius, construct an arc which intersects the arc OP at the point M. 
Now measure XY with the help of compass. 
Taking M as centre and XY as radius construct an arc which intersects the arc which is drawn from O and name it 
as point N. 
Now join the points O and N and extend it to the point Q. 
Here, ?POQ is the required angle. 
 
 
2. Draw an obtuse angle. Bisect it. Measure each of the angles so obtained. 
Solution: 
 
We know that obtuse angles are those which are greater than 90
o
 and less than 180
o
. 
Construct an obtuse angle ?BAC. 
Taking A as centre with appropriate radius construct an arc which intersects AB and AC at the points P and Q. 
Taking P as centre and radius which is more than half of PQ construct an arc. 
Taking Q as centre and same radius construct another arc which intersects the previous arc at the point R. 
Now join A and R and extend it to the point X. 
So the ray AX is the required bisector of ?BAC. 
By measuring ?BAR and ?CAR we get ?BAR = ?CAR = 65
o
. 
 
 
3. Using your protractor, draw an angle of measure 108
o
. With this angle as given, drawn an angle of 54
o
. 
Solution: 
 
 
 
 
 
 
 
Construct a ray OA. 
Using protractor, draw an angle ?AOB of 108
o 
where 108/2 = 54
o
 
Hence, 54
o
 is half of 108
o
. 
In order to get angle 54
o
, we must bisect the angle of 108
o
. 
Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. 
Taking P as centre and radius which is more than half of PQ construct an arc. 
Taking Q as centre and same radius construct another arc which intersects the previous arc and name it as point R. 
Now join the points O and R and extend it to the point X. 
Here, ?AOX is the required angle of 54
o
. 
 
 
4. Using protractor, draw a right angle. Bisect it to get an angle of measure 45
o
. 
Solution: 
 
Construct a ray OA. 
Using a protractor construct ?AOB of 90
o
. 
Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. 
Taking P as centre and radius which is more than half of PQ, construct an arc. 
Taking Q as centre and same radius, construct another arc which intersects the previous arc and name it as point 
R. 
Now join the points O and R and extend it to the point X. 
Here, ?AOX is the required angle of 45
o
 where ?AOB = 90
o
 and ?AOX = 45
o
. 
 
 
5. Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are 
perpendicular to each other. 
 
 
 
 
 
 
Solution: 
We know that the two angles which are adjacent and supplementary are known as linear pair of angles. 
Construct a line AB and mark a point O on it. 
By constructing an angle ?AOC we get another angle ?BOC. 
Now bisect ?AOC using a compass and a ruler and get the ray OX. 
In the same way bisect ?BOC and get the ray OY. 
We know that  
?XOY = ?XOC + ?COY 
It can be written as 
?XOY = 1/2 ?AOC + 1/2 ?BOC 
So we get 
?XOY = 1/2 (?AOC + ?BOC) 
We know that ?AOC and ?BOC are supplementary angles 
?XOY = 1/2 (180) = 90
o
 
 
 
6. Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are 
in the same line. 
Solution: 
 
Construct two lines AB and CD which intersects each other at the point O 
Since vertically opposite angles are equal we get  
?BOC = ?AOD and ?AOC = ?BOD 
Now bisect angle AOC and construct the bisecting ray as OX. 
In the same way, we bisect ?BOD and construct bisecting ray OY. 
We get  
?XOA + ?AOD + ?DOY = 1/2 ?AOC + ?AOD + 1/2 ?BOD 
We know that ?AOC = ?BOD 
?XOA + ?AOD + ?DOY = 1/2 ?BOD + ?AOD + 1/2 ?BOD 
So we get  
?XOA + ?AOD + ?DOY = ?AOD + ?BOD 
 
AB is a line 
We know that ?AOD and ?BOD are supplementary angles whose sum is equal to 180
o
. 
?XOA + ?AOD + ?DOY = 180
o
 
The angles on one side of a straight line is always 180
o
 and also the sum of angles is 180
o
 
Here, XY is a straight line where OX and OY are in the same line. 
Page 4


 
 
 
 
 
 
                                                                           
1. Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC. 
Solution: 
 
Construct an angle ?BAC and draw a ray OP. 
Taking A as centre and suitable radius, construct an arc which intersects AB and AC at points X and Y. 
Taking O as centre and same radius, construct an arc which intersects the arc OP at the point M. 
Now measure XY with the help of compass. 
Taking M as centre and XY as radius construct an arc which intersects the arc which is drawn from O and name it 
as point N. 
Now join the points O and N and extend it to the point Q. 
Here, ?POQ is the required angle. 
 
 
2. Draw an obtuse angle. Bisect it. Measure each of the angles so obtained. 
Solution: 
 
We know that obtuse angles are those which are greater than 90
o
 and less than 180
o
. 
Construct an obtuse angle ?BAC. 
Taking A as centre with appropriate radius construct an arc which intersects AB and AC at the points P and Q. 
Taking P as centre and radius which is more than half of PQ construct an arc. 
Taking Q as centre and same radius construct another arc which intersects the previous arc at the point R. 
Now join A and R and extend it to the point X. 
So the ray AX is the required bisector of ?BAC. 
By measuring ?BAR and ?CAR we get ?BAR = ?CAR = 65
o
. 
 
 
3. Using your protractor, draw an angle of measure 108
o
. With this angle as given, drawn an angle of 54
o
. 
Solution: 
 
 
 
 
 
 
 
Construct a ray OA. 
Using protractor, draw an angle ?AOB of 108
o 
where 108/2 = 54
o
 
Hence, 54
o
 is half of 108
o
. 
In order to get angle 54
o
, we must bisect the angle of 108
o
. 
Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. 
Taking P as centre and radius which is more than half of PQ construct an arc. 
Taking Q as centre and same radius construct another arc which intersects the previous arc and name it as point R. 
Now join the points O and R and extend it to the point X. 
Here, ?AOX is the required angle of 54
o
. 
 
 
4. Using protractor, draw a right angle. Bisect it to get an angle of measure 45
o
. 
Solution: 
 
Construct a ray OA. 
Using a protractor construct ?AOB of 90
o
. 
Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. 
Taking P as centre and radius which is more than half of PQ, construct an arc. 
Taking Q as centre and same radius, construct another arc which intersects the previous arc and name it as point 
R. 
Now join the points O and R and extend it to the point X. 
Here, ?AOX is the required angle of 45
o
 where ?AOB = 90
o
 and ?AOX = 45
o
. 
 
 
5. Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are 
perpendicular to each other. 
 
 
 
 
 
 
Solution: 
We know that the two angles which are adjacent and supplementary are known as linear pair of angles. 
Construct a line AB and mark a point O on it. 
By constructing an angle ?AOC we get another angle ?BOC. 
Now bisect ?AOC using a compass and a ruler and get the ray OX. 
In the same way bisect ?BOC and get the ray OY. 
We know that  
?XOY = ?XOC + ?COY 
It can be written as 
?XOY = 1/2 ?AOC + 1/2 ?BOC 
So we get 
?XOY = 1/2 (?AOC + ?BOC) 
We know that ?AOC and ?BOC are supplementary angles 
?XOY = 1/2 (180) = 90
o
 
 
 
6. Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are 
in the same line. 
Solution: 
 
Construct two lines AB and CD which intersects each other at the point O 
Since vertically opposite angles are equal we get  
?BOC = ?AOD and ?AOC = ?BOD 
Now bisect angle AOC and construct the bisecting ray as OX. 
In the same way, we bisect ?BOD and construct bisecting ray OY. 
We get  
?XOA + ?AOD + ?DOY = 1/2 ?AOC + ?AOD + 1/2 ?BOD 
We know that ?AOC = ?BOD 
?XOA + ?AOD + ?DOY = 1/2 ?BOD + ?AOD + 1/2 ?BOD 
So we get  
?XOA + ?AOD + ?DOY = ?AOD + ?BOD 
 
AB is a line 
We know that ?AOD and ?BOD are supplementary angles whose sum is equal to 180
o
. 
?XOA + ?AOD + ?DOY = 180
o
 
The angles on one side of a straight line is always 180
o
 and also the sum of angles is 180
o
 
Here, XY is a straight line where OX and OY are in the same line. 
 
 
 
 
 
 
 
 
7. Using ruler and compasses only, draw a right angle. 
Solution: 
 
Construct a ray OA. 
Taking O as centre and convenient radius construct an arc PQ using a compass intersecting the ray OA at the 
point Q. 
Taking P as centre and same radius construct another arc which intersects the arc PQ at the point R. 
Taking R as centre and same radius, construct an arc which cuts the arc PQ at the point C opposite to P. 
Using C and R as the centre construct two arcs of radius which is more than half of CR intersecting each other at 
the point S. 
Now join the points O and S and extend it to the point B. 
Here, ?AOB is the required angle of 90
o
. 
 
 
8. Using ruler and compasses only, draw an angle of measure 135
o
. 
Solution: 
 
Construct a line AB and mark a point O on it. 
Taking O as centre and convenient radius, construct an arc PQ using a compass which intersects the line AB at the 
point P and Q. 
Taking P as centre and same radius, construct another arc which intersects the arc PQ at the point R. 
Taking Q as centre and same radius, construct another arc which intersects the arc PQ at the point S which is 
opposite to P. 
Considering S and R as centres and radius which is more than half of SR, construct two arcs which intersects each 
other at the point T. 
Page 5


 
 
 
 
 
 
                                                                           
1. Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC. 
Solution: 
 
Construct an angle ?BAC and draw a ray OP. 
Taking A as centre and suitable radius, construct an arc which intersects AB and AC at points X and Y. 
Taking O as centre and same radius, construct an arc which intersects the arc OP at the point M. 
Now measure XY with the help of compass. 
Taking M as centre and XY as radius construct an arc which intersects the arc which is drawn from O and name it 
as point N. 
Now join the points O and N and extend it to the point Q. 
Here, ?POQ is the required angle. 
 
 
2. Draw an obtuse angle. Bisect it. Measure each of the angles so obtained. 
Solution: 
 
We know that obtuse angles are those which are greater than 90
o
 and less than 180
o
. 
Construct an obtuse angle ?BAC. 
Taking A as centre with appropriate radius construct an arc which intersects AB and AC at the points P and Q. 
Taking P as centre and radius which is more than half of PQ construct an arc. 
Taking Q as centre and same radius construct another arc which intersects the previous arc at the point R. 
Now join A and R and extend it to the point X. 
So the ray AX is the required bisector of ?BAC. 
By measuring ?BAR and ?CAR we get ?BAR = ?CAR = 65
o
. 
 
 
3. Using your protractor, draw an angle of measure 108
o
. With this angle as given, drawn an angle of 54
o
. 
Solution: 
 
 
 
 
 
 
 
Construct a ray OA. 
Using protractor, draw an angle ?AOB of 108
o 
where 108/2 = 54
o
 
Hence, 54
o
 is half of 108
o
. 
In order to get angle 54
o
, we must bisect the angle of 108
o
. 
Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. 
Taking P as centre and radius which is more than half of PQ construct an arc. 
Taking Q as centre and same radius construct another arc which intersects the previous arc and name it as point R. 
Now join the points O and R and extend it to the point X. 
Here, ?AOX is the required angle of 54
o
. 
 
 
4. Using protractor, draw a right angle. Bisect it to get an angle of measure 45
o
. 
Solution: 
 
Construct a ray OA. 
Using a protractor construct ?AOB of 90
o
. 
Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. 
Taking P as centre and radius which is more than half of PQ, construct an arc. 
Taking Q as centre and same radius, construct another arc which intersects the previous arc and name it as point 
R. 
Now join the points O and R and extend it to the point X. 
Here, ?AOX is the required angle of 45
o
 where ?AOB = 90
o
 and ?AOX = 45
o
. 
 
 
5. Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are 
perpendicular to each other. 
 
 
 
 
 
 
Solution: 
We know that the two angles which are adjacent and supplementary are known as linear pair of angles. 
Construct a line AB and mark a point O on it. 
By constructing an angle ?AOC we get another angle ?BOC. 
Now bisect ?AOC using a compass and a ruler and get the ray OX. 
In the same way bisect ?BOC and get the ray OY. 
We know that  
?XOY = ?XOC + ?COY 
It can be written as 
?XOY = 1/2 ?AOC + 1/2 ?BOC 
So we get 
?XOY = 1/2 (?AOC + ?BOC) 
We know that ?AOC and ?BOC are supplementary angles 
?XOY = 1/2 (180) = 90
o
 
 
 
6. Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are 
in the same line. 
Solution: 
 
Construct two lines AB and CD which intersects each other at the point O 
Since vertically opposite angles are equal we get  
?BOC = ?AOD and ?AOC = ?BOD 
Now bisect angle AOC and construct the bisecting ray as OX. 
In the same way, we bisect ?BOD and construct bisecting ray OY. 
We get  
?XOA + ?AOD + ?DOY = 1/2 ?AOC + ?AOD + 1/2 ?BOD 
We know that ?AOC = ?BOD 
?XOA + ?AOD + ?DOY = 1/2 ?BOD + ?AOD + 1/2 ?BOD 
So we get  
?XOA + ?AOD + ?DOY = ?AOD + ?BOD 
 
AB is a line 
We know that ?AOD and ?BOD are supplementary angles whose sum is equal to 180
o
. 
?XOA + ?AOD + ?DOY = 180
o
 
The angles on one side of a straight line is always 180
o
 and also the sum of angles is 180
o
 
Here, XY is a straight line where OX and OY are in the same line. 
 
 
 
 
 
 
 
 
7. Using ruler and compasses only, draw a right angle. 
Solution: 
 
Construct a ray OA. 
Taking O as centre and convenient radius construct an arc PQ using a compass intersecting the ray OA at the 
point Q. 
Taking P as centre and same radius construct another arc which intersects the arc PQ at the point R. 
Taking R as centre and same radius, construct an arc which cuts the arc PQ at the point C opposite to P. 
Using C and R as the centre construct two arcs of radius which is more than half of CR intersecting each other at 
the point S. 
Now join the points O and S and extend it to the point B. 
Here, ?AOB is the required angle of 90
o
. 
 
 
8. Using ruler and compasses only, draw an angle of measure 135
o
. 
Solution: 
 
Construct a line AB and mark a point O on it. 
Taking O as centre and convenient radius, construct an arc PQ using a compass which intersects the line AB at the 
point P and Q. 
Taking P as centre and same radius, construct another arc which intersects the arc PQ at the point R. 
Taking Q as centre and same radius, construct another arc which intersects the arc PQ at the point S which is 
opposite to P. 
Considering S and R as centres and radius which is more than half of SR, construct two arcs which intersects each 
other at the point T. 
 
 
 
 
 
 
Now join the points O and T which intersects the arc PQ at the point C. 
Considering C and Q as centres and radius which is more than half of CQ, construct two arcs which intersects 
each other at the point D. 
Now join the points O and D and extend it to point X to form the ray OX. 
Here, ?AOX is the required angle of 135
o
. 
 
 
9. Using a protractor, draw an angle of measure 72
o
. With this angle as given, draw angles of measure 36
o
 
and 54
o
. 
Solution: 
 
Construct a ray OA. 
Using protractor construct ?AOB of 72
o
 
Taking O as centre and convenient radius, construct an arc which cut sides OA and OB at the point P and Q. 
Taking P and Q as centres and radius which is more than half of PQ, construct two arcs which cuts each other at 
the point R. 
Now join the points O and R and extend it to the point X. 
Here, OR intersects the arc PQ at the point C. 
Taking C and Q as centres and radius which is more than half of CQ, construct two arcs which cuts each other at 
point T. 
Now join the points O and T and extend it to the point Y. 
 
OX bisects ?AOB 
It can be written as 
?AOX = ?BOX = 72/2 = 36
o
 
 
OY bisects ?BOX 
It can be written as 
?XOY = ?BOY = 36/2 = 18
o
 
 
We know that   
?AOY = ?AOX + ?XOY = 36
o
 + 18
o
 = 54
o
 
 
Here, ?AOX is the required angle of 36
o
 and ?AOY is the required angle of 54
o
.