RD Sharma Solutions: Geometrical Constructions (Exercise 19.5)

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

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``` 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
.
```

## Mathematics (Maths) Class 6

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## Mathematics (Maths) Class 6

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