Parallel Lines :
The lines in a plane which do not intersect
to each other are called parallel lines.
As shown in above figures the line l and
line m are parallel to each other. to each other are called parallel lines.
As shown in above figures the line l and
This is written as line l || line m.
1. To draw a line parallel to a given line through a point outside the given line.
The procedure is as below:
1) First place ruler and setsquare as shown in the
adjacent figure.
2) Slide the set square XYZ upwords till its
side XY passes through point P.
3) Draw the line m passing through point P
and extend it. So line l || line m.
2. Line parallel to same lines:
The line l , line m and line n are parallel to each other.
Examples : 1) Notebook page lines are parallel to each other.
2) Two electric poles
3) Railway lines
3. Lines perpendicular to the same lines.
In figure line m and line n are perpendicular to line l.
This tells us that lines perpendicular to same lines
are parallel to each other.
4. Transversals and intercepts:
In figure line l , line m and line n are
parallel to each other. Line p which intersect to
these lines in points A, B and C, is called as tranversal
which makes two intercepts as seg Ab and seg BC.
5. Properties of parallel lines and their transversals
with respect to intercepts.
As shown in the adjacent figure,
the intercepts made by line l and
m on transversals p, q are as seg AB,
seg BC and seg DE, seg EF respectively.
If the seg AB and seg BC have the equal
length then we write seg AB = seg BC.
The ratio of the lengths of intercepts made by three parallel lines on one transversl and the ratio of the lengths of the corresponding intercepts made by the same lines on any other transversal are equal. In above figure
we have
Exercise : In adjoining figure, line k || line l || line m.
Their transversals lines c and line d, cut each other
at points X, Y, Z and P, Q, R respectively.
If l(XY) = 5, l(YZ)= 3, l(PQ) = 5.5, find
l(QR) .
Solution: By the property of three parallel lines,
6. Dividing a line segment into equal parts:
i) Dividing a line segment into a given number of equal parts.
Exercise : Divide a line segment PQ of length 5.5 cm into four congruent parts.
Solution :
Procedure :
1. Draw seg PQ of length 5.5 cm.
2. Draw acute angle <SPQ and <PQR at the
points P and Q of same measure but sides
as shown in figure.
3. Take convenient radius and with the tip of
your compass at P make a point P1 on ray PS.
Now with compass point on P1 and at the same
distance PP1 mark another point P2. Similarly
mark two more points P3 and P4.
Keeping the same radius mark points Q1, Q2,
Q3, Q4 on ray QR as you did on ray PS.
4. Draw line segments PQ4, P1Q3, P2Q2, P3Q1,
P4Q using the meter scale.
ii) Dividing a line segment in a given ratio :
Exercise : Divide XY segment of length 4.5 cm
at point L so that l(XL) : l(LY) = 1 : 2.
Procedure :
1. Draw seg XY of length 4.5 cm.
2. Divide the seg XY into three equal parts.
3. Name the points of division of the segment
as L and M.
2. Line parallel to same lines:
The line l , line m and line n are parallel to each other.
Examples : 1) Notebook page lines are parallel to each other.
2) Two electric poles
3) Railway lines
3. Lines perpendicular to the same lines.
In figure line m and line n are perpendicular to line l.
This tells us that lines perpendicular to same lines
are parallel to each other.
4. Transversals and intercepts:
In figure line l , line m and line n areparallel to each other. Line p which intersect to
these lines in points A, B and C, is called as tranversal
which makes two intercepts as seg Ab and seg BC.
5. Properties of parallel lines and their transversals
with respect to intercepts.
As shown in the adjacent figure,the intercepts made by line l and
m on transversals p, q are as seg AB,
seg BC and seg DE, seg EF respectively.
If the seg AB and seg BC have the equal
length then we write seg AB = seg BC.
The ratio of the lengths of intercepts made by three parallel lines on one transversl and the ratio of the lengths of the corresponding intercepts made by the same lines on any other transversal are equal. In above figure
we have
Exercise : In adjoining figure, line k || line l || line m.
Their transversals lines c and line d, cut each otherat points X, Y, Z and P, Q, R respectively.
If l(XY) = 5, l(YZ)= 3, l(PQ) = 5.5, find
l(QR) .
Solution: By the property of three parallel lines,
6. Dividing a line segment into equal parts:
i) Dividing a line segment into a given number of equal parts.
Exercise : Divide a line segment PQ of length 5.5 cm into four congruent parts.
Solution :
Procedure :
1. Draw seg PQ of length 5.5 cm.
2. Draw acute angle <SPQ and <PQR at thepoints P and Q of same measure but sides
as shown in figure.
3. Take convenient radius and with the tip of
your compass at P make a point P1 on ray PS.
Now with compass point on P1 and at the same
distance PP1 mark another point P2. Similarly
mark two more points P3 and P4.
Keeping the same radius mark points Q1, Q2,
Q3, Q4 on ray QR as you did on ray PS.
4. Draw line segments PQ4, P1Q3, P2Q2, P3Q1,
P4Q using the meter scale.
ii) Dividing a line segment in a given ratio :
Exercise : Divide XY segment of length 4.5 cmat point L so that l(XL) : l(LY) = 1 : 2.
Procedure :
1. Draw seg XY of length 4.5 cm.
2. Divide the seg XY into three equal parts.
3. Name the points of division of the segment
as L and M.