6. Lines and Angles
NCERT
6. Lines and Angles
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Important Note :
1. If two lines intersect each other, then the vertically opposite angles are equal.
2. If a transversal intersects two parallel lines, then
(i) each pair of corresponding angles is equal,
(ii) each pair of alternate interior angles is equal,
(iii) each pair of interior angles on the same side of the transversal is supplementary.
3. If a transversal intersects two lines such that, either
(i) any one pair of corresponding angles is equal, or
(ii) any one pair of alternate interior angles is equal, or
(iii) any one pair of interior angles on the same side of the transversal is supplementary, then the lines are parallel.
4. Lines which are parallel to a given line are parallel to each other.
5. The sum of the three angles of a triangle is 180°.
6. If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
EXERCISE 6.1
and
, find 
and reflex 
. 
Solution: Given, 
and 
.
Since , 
is a straight line .
and 
is a straight line .
Reflex 
and 
, find
.
Solution: Given, 
and .
Let 
and 
.
Since, 
is a straight line .
and 
and 
is a straight line .
Therefore, the value of 
is 126° .
, then prove that 
.
Solution: Given , 
, then we prove that .
Proof : Since, ray 
stands on line 
.
………. (i)
Ray 
stands on line 
.
…………. (ii)
From (i) and (ii) , we get 
But, 
Proved .
, then prove that AOB is a line .
Solution : Given, , then we that prove that
is a line .
Ray 
stands on line .
…….(i) [Linear pair of angles]
Ray 
stands on line .
……(ii) [Linear pair of angles]
Adding (i) and (ii) , we get 
[Linear pair of angles]
Therefore, 
is a line . Proved .
.
Solution: Given , POQ is a line . Ray OR is perpendicular to line PQ . OS is another ray lying between rays OP and OR .
To Prove :
Proof : Since, ray OR is perpendicular to line PQ .
i.e., 
……….. (i)
And 
[Add both sides 
]
……….. (ii)
From (i) and (ii) , we get 
Proved .
and XY is produced to point P . Draw a figure from the given information . If ray YQ bisects 
and find 
and reflex 
.Solution : Given,
Ray YQ bisects , then 
.
Let 
Since, 
is a straight line .
And 
EXERCISE 6.2
1. In Fig. 
, find the values of 
and 
and then show that 
.
2. In Fig. 
,if ,
and 
find .
3. In Fig .6.30, if 
and 
, find 
and 
.
4. In Fig. 6.31, if 
and 
find 
5. In Fig . 6.32,if 
and , 
find and 
.
6. In Fig .6.33, 
and 
are two mirrors placed parallel to each other .An incident ray 
strikes the mirror at 
, the reflected ray moves along the path 
and strikes the mirror at 
and again reflects back along 
. prove that .
EXERCISE 6.3
1. In Fig. 6.39, sides
and 
of 
are produced to points 
and 
respectively . If 
and 
find 
2.In Fig .6.40 , 
If 
and 
are the bisectors of 
and 
respectively of 
find 
and 
.
3. In Fig .6.41 ,if 
and 
, find 
.
4. In Fig .6.42 , if lines and intersect at point 
,such that 
and 
find 
5. In Fig. 6.43 , if 
and 
then find the values of and .
6. In Fig . 6.44, the sides 
of is produced to a point .If the bisectors of 
and 
meet at point 
, then prove that 
Posted 3 years ago