Lines Angles and Triangles


   
 
Question (1): Can two vertically opposite angles be supplement to each other? Draw a diagram to illustrate your answer.

Answer: 
Let AB and CD bisect each other at right angles as shown in figure.


image


( Vertically opposite angles can be
supplements to each other)
Question (1):

Answer: 



x and y


image 146 + y = 180

y = 180 - 146
= 340
(vertically opposite angles)
2x = 34

x = 17o
Question (3):

Answer: 
5x + 10 = 90o
5x = 90o - 10o
= 80o






Verification:


Question (4): In the following diagrams parallel lines are marked by arrows in the same direction. Transversal is also drawn. Find the missing angles and provide the reason for each.
Note: (No proof is required but the essential steps of working must be given)
Answer:  (i) Find a, b, c, d, e, f, g.








(ii) In the adjoining diagram AB||CD; PQ and RS are transversals. Find angles 1, 2, .....16.








image






Question (5): If two straight lines are each perpendicular to a third straight line then they are parallel to each other. Prove.

Answer: 

Straight line AB and CD are perpendicular to line l.

AB||CD








Question (6):

Answer: 







From (i), (ii) and (iii)

are supplementary angles.
Question (7):

Answer: 
AB||CD, BC||DE




AB||CD (given)
BC is a transversal

BC||DE (given)
DC is a transversal


Question (8): In the figure, show that AB||EF.

Answer:  [given]




i.e.,





Question (9): If a line is perpendicular to one of two given parallel lines, show that it is also perpendicular to the other line.

Answer: 



image



[corresponding angles]
But,




Question (10):
are the bisectors of the corresponding angles BGE and DHE respectively.
Show that



Answer: 

angles BGE and DHE respectively.




i)











Question (11): Two lines 'l', 'm' are perpendicular to the lines 'n', 'p'. If 'n' and 'p' intersect, prove that 'l' and 'm' must also intersect.

Answer: 
l ^ 'n' and 'm' ^ 'p' [given]
Let 'n' and 'p' intersect at 'A'

' l' and 'm' intersect.

Let us assume that 'l' and 'm' do not intersect. Then 'l' must be parallel to 'm'.
But l ^ n [given]
\m ^ n ..... (i)
now m ^ p ..... (ii)
from (i) and (ii) we conclude that n||p.
But it is given that 'n' and 'p' are intersecting lines. So our assumption that 'l' and 'm' do not intersect is wrong.
Hence 'l' and 'm' intersect.
Question (12): In the given figure 'm' and 'n' are two plane mirrors parallel to


Answer: 
m||n [given]












Question (13):
Answer: 
(Angle sum property)





Question (14): The sum of two angles of a triangle is 80o and their difference is 20o. Find all the angles of the triangle.
Answer:  Let the two angles of the triangle be xo and yo.
Then xo+ yo= 80o ...(i)
xo- yo= 20o ...(ii)
From equations (i) + (ii) we get 2xo= 100o

xo= 50o
Substituting xo= 50o in (i)
xo+ yo= 80o

50o+ yo= 80o
yo = 80o- 50o
yo = 30o
The two angles of the triangle are 50o and 30o.
The third angle = 180o - (50o +30o) (sum of angles in a triangle is 180o)
= 180o - 80o
= 100o
Question (15):
Answer: 
[angle sum property]


...(i)




...(ii)



Question (16):

Answer: 





















Thus the three angles are



Question (17): An exterior angle of a triangle is 110o and one of the interior opposite angles is 30o. Find the other two angles of the triangle.
Answer: 



= 80o




= 70o

Question (18):
Answer: 





[ Sum of angle of a triangle = 180o]





Question (19):
Answer: 








Adding (i), (ii) and (iii)



=360o

Question (20):
Answer: 











Multiplying both sides by 2.

Subtracting (i) from (ii)


Question (21): Three angles of a quadrilateral are 100o, 48o and 92o. Find the 4th angle.
Answer: 


quadrilateral is 360o]




=120o
The 4th angle measures 120o.
Question (22):
Answer: 


Adding (i) and (ii)
...(iii)

...(iv) [ sum of the angles of a
quadrilateral equals 360o]
Equating (iii) and (iv)


Question (23): ABCDE is a regular pentagon. The bisectors of of the pentagon meets

[Hint: Each angle of a regular pentagon is 108o]
Answer: 

ABCDE is a regular pentagon.



Number of sides of the regular pentagon, n=5.
Each interior angle






=108o


=54o
In quadrilateral AMCD,





Question (24): In the figure, lines 'l' and 'm' intersect at O, forming angles as shown in the figure. If x=45o, find the values of y, z and u.

Answer:  xo= 45o [given]
zo= 45o [ vertically opposite angles]
xo = yo = 180o [Linear pair]
45o + yo = 180o
yo = 180o - 45o

yo = 135o
If yo = 135o, then xo = 135o [ vertically opposite angles]
Hence the values are: y=135o, z=45o and x=135o
Question (25): In the adjoining figure, determine the value of y.

Answer: 
ao = 5yo [ vertically opposite angles]
Now 5yo + ao + 2yo = 180o
i.e., 5yo + 5yo + 2yo = 180o [ Linear pair]

12yo = 180o

yo = 15o
Question (26): In the adjoining figure, three coplanar lines intersect at a common point, forming angles as shown. Given a=50o and b=90o. Find the values of c, d, e and f.

Answer:  d = 50o [ ao=50o, vertically opposite angles]
e = 90o [ bo=90o, vertically opposite angles]
b + c + d = 180o [ They form a linear pair]
90 + c + 50 = 180o
140o+ c = 180o
c = 40o
Similarly a + e + f = 180o [ They form a linear pair]
50o+ 90o+ f = 180o
140o+ f = 180o
f = 40o [c=40o, vertically opposite angles]
Hence c = 40o; d = 50o; e = 90o and f = 40o
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