Triangle and its Angles- 2 RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

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Question:13
The exterior angles, obtained on producing the base of a triangle both way are 104° and 136°. Find all the angles of the triangle.
Solution:
In the given problem, the exterior angles obtained on producing the base of a triangle both ways are and . So, let us draw ?ABC and extend the base BC, such that:
Page 2


                   
                            
     
                
                      
                
                     
 
       
                            
               
                     
 
        
                       
 
                     
     
             
             
     
       
     
Question:13
The exterior angles, obtained on producing the base of a triangle both way are 104° and 136°. Find all the angles of the triangle.
Solution:
In the given problem, the exterior angles obtained on producing the base of a triangle both ways are and . So, let us draw ?ABC and extend the base BC, such that:
Here, we need to find all the three angles of the triangle.
Now, since BCD is a straight line, using the property, “angles forming a linear pair are supplementary”, we get
Similarly, EBC is a straight line, so we get,
Further, using angle sum property in ?ABC
Therefore, .
Question:14
In the given figure, the sides BC, CA and AB of a ? ABC have been produced to D, E and F respectively. If ?ACD = 105° and ?EAF = 45°, find all the angles of the ? ABC.
Solution:
In the given ?ABC, and . We need to find .
Here, are vertically opposite angles. So, using the property, “vertically opposite angles are equal”, we get,
Further, BCD is a straight line. So, using linear pair property, we get,
Now, in ?ABC, using “the angle sum property”, we get,
Therefore, .
Question:15
Compute the value of x in each of the following figures:
i
Page 3


                   
                            
     
                
                      
                
                     
 
       
                            
               
                     
 
        
                       
 
                     
     
             
             
     
       
     
Question:13
The exterior angles, obtained on producing the base of a triangle both way are 104° and 136°. Find all the angles of the triangle.
Solution:
In the given problem, the exterior angles obtained on producing the base of a triangle both ways are and . So, let us draw ?ABC and extend the base BC, such that:
Here, we need to find all the three angles of the triangle.
Now, since BCD is a straight line, using the property, “angles forming a linear pair are supplementary”, we get
Similarly, EBC is a straight line, so we get,
Further, using angle sum property in ?ABC
Therefore, .
Question:14
In the given figure, the sides BC, CA and AB of a ? ABC have been produced to D, E and F respectively. If ?ACD = 105° and ?EAF = 45°, find all the angles of the ? ABC.
Solution:
In the given ?ABC, and . We need to find .
Here, are vertically opposite angles. So, using the property, “vertically opposite angles are equal”, we get,
Further, BCD is a straight line. So, using linear pair property, we get,
Now, in ?ABC, using “the angle sum property”, we get,
Therefore, .
Question:15
Compute the value of x in each of the following figures:
i
ii
iii
Solution:
In the given problem, we need to find the value of x
i In the given ?ABC, and 
Now, BCD is a straight line. So, using the property, “the angles forming a linear pair are supplementary”, we get,
Similarly, EAC is a straight line. So, we get,
Further, using the angle sum property of a triangle,
In ?ABC
Therefore, 
ii In the given ?ABC, and 
Here, BCD is a straight line. So, using the property, “the angles forming a linear pair are supplementary” we get,
Page 4


                   
                            
     
                
                      
                
                     
 
       
                            
               
                     
 
        
                       
 
                     
     
             
             
     
       
     
Question:13
The exterior angles, obtained on producing the base of a triangle both way are 104° and 136°. Find all the angles of the triangle.
Solution:
In the given problem, the exterior angles obtained on producing the base of a triangle both ways are and . So, let us draw ?ABC and extend the base BC, such that:
Here, we need to find all the three angles of the triangle.
Now, since BCD is a straight line, using the property, “angles forming a linear pair are supplementary”, we get
Similarly, EBC is a straight line, so we get,
Further, using angle sum property in ?ABC
Therefore, .
Question:14
In the given figure, the sides BC, CA and AB of a ? ABC have been produced to D, E and F respectively. If ?ACD = 105° and ?EAF = 45°, find all the angles of the ? ABC.
Solution:
In the given ?ABC, and . We need to find .
Here, are vertically opposite angles. So, using the property, “vertically opposite angles are equal”, we get,
Further, BCD is a straight line. So, using linear pair property, we get,
Now, in ?ABC, using “the angle sum property”, we get,
Therefore, .
Question:15
Compute the value of x in each of the following figures:
i
ii
iii
Solution:
In the given problem, we need to find the value of x
i In the given ?ABC, and 
Now, BCD is a straight line. So, using the property, “the angles forming a linear pair are supplementary”, we get,
Similarly, EAC is a straight line. So, we get,
Further, using the angle sum property of a triangle,
In ?ABC
Therefore, 
ii In the given ?ABC, and 
Here, BCD is a straight line. So, using the property, “the angles forming a linear pair are supplementary” we get,
Similarly, EBC is a straight line. So, we get
Further, using the angle sum property of a triangle,
In ?ABC
Therefore, 
iii In the given figure, and 
      
Here, and AD is the transversal, so form a pair of alternate interior angles. Therefore, using the property, “alternate interior angles are equal”, we get,
Further, applying angle sum property of the triangle
In ?DEC
Therefore, 
Question:16
In the given figure, AC ? CE and ?A : ?B : ?C = 3 : 2 : 1, find the value of ?ECD.
 
Solution:
In the given figure, and . We need to find the value of 
 
Since, 
Let, 
Applying the angle sum property of the triangle, in ?ABC, we get,
Thus,
Page 5


                   
                            
     
                
                      
                
                     
 
       
                            
               
                     
 
        
                       
 
                     
     
             
             
     
       
     
Question:13
The exterior angles, obtained on producing the base of a triangle both way are 104° and 136°. Find all the angles of the triangle.
Solution:
In the given problem, the exterior angles obtained on producing the base of a triangle both ways are and . So, let us draw ?ABC and extend the base BC, such that:
Here, we need to find all the three angles of the triangle.
Now, since BCD is a straight line, using the property, “angles forming a linear pair are supplementary”, we get
Similarly, EBC is a straight line, so we get,
Further, using angle sum property in ?ABC
Therefore, .
Question:14
In the given figure, the sides BC, CA and AB of a ? ABC have been produced to D, E and F respectively. If ?ACD = 105° and ?EAF = 45°, find all the angles of the ? ABC.
Solution:
In the given ?ABC, and . We need to find .
Here, are vertically opposite angles. So, using the property, “vertically opposite angles are equal”, we get,
Further, BCD is a straight line. So, using linear pair property, we get,
Now, in ?ABC, using “the angle sum property”, we get,
Therefore, .
Question:15
Compute the value of x in each of the following figures:
i
ii
iii
Solution:
In the given problem, we need to find the value of x
i In the given ?ABC, and 
Now, BCD is a straight line. So, using the property, “the angles forming a linear pair are supplementary”, we get,
Similarly, EAC is a straight line. So, we get,
Further, using the angle sum property of a triangle,
In ?ABC
Therefore, 
ii In the given ?ABC, and 
Here, BCD is a straight line. So, using the property, “the angles forming a linear pair are supplementary” we get,
Similarly, EBC is a straight line. So, we get
Further, using the angle sum property of a triangle,
In ?ABC
Therefore, 
iii In the given figure, and 
      
Here, and AD is the transversal, so form a pair of alternate interior angles. Therefore, using the property, “alternate interior angles are equal”, we get,
Further, applying angle sum property of the triangle
In ?DEC
Therefore, 
Question:16
In the given figure, AC ? CE and ?A : ?B : ?C = 3 : 2 : 1, find the value of ?ECD.
 
Solution:
In the given figure, and . We need to find the value of 
 
Since, 
Let, 
Applying the angle sum property of the triangle, in ?ABC, we get,
Thus,
Further, BCD is a straight line. So, applying the property, “the angles forming a linear pair are supplementary”, we get,
Therefore, .
Question:17
In the given figure, AB || DE. Find ?ACD.
Solution:
In the given problem,
We need to find
Now, and AE is the transversal, so using the property, “alternate interior angles are equal”, we get,
Further, applying angle sum property of the triangle
In ?DCE
Further, ACE is a straight line, so using the property, “the angles forming a linear pair are supplementary”, we get,
Therefore, .
Question:18
Which of the following statements are true T
and which are false F
:
i
Sum of the three angles of a triangle is 180°.
ii
A triangle can have two right angles.
iii
All the angles of a triangle can be less than 60°
iv
All the angles of a triangle  can be greater than 60°.
v
All the angles of a triangle can be equal to 60°.
vi
A triangle can have two obtuse angles.
vii
A triangle can have at most one obtuse angles.
viii
If one angle of a triangle is obtuse, then it cannot be a right angled triangle.