Class 9 Maths Question Answers - Triangle and its Angles

 Page 1


   
          
                      
                 
 
Question:28
Define a triangle.
Solution:
A plane figure bounded by three lines in a plane is called a triangle. The figure below represents a ?ABC, with AB, AC and BC as the three line segments.
Question:29
Write the sum of the angles of an obtuse triangle.
Solution:
In the given problem, ?ABC is an obtuse triangle, with as the obtuse angle.
So, according to “the angle sum property of the triangle”, for any kind of triangle, the sum of its angles is 180°. So,
Therefore, sum of the angles of an obtuse triangle is .
Question:30
In ? ABC, if u ?B = 60°, ?C = 80° and the bisectors of angles ?ABC and ?ACB meet at a point O, then find the measure of ?BOC.
Solution:
In ?ABC, , and the bisectors of and meet at O.
We need to find the measure of 
Page 2


   
          
                      
                 
 
Question:28
Define a triangle.
Solution:
A plane figure bounded by three lines in a plane is called a triangle. The figure below represents a ?ABC, with AB, AC and BC as the three line segments.
Question:29
Write the sum of the angles of an obtuse triangle.
Solution:
In the given problem, ?ABC is an obtuse triangle, with as the obtuse angle.
So, according to “the angle sum property of the triangle”, for any kind of triangle, the sum of its angles is 180°. So,
Therefore, sum of the angles of an obtuse triangle is .
Question:30
In ? ABC, if u ?B = 60°, ?C = 80° and the bisectors of angles ?ABC and ?ACB meet at a point O, then find the measure of ?BOC.
Solution:
In ?ABC, , and the bisectors of and meet at O.
We need to find the measure of 
Since,BO is the bisector of 
Similarly,CO is the bisector of 
Now, applying angle sum property of the triangle, in ?BOC, we get,
Therefore, .
Question:31
If the angles of a triangle are in the ratio 2 : 1 : 3, then find the measure of smallest angle.
Solution:
In the given problem, angles of ?ABC are in the ratio 2:1:3
We need to find the measure of the smallest angle.
Let,
According to the angle sum property of the triangle, in ?ABC, we get,
Thus, 
Since, the measure of  is the smallest of all the three angles.
Therefore, the measure of the smallest angle is .
Question:32
State exterior angle theorem.
Solution:
Exterior angle theorem states that, if a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
Thus, in ?ABC
Question:33
The sum of two angles of a triangle is equal to its third angle. Determine the measure of the third angle.
Solution:
Page 3


   
          
                      
                 
 
Question:28
Define a triangle.
Solution:
A plane figure bounded by three lines in a plane is called a triangle. The figure below represents a ?ABC, with AB, AC and BC as the three line segments.
Question:29
Write the sum of the angles of an obtuse triangle.
Solution:
In the given problem, ?ABC is an obtuse triangle, with as the obtuse angle.
So, according to “the angle sum property of the triangle”, for any kind of triangle, the sum of its angles is 180°. So,
Therefore, sum of the angles of an obtuse triangle is .
Question:30
In ? ABC, if u ?B = 60°, ?C = 80° and the bisectors of angles ?ABC and ?ACB meet at a point O, then find the measure of ?BOC.
Solution:
In ?ABC, , and the bisectors of and meet at O.
We need to find the measure of 
Since,BO is the bisector of 
Similarly,CO is the bisector of 
Now, applying angle sum property of the triangle, in ?BOC, we get,
Therefore, .
Question:31
If the angles of a triangle are in the ratio 2 : 1 : 3, then find the measure of smallest angle.
Solution:
In the given problem, angles of ?ABC are in the ratio 2:1:3
We need to find the measure of the smallest angle.
Let,
According to the angle sum property of the triangle, in ?ABC, we get,
Thus, 
Since, the measure of  is the smallest of all the three angles.
Therefore, the measure of the smallest angle is .
Question:32
State exterior angle theorem.
Solution:
Exterior angle theorem states that, if a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
Thus, in ?ABC
Question:33
The sum of two angles of a triangle is equal to its third angle. Determine the measure of the third angle.
Solution:
In the given problem, the sum of two angles of a triangle is equal to its third angle.
We need to find the measure of the third angle.
Thus, it is given, in 
                                 ........
1
Now, according to the angle sum property of the triangle, we get,
Using1
Therefore, the measure of the third angle is .
Question:34
In the given figure, if AB || CD, EF || BC, ?BAC = 65° and ?DHF = 35°, find ?AGH.
Solution:
In the given figure, , , and 
We need to find
Here, GF and CD are straight lines intersecting at point H, so using the property, “vertically opposite angles are equal”, we get,
Further, as and AC is the transversal
Using the property, “alternate interior angles are equal”
Further applying angle sum property of the triangle
In ?GHC
Hence, applying the property, “angles forming a linear pair are supplementary”
As AGC is a straight line
Therefore,
Question:35
In the given figure, if AB || DE and BD || FG such that ?FGH = 125° and ?B = 55°, find x and y.
 
Page 4


   
          
                      
                 
 
Question:28
Define a triangle.
Solution:
A plane figure bounded by three lines in a plane is called a triangle. The figure below represents a ?ABC, with AB, AC and BC as the three line segments.
Question:29
Write the sum of the angles of an obtuse triangle.
Solution:
In the given problem, ?ABC is an obtuse triangle, with as the obtuse angle.
So, according to “the angle sum property of the triangle”, for any kind of triangle, the sum of its angles is 180°. So,
Therefore, sum of the angles of an obtuse triangle is .
Question:30
In ? ABC, if u ?B = 60°, ?C = 80° and the bisectors of angles ?ABC and ?ACB meet at a point O, then find the measure of ?BOC.
Solution:
In ?ABC, , and the bisectors of and meet at O.
We need to find the measure of 
Since,BO is the bisector of 
Similarly,CO is the bisector of 
Now, applying angle sum property of the triangle, in ?BOC, we get,
Therefore, .
Question:31
If the angles of a triangle are in the ratio 2 : 1 : 3, then find the measure of smallest angle.
Solution:
In the given problem, angles of ?ABC are in the ratio 2:1:3
We need to find the measure of the smallest angle.
Let,
According to the angle sum property of the triangle, in ?ABC, we get,
Thus, 
Since, the measure of  is the smallest of all the three angles.
Therefore, the measure of the smallest angle is .
Question:32
State exterior angle theorem.
Solution:
Exterior angle theorem states that, if a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
Thus, in ?ABC
Question:33
The sum of two angles of a triangle is equal to its third angle. Determine the measure of the third angle.
Solution:
In the given problem, the sum of two angles of a triangle is equal to its third angle.
We need to find the measure of the third angle.
Thus, it is given, in 
                                 ........
1
Now, according to the angle sum property of the triangle, we get,
Using1
Therefore, the measure of the third angle is .
Question:34
In the given figure, if AB || CD, EF || BC, ?BAC = 65° and ?DHF = 35°, find ?AGH.
Solution:
In the given figure, , , and 
We need to find
Here, GF and CD are straight lines intersecting at point H, so using the property, “vertically opposite angles are equal”, we get,
Further, as and AC is the transversal
Using the property, “alternate interior angles are equal”
Further applying angle sum property of the triangle
In ?GHC
Hence, applying the property, “angles forming a linear pair are supplementary”
As AGC is a straight line
Therefore,
Question:35
In the given figure, if AB || DE and BD || FG such that ?FGH = 125° and ?B = 55°, find x and y.
 
Solution:
In the given figure, , , and 
We need to find the value of x and y
Here, as and BD is the transversal, so according to the property, “alternate interior angles are equal”, we get
Similarly, as and DF is the transversal
Using1
Further, EGH is a straight line. So, using the property, angles forming a linear pair are supplementary
Also, using the property, “an exterior angle of a triangle is equal to the sum of the two opposite interior angles”, we get,
In  with as its exterior angle
Thus,
Question:36
If the angles A, B and C of ?ABC satisfy the relation B - A = C - B, then find the measure of ?B.
Solution:
In the given ?ABC, 
, and satisfy the relation 
We need to fine the measure of .
As,
          ........
1
Now, using the angle sum property of the triangle, we get,
Using1
Therefore,
Page 5


   
          
                      
                 
 
Question:28
Define a triangle.
Solution:
A plane figure bounded by three lines in a plane is called a triangle. The figure below represents a ?ABC, with AB, AC and BC as the three line segments.
Question:29
Write the sum of the angles of an obtuse triangle.
Solution:
In the given problem, ?ABC is an obtuse triangle, with as the obtuse angle.
So, according to “the angle sum property of the triangle”, for any kind of triangle, the sum of its angles is 180°. So,
Therefore, sum of the angles of an obtuse triangle is .
Question:30
In ? ABC, if u ?B = 60°, ?C = 80° and the bisectors of angles ?ABC and ?ACB meet at a point O, then find the measure of ?BOC.
Solution:
In ?ABC, , and the bisectors of and meet at O.
We need to find the measure of 
Since,BO is the bisector of 
Similarly,CO is the bisector of 
Now, applying angle sum property of the triangle, in ?BOC, we get,
Therefore, .
Question:31
If the angles of a triangle are in the ratio 2 : 1 : 3, then find the measure of smallest angle.
Solution:
In the given problem, angles of ?ABC are in the ratio 2:1:3
We need to find the measure of the smallest angle.
Let,
According to the angle sum property of the triangle, in ?ABC, we get,
Thus, 
Since, the measure of  is the smallest of all the three angles.
Therefore, the measure of the smallest angle is .
Question:32
State exterior angle theorem.
Solution:
Exterior angle theorem states that, if a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
Thus, in ?ABC
Question:33
The sum of two angles of a triangle is equal to its third angle. Determine the measure of the third angle.
Solution:
In the given problem, the sum of two angles of a triangle is equal to its third angle.
We need to find the measure of the third angle.
Thus, it is given, in 
                                 ........
1
Now, according to the angle sum property of the triangle, we get,
Using1
Therefore, the measure of the third angle is .
Question:34
In the given figure, if AB || CD, EF || BC, ?BAC = 65° and ?DHF = 35°, find ?AGH.
Solution:
In the given figure, , , and 
We need to find
Here, GF and CD are straight lines intersecting at point H, so using the property, “vertically opposite angles are equal”, we get,
Further, as and AC is the transversal
Using the property, “alternate interior angles are equal”
Further applying angle sum property of the triangle
In ?GHC
Hence, applying the property, “angles forming a linear pair are supplementary”
As AGC is a straight line
Therefore,
Question:35
In the given figure, if AB || DE and BD || FG such that ?FGH = 125° and ?B = 55°, find x and y.
 
Solution:
In the given figure, , , and 
We need to find the value of x and y
Here, as and BD is the transversal, so according to the property, “alternate interior angles are equal”, we get
Similarly, as and DF is the transversal
Using1
Further, EGH is a straight line. So, using the property, angles forming a linear pair are supplementary
Also, using the property, “an exterior angle of a triangle is equal to the sum of the two opposite interior angles”, we get,
In  with as its exterior angle
Thus,
Question:36
If the angles A, B and C of ?ABC satisfy the relation B - A = C - B, then find the measure of ?B.
Solution:
In the given ?ABC, 
, and satisfy the relation 
We need to fine the measure of .
As,
          ........
1
Now, using the angle sum property of the triangle, we get,
Using1
Therefore,
Question:37
In ?ABC, if bisectors of ?ABC and ?ACB intersect at O at angle of 120°, then find the measure of ?A.
Solution:
In the given ?ABC, , the bisectors of and meet at O and 
We need to find the measure of 
So here, using the corollary, “if the bisectors of and of a meet at a point O, then ”
Thus, in ?ABC
Thus,
Question:38
If the side BC of ?ABC is produced on both sides, then write the difference between the sum of the exterior angles so formed and ?A.
Solution:
In the given problem, we need to find the difference between the sum of the exterior angles and .
Now, according to the exterior angle theorem
               .........1
Also,
              .........2
Further, adding
1 and
2
              .........3
Also, according to the angle sum property of the triangle, we get,
        .........4
Now, we need to find the difference between the sum of the exterior angles and .
Thus, 
Using4
Therefore, 
Question:39
In a triangle ABC, if AB =  AC and AB is produced to D such that BD =  BC, find ?ACD: ?ADC.
Solution:
In the given , and AB is produced to D such that 
We need to find