Class 9 Maths Question Answers - Congruent Triangles

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Question:47
In two congruent triangles ABC and DEF, if AB = DE and BC = EF. Name the pairs of equal angles.
Solution:
It is given that
Since, the triangles ABC and DEF are congruent, therefore,
Question:48
In two triangles ABC and DEF, it is given that ?A = ?D, ?B = ?E and ?C = ?F. Are the two triangles necessarily congruent?
Solution:
It is given that
For necessarily triangle to be congruent, sides should also be equal.
 
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Question:47
In two congruent triangles ABC and DEF, if AB = DE and BC = EF. Name the pairs of equal angles.
Solution:
It is given that
Since, the triangles ABC and DEF are congruent, therefore,
Question:48
In two triangles ABC and DEF, it is given that ?A = ?D, ?B = ?E and ?C = ?F. Are the two triangles necessarily congruent?
Solution:
It is given that
For necessarily triangle to be congruent, sides should also be equal.
 
Question:49
If ABC and DEF are two triangles such that AC = 2.5 cm, BC = 5 cm, ?C = 75°, DE = 2.5 cm, DF = 5cm and ?D = 75°. Are two
triangles congruent?
Solution:
It is given that
Since, two sides and angle between them are equal, therefore triangle ABC and DEF are congruent.
Question:50
In two triangles ABC and ADC, if AB = AD and BC = CD. Are they congruent?
Solution:
The given information and corresponding figure is given below
From the figure, we have
And,
Hence, triangles ABC and ADC are congruent to each other.
Question:51
In triangles ABC and CDE, if AC = CE, BC = CD, ?A = 60°, ?C = 30° and ?D = 90°.
Are two triangles congruent?
Solution:
For the triangles ABC and ECD, we have the following information and corresponding figure:
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Question:47
In two congruent triangles ABC and DEF, if AB = DE and BC = EF. Name the pairs of equal angles.
Solution:
It is given that
Since, the triangles ABC and DEF are congruent, therefore,
Question:48
In two triangles ABC and DEF, it is given that ?A = ?D, ?B = ?E and ?C = ?F. Are the two triangles necessarily congruent?
Solution:
It is given that
For necessarily triangle to be congruent, sides should also be equal.
 
Question:49
If ABC and DEF are two triangles such that AC = 2.5 cm, BC = 5 cm, ?C = 75°, DE = 2.5 cm, DF = 5cm and ?D = 75°. Are two
triangles congruent?
Solution:
It is given that
Since, two sides and angle between them are equal, therefore triangle ABC and DEF are congruent.
Question:50
In two triangles ABC and ADC, if AB = AD and BC = CD. Are they congruent?
Solution:
The given information and corresponding figure is given below
From the figure, we have
And,
Hence, triangles ABC and ADC are congruent to each other.
Question:51
In triangles ABC and CDE, if AC = CE, BC = CD, ?A = 60°, ?C = 30° and ?D = 90°.
Are two triangles congruent?
Solution:
For the triangles ABC and ECD, we have the following information and corresponding figure:
In triangles ABC and ECD, we have
The SSA criteria for two triangles to be congruent are being followed. So both the triangles are congruent.
Question:52
ABC is an isosceles triangle in which AB = AC. BE and CF are its two medians. Show that BE = CF.
Solution:
In the triangle ABC it is given that 
,  and  are medians.
We have to show that 
To show we will show that 
In triangle ?BFC and ?BEC
As , so 
             .........1
BC=BC
commonsides   ........
2
Since,
As F and E are mid points of sides AB and AC respectively, so
BF = CE         ..........
3
From equation
1,
2, and
3
Hence Proved.
Question:53
Find the measure of each angle of an equilateral triangle.
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Question:47
In two congruent triangles ABC and DEF, if AB = DE and BC = EF. Name the pairs of equal angles.
Solution:
It is given that
Since, the triangles ABC and DEF are congruent, therefore,
Question:48
In two triangles ABC and DEF, it is given that ?A = ?D, ?B = ?E and ?C = ?F. Are the two triangles necessarily congruent?
Solution:
It is given that
For necessarily triangle to be congruent, sides should also be equal.
 
Question:49
If ABC and DEF are two triangles such that AC = 2.5 cm, BC = 5 cm, ?C = 75°, DE = 2.5 cm, DF = 5cm and ?D = 75°. Are two
triangles congruent?
Solution:
It is given that
Since, two sides and angle between them are equal, therefore triangle ABC and DEF are congruent.
Question:50
In two triangles ABC and ADC, if AB = AD and BC = CD. Are they congruent?
Solution:
The given information and corresponding figure is given below
From the figure, we have
And,
Hence, triangles ABC and ADC are congruent to each other.
Question:51
In triangles ABC and CDE, if AC = CE, BC = CD, ?A = 60°, ?C = 30° and ?D = 90°.
Are two triangles congruent?
Solution:
For the triangles ABC and ECD, we have the following information and corresponding figure:
In triangles ABC and ECD, we have
The SSA criteria for two triangles to be congruent are being followed. So both the triangles are congruent.
Question:52
ABC is an isosceles triangle in which AB = AC. BE and CF are its two medians. Show that BE = CF.
Solution:
In the triangle ABC it is given that 
,  and  are medians.
We have to show that 
To show we will show that 
In triangle ?BFC and ?BEC
As , so 
             .........1
BC=BC
commonsides   ........
2
Since,
As F and E are mid points of sides AB and AC respectively, so
BF = CE         ..........
3
From equation
1,
2, and
3
Hence Proved.
Question:53
Find the measure of each angle of an equilateral triangle.
Solution:
In equilateral triangle we know that each angle is equal
So
Now 
bytriangleproperty
Hence .
Question:54
CDE is an equilateral triangle formed on a side CD of a square ABCD. Show that ?ADE ? ?BCE.
Solution:
We have to prove that 
Given  is a square
So 
Now in is equilateral triangle.
So 
In  and 
Sideoftriangle
Sideofequilateraltriangle
And,
So 
Hence from congruence Proved.
Question:55
Prove that the sum of three altitudes of a triangle is less than the sum of its sides.
Solution:
We have to prove that the sum of three altitude of the triangle is less than the sum of its sides.
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Question:47
In two congruent triangles ABC and DEF, if AB = DE and BC = EF. Name the pairs of equal angles.
Solution:
It is given that
Since, the triangles ABC and DEF are congruent, therefore,
Question:48
In two triangles ABC and DEF, it is given that ?A = ?D, ?B = ?E and ?C = ?F. Are the two triangles necessarily congruent?
Solution:
It is given that
For necessarily triangle to be congruent, sides should also be equal.
 
Question:49
If ABC and DEF are two triangles such that AC = 2.5 cm, BC = 5 cm, ?C = 75°, DE = 2.5 cm, DF = 5cm and ?D = 75°. Are two
triangles congruent?
Solution:
It is given that
Since, two sides and angle between them are equal, therefore triangle ABC and DEF are congruent.
Question:50
In two triangles ABC and ADC, if AB = AD and BC = CD. Are they congruent?
Solution:
The given information and corresponding figure is given below
From the figure, we have
And,
Hence, triangles ABC and ADC are congruent to each other.
Question:51
In triangles ABC and CDE, if AC = CE, BC = CD, ?A = 60°, ?C = 30° and ?D = 90°.
Are two triangles congruent?
Solution:
For the triangles ABC and ECD, we have the following information and corresponding figure:
In triangles ABC and ECD, we have
The SSA criteria for two triangles to be congruent are being followed. So both the triangles are congruent.
Question:52
ABC is an isosceles triangle in which AB = AC. BE and CF are its two medians. Show that BE = CF.
Solution:
In the triangle ABC it is given that 
,  and  are medians.
We have to show that 
To show we will show that 
In triangle ?BFC and ?BEC
As , so 
             .........1
BC=BC
commonsides   ........
2
Since,
As F and E are mid points of sides AB and AC respectively, so
BF = CE         ..........
3
From equation
1,
2, and
3
Hence Proved.
Question:53
Find the measure of each angle of an equilateral triangle.
Solution:
In equilateral triangle we know that each angle is equal
So
Now 
bytriangleproperty
Hence .
Question:54
CDE is an equilateral triangle formed on a side CD of a square ABCD. Show that ?ADE ? ?BCE.
Solution:
We have to prove that 
Given  is a square
So 
Now in is equilateral triangle.
So 
In  and 
Sideoftriangle
Sideofequilateraltriangle
And,
So 
Hence from congruence Proved.
Question:55
Prove that the sum of three altitudes of a triangle is less than the sum of its sides.
Solution:
We have to prove that the sum of three altitude of the triangle is less than the sum of its sides.
In  we have
,  and 
We have to prove 
As we know perpendicular line segment is shortest in length
Since 
So      ........1
And 
        ........2
Adding
1 and
2 we get
    ........3
Now , so
        .......4
And again , this implies that
     ........5
Adding
3 &
4 and
5 we have
Hence Proved.
Question:56
In the given figure, if AB = AC and ?B = ?C. Prove that BQ = CP.