Congruent Triangles - Exercise 12.5 | Extra Documents & Tests for Class 9 PDF Download

 Page 1


 
       
 
 
Question:29
ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is-
isosceles.
Solution:
We have to prove that is isosceles.
Let  and  be perpendicular from D on AB and AC respectively.
In order to prove that 
We will prove that 
Now in  and  we have
(Since D is mid point of BC)
Given
So by congruence criterion we have 
And 
Hence is isosceles.
Question:30
ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ?ABC
is isosceles.
Solution:
It is given that 
Page 2


 
       
 
 
Question:29
ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is-
isosceles.
Solution:
We have to prove that is isosceles.
Let  and  be perpendicular from D on AB and AC respectively.
In order to prove that 
We will prove that 
Now in  and  we have
(Since D is mid point of BC)
Given
So by congruence criterion we have 
And 
Hence is isosceles.
Question:30
ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ?ABC
is isosceles.
Solution:
It is given that 
, and 
And .
 
We have to prove  is isosceles.
To prove  is isosceles we will prove 
For this we have to prove 
Now comparing  and  we have
Given
Commonside
So, by right hand side congruence criterion we have 
So 
sincesidesoppositetoequalangleareequal
Hence  is isosceles.
Question:31
If perpendiculars from any point within an angle on its arms are congruent, prove that it lies on the bisector of that angle.
Solution:
Let P be a point within such that
 
 
We have to prove that P lies on the bisector of 
In  and  we have
Wehave
Common
Page 3


 
       
 
 
Question:29
ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is-
isosceles.
Solution:
We have to prove that is isosceles.
Let  and  be perpendicular from D on AB and AC respectively.
In order to prove that 
We will prove that 
Now in  and  we have
(Since D is mid point of BC)
Given
So by congruence criterion we have 
And 
Hence is isosceles.
Question:30
ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ?ABC
is isosceles.
Solution:
It is given that 
, and 
And .
 
We have to prove  is isosceles.
To prove  is isosceles we will prove 
For this we have to prove 
Now comparing  and  we have
Given
Commonside
So, by right hand side congruence criterion we have 
So 
sincesidesoppositetoequalangleareequal
Hence  is isosceles.
Question:31
If perpendiculars from any point within an angle on its arms are congruent, prove that it lies on the bisector of that angle.
Solution:
Let P be a point within such that
 
 
We have to prove that P lies on the bisector of 
In  and  we have
Wehave
Common
So by right hand side congruence criterion, we have 
So, 
Hence P lies on the bisector of  proved.
Question:32
In the given figure, AD ? CD and CB ? CD. If AQ = BP and DP = CQ, prove that
?DAQ = ?CBP.
 
Solution:
It is given that 
, and 
If  and 
We have to prove that
In triangles and we have
(Since  given)
So
And 
given
So by right hand side congruence criterion we have
So
Hence Proved.
Question:33
Which of the following statements are true T
and which are false F
:
i
Sides opposite to equal angles of a triangle may be unequal.
Page 4


 
       
 
 
Question:29
ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is-
isosceles.
Solution:
We have to prove that is isosceles.
Let  and  be perpendicular from D on AB and AC respectively.
In order to prove that 
We will prove that 
Now in  and  we have
(Since D is mid point of BC)
Given
So by congruence criterion we have 
And 
Hence is isosceles.
Question:30
ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ?ABC
is isosceles.
Solution:
It is given that 
, and 
And .
 
We have to prove  is isosceles.
To prove  is isosceles we will prove 
For this we have to prove 
Now comparing  and  we have
Given
Commonside
So, by right hand side congruence criterion we have 
So 
sincesidesoppositetoequalangleareequal
Hence  is isosceles.
Question:31
If perpendiculars from any point within an angle on its arms are congruent, prove that it lies on the bisector of that angle.
Solution:
Let P be a point within such that
 
 
We have to prove that P lies on the bisector of 
In  and  we have
Wehave
Common
So by right hand side congruence criterion, we have 
So, 
Hence P lies on the bisector of  proved.
Question:32
In the given figure, AD ? CD and CB ? CD. If AQ = BP and DP = CQ, prove that
?DAQ = ?CBP.
 
Solution:
It is given that 
, and 
If  and 
We have to prove that
In triangles and we have
(Since  given)
So
And 
given
So by right hand side congruence criterion we have
So
Hence Proved.
Question:33
Which of the following statements are true T
and which are false F
:
i
Sides opposite to equal angles of a triangle may be unequal.
ii
Angles opposite to equal sides of a triangle are equal.
iii
The measure of each angle of an equilateral triangle is 60°.
iv
If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.
v
The bisectors of two equal angles of a triangle are equal.
vi
If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.
vii
The two altitudes corresponding to two equal sides of a triangle need not be equal.
viii
If any two sides of a right triangle are respectively equal to two sides of other right triangle, then the two triangles are congruent.
ix
Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal equal to the hypotenuse and a
side of the other triangle.
 
 
 
Solution:
1 
2 
3 
4 
5 
6 
7 
8 
9 
Question:34
Fill in the blanks in the following so that each of the following statements is true.
i
Sides opposite to equal angles of a triangle are .......
ii
Angle opposite to equal sides of a triangle are .......
iii
In an equilateral triangle all angles are ........
iv
In a ABC if ?A = ?C, then AB =  ............
v
If altitudes CE and BF of a triangle ABC are equal, then AB = ........
Page 5


 
       
 
 
Question:29
ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is-
isosceles.
Solution:
We have to prove that is isosceles.
Let  and  be perpendicular from D on AB and AC respectively.
In order to prove that 
We will prove that 
Now in  and  we have
(Since D is mid point of BC)
Given
So by congruence criterion we have 
And 
Hence is isosceles.
Question:30
ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ?ABC
is isosceles.
Solution:
It is given that 
, and 
And .
 
We have to prove  is isosceles.
To prove  is isosceles we will prove 
For this we have to prove 
Now comparing  and  we have
Given
Commonside
So, by right hand side congruence criterion we have 
So 
sincesidesoppositetoequalangleareequal
Hence  is isosceles.
Question:31
If perpendiculars from any point within an angle on its arms are congruent, prove that it lies on the bisector of that angle.
Solution:
Let P be a point within such that
 
 
We have to prove that P lies on the bisector of 
In  and  we have
Wehave
Common
So by right hand side congruence criterion, we have 
So, 
Hence P lies on the bisector of  proved.
Question:32
In the given figure, AD ? CD and CB ? CD. If AQ = BP and DP = CQ, prove that
?DAQ = ?CBP.
 
Solution:
It is given that 
, and 
If  and 
We have to prove that
In triangles and we have
(Since  given)
So
And 
given
So by right hand side congruence criterion we have
So
Hence Proved.
Question:33
Which of the following statements are true T
and which are false F
:
i
Sides opposite to equal angles of a triangle may be unequal.
ii
Angles opposite to equal sides of a triangle are equal.
iii
The measure of each angle of an equilateral triangle is 60°.
iv
If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.
v
The bisectors of two equal angles of a triangle are equal.
vi
If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.
vii
The two altitudes corresponding to two equal sides of a triangle need not be equal.
viii
If any two sides of a right triangle are respectively equal to two sides of other right triangle, then the two triangles are congruent.
ix
Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal equal to the hypotenuse and a
side of the other triangle.
 
 
 
Solution:
1 
2 
3 
4 
5 
6 
7 
8 
9 
Question:34
Fill in the blanks in the following so that each of the following statements is true.
i
Sides opposite to equal angles of a triangle are .......
ii
Angle opposite to equal sides of a triangle are .......
iii
In an equilateral triangle all angles are ........
iv
In a ABC if ?A = ?C, then AB =  ............
v
If altitudes CE and BF of a triangle ABC are equal, then AB = ........
vi
In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is ...... CE.
vii
  In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then ?ABC ? ? .........
Solution:
1 
2 
3 
4 
5 
6 
7 
Question:35
ABCD is a square, X and Y are points on sides AD and BC respectively such that AY= BX. Prove that BY = AX and ?BAY =
?ABX.
Solution:
It is given ABCD is a square and 
We have to prove that and
In right angled triangles and ? we have
And , and 
So by right hand side congruence criterion we have 
So 
sincetriangleiscongruent
Hence Proved.