# NCERT Solution - Triangles Notes - Class 9

## Class 9: NCERT Solution - Triangles Notes - Class 9

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Class IX  Chapter 7 – Triangles
Maths

Exercise 7.1 Question
1:
In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that
?ABC  ?ABD. What can you say about BC and BD?

In ?ABC and ?ABD,
CAB = DAB (AB bisects  A)
AB = AB (Common)
?ABC  ?ABD (By SAS congruence rule)
BC = BD (By CPCT)
Therefore, BC and BD are of equal lengths.
Question 2:
ABCD is a quadrilateral in which AD = BC and  DAB =  CBA (See the given figure).
Page 2

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1

Class IX  Chapter 7 – Triangles
Maths

Exercise 7.1 Question
1:
In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that
?ABC  ?ABD. What can you say about BC and BD?

In ?ABC and ?ABD,
CAB = DAB (AB bisects  A)
AB = AB (Common)
?ABC  ?ABD (By SAS congruence rule)
BC = BD (By CPCT)
Therefore, BC and BD are of equal lengths.
Question 2:
ABCD is a quadrilateral in which AD = BC and  DAB =  CBA (See the given figure).
Cbse-spot.blogspot.com
2

Prove that
(i) ?ABD  ?BAC
(ii) BD = AC

(iii) ABD =

BAC.

In ?ABD and ?BAC,

DAB = CBA (Given)
AB = BA (Common)
?ABD  ?BAC (By SAS congruence rule)
BD = AC (By CPCT) And,  ABD
=  BAC (By CPCT)
Question 3:
AD and BC are equal perpendiculars to a line segment AB (See the given figure).
Show that CD bisects AB.
Page 3

Cbse-spot.blogspot.com
1

Class IX  Chapter 7 – Triangles
Maths

Exercise 7.1 Question
1:
In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that
?ABC  ?ABD. What can you say about BC and BD?

In ?ABC and ?ABD,
CAB = DAB (AB bisects  A)
AB = AB (Common)
?ABC  ?ABD (By SAS congruence rule)
BC = BD (By CPCT)
Therefore, BC and BD are of equal lengths.
Question 2:
ABCD is a quadrilateral in which AD = BC and  DAB =  CBA (See the given figure).
Cbse-spot.blogspot.com
2

Prove that
(i) ?ABD  ?BAC
(ii) BD = AC

(iii) ABD =

BAC.

In ?ABD and ?BAC,

DAB = CBA (Given)
AB = BA (Common)
?ABD  ?BAC (By SAS congruence rule)
BD = AC (By CPCT) And,  ABD
=  BAC (By CPCT)
Question 3:
AD and BC are equal perpendiculars to a line segment AB (See the given figure).
Show that CD bisects AB.
Cbse-spot.blogspot.com
3

In ?BOC and ?AOD,
BOC =  AOD (Vertically opposite angles)
CBO =  DAO (Each 90º)
?BOC  ?AOD (AAS congruence rule)
BO = AO (By CPCT)
CD bisects AB.
Question 4:  l and m are two parallel lines intersected by another pair of parallel lines
p and q (see

the given figure). Show that ?ABC  ?CDA.

In ?ABC and ?CDA,
Page 4

Cbse-spot.blogspot.com
1

Class IX  Chapter 7 – Triangles
Maths

Exercise 7.1 Question
1:
In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that
?ABC  ?ABD. What can you say about BC and BD?

In ?ABC and ?ABD,
CAB = DAB (AB bisects  A)
AB = AB (Common)
?ABC  ?ABD (By SAS congruence rule)
BC = BD (By CPCT)
Therefore, BC and BD are of equal lengths.
Question 2:
ABCD is a quadrilateral in which AD = BC and  DAB =  CBA (See the given figure).
Cbse-spot.blogspot.com
2

Prove that
(i) ?ABD  ?BAC
(ii) BD = AC

(iii) ABD =

BAC.

In ?ABD and ?BAC,

DAB = CBA (Given)
AB = BA (Common)
?ABD  ?BAC (By SAS congruence rule)
BD = AC (By CPCT) And,  ABD
=  BAC (By CPCT)
Question 3:
AD and BC are equal perpendiculars to a line segment AB (See the given figure).
Show that CD bisects AB.
Cbse-spot.blogspot.com
3

In ?BOC and ?AOD,
BOC =  AOD (Vertically opposite angles)
CBO =  DAO (Each 90º)
?BOC  ?AOD (AAS congruence rule)
BO = AO (By CPCT)
CD bisects AB.
Question 4:  l and m are two parallel lines intersected by another pair of parallel lines
p and q (see

the given figure). Show that ?ABC  ?CDA.

In ?ABC and ?CDA,
Cbse-spot.blogspot.com
4

BAC = DCA (Alternate interior angles, as p || q)
AC = CA (Common)
BCA = DAC (Alternate interior angles, as l || m)
?ABC  ?CDA (By ASA congruence rule)
Question 5:
Line l A is the bisector of an angle and B is any point on l. BP and BQ are
perpendiculars from B to the arms of A (see the given figure). Show that: i) ?APB
?AQB ( ii) BP = BQ or B is equidistant from the arms of (  A.

In ?APB and ?AQB,
APB = AQB (Each 90º)
PAB = QAB (l is the angle bisector of A)
AB = AB (Common)
?APB  ?AQB (By AAS congruence rule)
BP = BQ (By CPCT)
rms of A. Or,
it can be said that B is equidistant from the a
Question 6:
In the given figure, AC = AE, AB = AD and BAD = EAC. Show that BC = DE.
Page 5

Cbse-spot.blogspot.com
1

Class IX  Chapter 7 – Triangles
Maths

Exercise 7.1 Question
1:
In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that
?ABC  ?ABD. What can you say about BC and BD?

In ?ABC and ?ABD,
CAB = DAB (AB bisects  A)
AB = AB (Common)
?ABC  ?ABD (By SAS congruence rule)
BC = BD (By CPCT)
Therefore, BC and BD are of equal lengths.
Question 2:
ABCD is a quadrilateral in which AD = BC and  DAB =  CBA (See the given figure).
Cbse-spot.blogspot.com
2

Prove that
(i) ?ABD  ?BAC
(ii) BD = AC

(iii) ABD =

BAC.

In ?ABD and ?BAC,

DAB = CBA (Given)
AB = BA (Common)
?ABD  ?BAC (By SAS congruence rule)
BD = AC (By CPCT) And,  ABD
=  BAC (By CPCT)
Question 3:
AD and BC are equal perpendiculars to a line segment AB (See the given figure).
Show that CD bisects AB.
Cbse-spot.blogspot.com
3

In ?BOC and ?AOD,
BOC =  AOD (Vertically opposite angles)
CBO =  DAO (Each 90º)
?BOC  ?AOD (AAS congruence rule)
BO = AO (By CPCT)
CD bisects AB.
Question 4:  l and m are two parallel lines intersected by another pair of parallel lines
p and q (see

the given figure). Show that ?ABC  ?CDA.

In ?ABC and ?CDA,
Cbse-spot.blogspot.com
4

BAC = DCA (Alternate interior angles, as p || q)
AC = CA (Common)
BCA = DAC (Alternate interior angles, as l || m)
?ABC  ?CDA (By ASA congruence rule)
Question 5:
Line l A is the bisector of an angle and B is any point on l. BP and BQ are
perpendiculars from B to the arms of A (see the given figure). Show that: i) ?APB
?AQB ( ii) BP = BQ or B is equidistant from the arms of (  A.

In ?APB and ?AQB,
APB = AQB (Each 90º)
PAB = QAB (l is the angle bisector of A)
AB = AB (Common)
?APB  ?AQB (By AAS congruence rule)
BP = BQ (By CPCT)
rms of A. Or,
it can be said that B is equidistant from the a
Question 6:
In the given figure, AC = AE, AB = AD and BAD = EAC. Show that BC = DE.
Cbse-spot.blogspot.com
5

It is given that BAD = EAC
BAD + DAC = EAC + DAC
BAC = DAE
In ?BAC and ?DAE,  AB
DAE (Proved above)
AC = AE (Given)
?BAC  ?DAE (By SAS congruence rule)
BC = DE (By CPCT)
Question 7:
AB is a line segment and P is its mid-point. D and E are points on the same side of AB
such that BAD = ABE and EPA = DPB (See the given figure). Show that  i)
?DAP  ?EBP (

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