ML Aggarwal Triangles Exe-10.1 Class 9 ICSE Maths Solutions Ch-10

ML Aggarwal Triangles Exe-10.1 Class 9 ICSE Maths Solutions Ch-10. Step by Step Answer of Exercise-10.1 of Triangles of ML Aggarwal for ICSE Class 9th Mathematics Questions. Visit official website CISCE for detail information about ICSE Board Class-9.

ML Aggarwal Triangles Exe-10.1 Class 9 ICSE Maths Solutions Ch-10

Board ICSE
Subject Maths
Class 9th
Chapter-10 Triangles
Topics Solution of Exe-10.1 Questions
Academic Session 2024-2025

Solution of Exe-10.1 Questions

ML Aggarwal Triangles for ICSE Class 9 Mathematics

Question 1. It is given that ∆ABC ≅ ∆RPQ. Is it true to say that BC = QR ? Why?

Solution: Given ∆ABC ≅ ∆RPQ

Therefore, their corresponding sides and angles are equal.

1. It is given that ∆ABC ∆R.Is it true to say that BC QR Why

Therefore BC = PQ

Hence it is not true to say that BC = QR

Question 2. “If two sides and an angle of one triangle are equal to two sides and an angle of another triangle, then the two triangles must be congruent.” Is the statement true? Why?

Answer:

No, it is not true statement as the angles should be included angle of there two given sides.

Question 3. In the given figure, AB=AC and AP=AQ. Prove that

(i) ∆APC ≅ ∆AQB
(ii) CP = BQ
(iii) ∠APC = ∠AQB.

3. In the given figure, AB=AC and AP=AQ. Prove that

Answer: (i) In △ APC and △AQB

AB=AC and AP=AQ [given]

From the given figure, ∠A = ∠A [common in both the triangles]

Therefore, using SAS axiom we have ∆APC ≅ ∆AQB

(ii) In △ APC and △AQB

AB=AC and AP=AQ [given]

From the given figure, ∠A = ∠A [common in both the triangles]

By using corresponding parts of congruent triangles concept we have

BQ = CP

(iii) In △ APC and △AQB

AB=AC and AP=AQ [given]

From the given figure, ∠A = ∠A [common in both the triangles]

By using corresponding parts of congruent triangles concept we have

∠APC = ∠AQB.

Question 4. In the given figure, AB = AC, P and Q are points on BA and CA respectively such that AP = AQ. Prove that

(i) ∆APC ≅ ∆AQB
(ii) CP = BQ
(iii) ∠ACP = ∠ABQ.

4. In the given figure, AB = AC, P and Q are points on BA and CA respectively such that AP = AQ. Prove that

Answer: (i) In the given figure AB = AC

P and Q are point on BA and CA produced respectively such that AP = AQ

Now we have to prove ∆APC ≅ ∆AQB

By using corresponding parts of congruent triangles concept we have

CP = BQ

∠ACP = ∠ABQ

(ii) CP = BQ
(iii) ∠ACP = ∠ABQ

In ∆ APC and ∆AQB

AC = AB (Given)

AP = AQ (Given)

∠PAC =∠QAB (Vertically opposite angle)

Question 5. In the given figure, AD = BC and BD = AC. Prove that :

∠ADB = ∠BCA and ∠DAB = ∠CBA.

5. In the given figure, AD = BC and BD = AC. Prove that

Answer:

Given: in the figure, AD = BC, BD = AC

5. In the given figure, AD = BC and BD = AC. Prove that

To prove :

(i) ∠ADB = ∠BCA

(ii) ∠DAB = ∠CBA

Proof : in ∆ADB and ∆ACB

AB = AB (Common)

AD = BC (given)

DB = AC (Given)

∆ADB = ∆ACD (SSS axiom)

∠ADB = ∠BCA

∠DAB = ∠CBA

Question 6. In the given figure, ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA. Prove that

(i) ∆ABD ≅ ∆BAC
(ii) BD = AC
(iii) ∠ABD = ∠BAC.

Answer:

Given : in the figure ABCD is a quadrilateral

In which AD = BC

∠DAB = ∠CBA

6. In the given figure, ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA. Prove that

To prove :

(i) ∆ABD = ∆BAC

(ii) ∠ABD = ∠BAC

Proof : in ∆ABD and ∆ABC

AB = AB (common)

∠DAB = ∠CBA (Given)

AD = BC

(i) ∆ABD ≅ ∆ABC (SAS axiom)

(ii) BD = AC

(ii) ∠ABD = ∠BAC

Question 7.In the given figure, AB = DC and AB || DC. Prove that AD = BC.

Answer:

Given: in the given figure.

AB = DC, AB ∥ DC

To prove : AD = BC

Proof : AB ∥ DC

∠ABD = ∠CDB (Alternate angles)

In ∆ABD and ∆CDB

AB = DC

7 triangle 10.1 ques 7

∠ABD = ∠CDB (Alternate angles)

BD = BD (common)

∆ABD ≅ CDB (SAS axiom)

AD = BC

Question 8. In the given figure. AC = AE, AB = AD and ∠BAD = ∠CAE. Show that BC = DE.

8. In the given figure. AC = AE, AB = AD and ∠BAD = ∠CAE. Show that BC = DE

Answer:

Given: in the figure, AC = AE, AB = AD

∠BAD = ∠CAE

To prove : BC = DE

8. In the given figure. AC = AE, AB = AD and ∠BAD = ∠CAE. Show that BC = DE

Proof : in ∆ABC and ∆ADE

AB = AD (given)

AC = AE (given)

∠BAD + ∠DAC + ∠CAE

∠BAC = ∠DAE

∆ABC = ∆ ADE (SAS axiom)

BC = DE

Question 9. In the adjoining figure, AB = CD, CE = BF and ∠ACE = ∠DBF. Prove that

(i) ∆ACE ≅ ∆DBF
(ii) AE = DF.

Answer:

Given : in the given figure

AB = CD

CE = BF

∠ACE = ∠DBF

9. In the adjoining figure, AB = CD, CE = BF and ∠ACE = ∠DBF. Prove that

To prove : (i) ∆ACE ≅ ∆DBF

(i) ∆ACE ≅ ∆DBF (SAS axiom)

AE = DE

(ii) AE = DF

Proof : AB = CD

Adding BC to both sides

AB + BC = BC + CD

AC = BD

Now in ∆ACE and ∆DBF

AC = BD (Proved)

CE = BF (Given)

∠ACE = ∠DBF (SAS axiom)

Question 10. In the adjoining figure, AB = AC and D is mid-point of BC. Use SSS rule of congruency to show that

(i) ∆ABD ≅ ∆ACD

(ii) AD is bisector of ∠A

(iii) AD is perpendicular to BC

In the given figure, AB = AC and D is mid-point of BC. Use SSS rule of congruency to show that (i) ∆ABD ≅ ∆ACD (ii) AD is bisector of ∠A (iii) AD is perpendicular to BC.

Answer: 

We will use the SSS congruency rule to prove this.

In ∆ABD and ∆ACD,

AB=AC (given)

AD=AD (common)

BD=CD (given AD is the bisector)

So, ∆ABD is congruent to ∆ACD on SSS rule.

By CPCT, angle ADB = angle ADC

But BDC is a straight line, so angle ADB + angle ADC = 180°

Which implies angle ADB=90°

Question 11. Two line segment AB and CD bisect each other at O. Prove that

(i) AC = BD

(ii) ∠CAB = ∠ABD

(iii) AD || CB

(iv) AD = CB

Two line segments AB and CD bisect each other at O. Prove that : (i) AC = BD (ii) ∠CAB = ∠ABD (iii) AD || CB (iv) AD = CB.

Answer: 

AB and CD bisect each other at O i.e, AO=BO and CO=DO

in ΔCOA and ΔDOB
Given CO=OD,∠COA=∠BOD [ vertically opposite angles]
AD=BD
∴ΔCOA≅ΔBOD

(i) ∴AC=BD[C.P.CT]
(ii) ∠CAB=∠ABD[C.P.CT]

again
in ΔCOB and ΔAOD
CO=OD [given]
BO=AO [given]
∠COB=∠AOD [vertically opposite angles]
∴ΔCOB≅ΔAOD
∴∠CBA=∠BAD [ C.P. C.T]

(iii) As we know the quadrilateral whose diagonals bisect each other is a parallelogram…..

So ABCD is a parallelogram

THEN, AB || BC

(iv) and so AD||CD [ ∵∠CBA=∠BAD which are alternate angles]

and AD=CB [C.P.C.T]

Question 12. In the adjoining figure, find the value of and y.

In the adjoining figure, find the value of x and y

Answer: 

as two sides are equal ,35= 2x+5

2x =30

x=15

y+5=46

y=41

—  : End of ML Aggarwal Triangles Exe-10.1 Class 9 ICSE Maths Solutions :–

Return to :-  ML Aggarawal Maths Solutions for ICSE  Class-9

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