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NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles

NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles

Edited By Ravindra Pindel | Updated on Feb 07, 2024 06:00 PM IST

NCERT Solutions for Class 7 Maths Chapter 7 - Congruence of triangles is one of the important topics of the geometry. In this chapter, there are two exercises and topic wise practice questions. The solutions of NCERT Class 7 chapter 7 congruence of triangles give detailed explanations to all these questions. Students can do their homework easily if they have a tool like NCERT Solutions for Class 7 in hand. In geometry, two objects or two figures are congruent if they have the same dimension and same shape, or in other words, we can say that two objects or figures are congruent if both are exact copies of one another. The relation of two objects or two figures being congruent is called congruence.

This Story also Contains
  1. NCERT Solutions for Maths Chapter 7 Congruence of Triangles Class 7- Important Formulae
  2. NCERT Solutions for Chapter 7 Maths Class 7 Congruence of Triangles - Important Points
  3. NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles
  4. NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles (Intext Questions and Exercise)
  5. NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles Topic 7.5
  6. NCERT Solutions for Chapter 7 Congruence of Triangles Class 7 Topic 7.6
  7. NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles Topic 7.7
  8. NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles Exercise 7.1
  9. NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles Exercise 7.2
  10. Congruence of Triangles Chapter 7 Maths Class 7-Topics

NCERT Chapter 7 Maths c 7 congruence of triangles deal with plane figures or 2D only, although congruence is a general concept applicable to 3D figures also. Congruence of plane figures, congruence among line segments, congruence of angles, congruence of triangles, some criteria for congruence of triangles like SSS congruence of two triangles, SAS congruence of two triangles, ASA congruence of two triangles, RHS congruence of two right-angled triangles are the concepts which are covered in this chapter. Questions on all these concepts are discussed in the NCERT solutions for Class 7 Maths chapter 7 congruence of triangles. The NCERT Solutions are prepared in such a manner that students are able to understand the concepts easily and prepare themselves very well for CBSE final exams to score higher marks. Here you will get solutions to two exercises in NCERT .

NCERT Solutions for Maths Chapter 7 Congruence of Triangles Class 7- Important Formulae

Criteria for Congruence of Triangles:

  • SSS Congruence: For triangles ABC and DEF, If AB = DE, BC = EF, and AC = DF, Then △ABC ≅ △DEF.
  • SAS Congruence: For triangles ABC and DEF, If AB = DE, ∠BAC = ∠EDF, and BC = EF, Then △ABC ≅ △DEF.
  • ASA Congruence: For triangles ABC and DEF, If ∠BAC = ∠EDF, ∠ABC = ∠DEF, and AC = DF, Then △ABC ≅ △DEF.
  • AAS Congruence: For triangles ABC and DEF, If ∠BAC = ∠EDF, ∠ACB = ∠EFD, and BC = EF, Then △ABC ≅ △DEF.
  • RHS Congruence: For right-angled triangles ABC and DEF, If ∠CAB = ∠FDE, AC = DF, and BC = EF, Then △ABC ≅ △DEF.

NCERT Solutions for Chapter 7 Maths Class 7 Congruence of Triangles - Important Points

Congruence of Lines: Two lines are congruent if they have the same length.

Congruence of Angles: Two angles are congruent if they have the same measure.

Congruence of Triangles: Two triangles are congruent if they are exact copies of each other and cover each other completely when superposed.

Criteria for Congruence of Triangles:

  • SSS Congruence Criterion: If the three sides of one triangle are equal to the three corresponding sides of another triangle.
  • SAS Congruence Criterion: If two sides and the angle between them of one triangle are equal to the two corresponding sides and angle between them of another triangle.
  • ASA Congruence Criterion: If two angles and the included side of one triangle are equal to the two corresponding angles and included side of another triangle.
  • AAS Congruence Criterion: If two angles and the non-included side of one triangle are equal to the two corresponding angles and the non-included side of another triangle.
  • RHS Congruence Criterion: If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and one side of another right-angled triangle (Applicable only to right-angled triangles).

Free download NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles PDF for CBSE Exam.

NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles

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NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles (Intext Questions and Exercise)

NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles Topic 7.5

1. When two triangles, say ABC and PQR are given, there are, in all, six possible matchings or correspondences. Two of them are:

(i)ABCPQR and (ii)ABCQRP

Find the other four correspondences by using two cutouts of triangles. Will all these correspondences lead to congruence? Think about it.

Answer:

the other four correspondences by using two cutouts of triangles are :

(i)ABCRPQ

(ii)BCAPQR

(iii)CABPQR

(iv)CBAPQR

NCERT Solutions for Chapter 7 Congruence of Triangles Class 7 Topic 7.6

1. In Fig 7.14, lengths of the sides of the triangles are indicated. By applying the SSS congruence rule, state which pairs of triangles are congruent. In case of congruent triangles, write the result in symbolic form:

Answer:

i) Since

AB = PQ

BC = QR

CA = PR

So, by SSS congruency rule both triangles are congruent to each other.

ΔABCΔPQR


ii) Since,

ED = MN

DF = NL

FE = LM

So, by SSS congruency rule both triangles are congruent to each other.

ΔEDFΔMNL .

iii) Since

AC = PR

BC = QR But

ABQR

So the given triangles are not congruent.

iv) Since,

AD = AD

AB = AC

BD = CD

So, By SSS Congruency rule, they both are congruent to each other.

ΔADBΔADC .

2. In Fig 7.15, AB=AC and D is the mid-point of BC. .

(i) State the three pairs of equal parts in ΔADB and ΔADC.
(ii) Is ΔADBΔADC? Give reasons.
(iii) Is B=C? Why?

Answer:

Here in ΔADB and ΔADC.

i) Three pair of equal parts are:

AD = AD ( common side )

BD = CD ( as d is the mid point of BC)

AB = AC (given in the question)

ii) Now,

by SSS Congruency rule,

ΔADBΔADC

iii) As both triangles are congruent to each other we can compare them and say

B=C .

3. In Fig 7.16, AC=BD and AD=BC. . Which of the following statements is meaningfully written?

(i)ΔABCΔABD (ii)ΔABCΔBAD

Answer:

Given,

AC=BD and

AD=BC. .

AB = AB ( common side )

So By SSS congruency rule,

ΔABCΔBAD .

So this statement is meaningfully written as all given criterions are satisfied in this.

1. ABC is an isosceles triangle with AB=AC (Fig 7.17). Take a trace-copy of ΔABC and also name it as ΔABC
(i) State the three pairs of equal parts in ΔABCandΔACB .
(ii) Is ΔABCΔACB ? Why or why not?
(iii) Is B=C ? Why or why not?

Answer:

Here, in ΔABCandΔACB .

i)the three pairs of equal parts in ΔABCandΔACB are

AB = AC

BC = CB

AC = AB

ii)

Hence By SSS Congruency rule, they both are congruent.

ΔABCΔACB

iii) Yes, B=C because ΔABCandΔACB are congruent and by equating the corresponding parts of the triangles we get,

B=C .

1. Which angle is included between the sides DE and EF of DEF ?

Answer:

Since both the sides DE and EF intersects at E,

E is included between the sides DE and EF of DEF .

2. By applying SAS congruence rule, you want to establish that PQRFED . It is given that PQ=FE and RP=DF . What additional information is needed to establish the congruence?

Answer:

To prove congruency by SAS rule, we need to equate two corresponding sides and one corresponding angle,

so in proving PQRFED we need,

PQ=FE

RP=DF

And

P=F .

Hence the extra information we need is P=F .

3. In Fig 7.24, measures of some parts of the triangles are indicated. By applying SAS congruence rule, state the pairs of congruent triangles, if any, in each case. In case of congruent triangles, write them in symbolic form.

Answer:

i) in ΔABC and ΔDEF

AB = DE

AC = DF

AD

Hence, they are not congruent.


ii) In ΔACB and ΔRPQ

AC = RP = 2.5 cm

CB = PQ = 3 cm

C=P=350

Hence by SAS congruency rule, they are congruent.

ΔACBΔRPQ .


iii) In ΔDFE and ΔPQR

DF= PQ = 3.5 cm

FE= QR = 3 cm

F=Q

Hence, by SAS congruency rule, they are congruent.

ΔDFEΔPQR


iv) In ΔQPR and ΔSRP

QP = SR = 3.5 cm

PR = RP (Common side)

QPR=SRP

Hence, by SAS congruency rule, they are congruent.

ΔQPRΔSRP .

1. What is the side included between the angles M and N of MNP ?

Answer:

The side MN is the side which is included between the angles M and N of MNP .

2. You want to establish DEFMNP , using the ASA congruence rule. You are given that D=M and F=P . What information is needed to establish the congruence? (Draw a rough figure and then try!)

Answer:

As we know, in ASA congruency two angles and one side is equated to their corresponding parts. So

To Prove DEFMNP

D=M

F=P

And The side joining these angles is

DF=MP .

So the information that is needed in order to prove congruency is DF=MP .

4. In Fig 7.25, AB and CD bisect each other at O .

(i) State the three pairs of equal parts in two triangles AOC and BOD .

(ii) Which of the following statements are true?

(a)AOCDOB
(b)AOCBOD

Answer:

i) The three pairs of equal parts in two triangles AOC and BOD are:

CO = DO (given)

OA = OB (given )

COA=DOB ( As opposite angles are equal when two lines intersect.)

ii) So by SAS congruency rule,

ΔCOAΔDOB

that is

AOCBOD

Hence, option B is correct.

3. In Fig 7.27, measures of some parts are indicated. By applying ASA congruence rule, state which pairs triangles are congruent. In case of congruence, write the result in symoblic form.

Answer:

i) in ΔABC and ΔFED

AB = FE = 3.5 cm

A=F=400

B=E=600

So by ASA congruency rule, both triangles are congruent.i.e.

ΔABCΔFED

ii) in ΔPQR and ΔFDE

Q=D=900

R=E=500

But,

EFRP

So, given triangles are not congruent.

iii) in ΔRPQ and ΔLMN

RQ = LN = 6 cm

R=L=600

Q=N=300

So by ASA congruency rule, both triangles are congruent.i.e.

ΔRPQΔLMN .

iv) in ΔADB and ΔBCA

AB = BA (common side)

CAB=DBA=300

D=C=1800300300450=750

So by ASA congruency rule, both triangles are congruent.i.e.

ΔADBΔBCA

4. Given below are measurements of some parts of two triangles. Examine whether the two triangles are congruent or not, by ASA congruence rule. In the case of congruence, write it in symbolic form.

DEF PQR

(i)D=60o,F=80o,DF=5cm Q=60o,R=80o,QR=5cm
(ii)D=60o,F=80o,DF=6cm Q=60o,R=80o,QP=6cm
(iii)D=80o,F=30o,EF=5cm P=80o,PQ=5cm,R=30o


Answer:

i)

Given in DEF and PQR .

D=Q=60oF=R=80oDF=QR=5cm

So, by ASA congruency criterion, they are congruent to each other.i.e.

DEFΔQPR .

ii)

Given in DEF and PQR .

D=Q=60oF=R=80oDF=QP=6cm

For congruency by ASA criterion, we need to be sure of equity of the side which is joining the two angles which are equal to their corresponding parts. Here the side QR is not given which is why we cannot conclude the congruency of both the triangles.

iii)

Given in DEF and PQR .

D=Q=60oF=R=80oDF=QP=6cm

For congruency by ASA criterion, we need to be sure of equity of the side which is joining the two angles which are equal to their corresponding parts. Here the side QR is not given which is why we cannot conclude the congruency of both the triangles.

5. In Fig 7.28, ray AZ bisects DAB as well as DCB .

(i) State the three pairs of equal parts in triangles BAC and DAC .
(ii) Is ΔBACΔDAC? Give reasons.
(iii) Is AB=AD? Justify your answer.
(iv) Is CD=CB? Give reasons.


Answer:

i)

Given in triangles BAC and DAC

DAC=BAC

DCA=BCA

AC=AC ( common side)

ii)

So, By ASA congruency criterion,triangles BAC and DAC are congruent.

ΔBACΔDAC

iii)

Since ΔBACΔDAC , all corresponding parts will be equal. So

AB=AD .

iv)

Since ΔBACΔDAC , all corresponding parts will be equal. So

CD=CB

NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles Topic 7.7

1. In Fig 7.32, measures of some parts of triangles are given.By applying RHS congruence rule, state which pairs of triangles are congruent. In case of congruent triangles, write the result in symbolic form.

Answer:

i) In ΔPQR and ΔDEF

Q=E

PR=DF=6cm

PQDE

Hence they are not congruent.

ii)

In ΔACB and ΔBDA

D=C

DB=AC=2cm

AB=BA ( same side )

So, by RHS congruency rule,

ΔACBΔBDA

iii)

In ΔABC and ΔADC

B=D

AB=AD=3.6cm

AC=AC ( same side )

So, by RHS congruency rule,

ΔABCΔADC

iv)

In ΔPSQ and ΔPSR

PSQ=PSR

PQ=PR=3cm

PS=PS ( same side )

So, by RHS congruency rule,

ΔPSQΔPSR


2. It is to be established by RHS congruence rule that ABCRPQ . What additional information is needed, if it is given that

B=P=900 and AB=RP?

Answer:

To prove congruency by RHS (Right angle, Hypotenuse, Side ) rule, we need hypotenuse and side equal to the corresponding hypotenuse and side of different angle.

So Given

B=P=900 ( Right angle )

AB=RP. ( Side )

So the third information we need is the equality of Hypotenuse of both triangles. i.e.

AC=RQ

Hence, if this information is given then we can say,

ABCRPQ .

3. In Fig 7.33, BD and CE are altitudes of ABC such that BD=CE .

(i) State the three pairs of equal parts in CBD and BCE .
(ii) Is CBDBCE ? Why or why not?
(iii) Is DCB=EBC ? Why or why not?

Answer:

i) Given, in CBD and BCE .

BD=CE

CEB=BDC=90o

BC=CB


ii) So, By RHS Rule of congruency, we conclude:

CBDBCE


iii) Since both the triangle are congruent, all parts of one triangle are equal to their corresponding part from another triangle.

So.

CBDBCE .

4. ABC is an isosceles triangle with AB=AC and AD is one of its altitudes (Fig 7.34).

(i) State the three pairs of equal parts in ADB and ADC .
(ii) Is ADBADC ? Why or why not?
(iii) Is B=C ? Why or why not?
(iv) Is BD=CD ? Why or why not?

Answer:

i) Given in ADB and ADC .

AB=AC

ADB=ADC=900

AD=AD ( Common side)


ii) So, by RHS Rule of congruency, we conclude

ADBADC


iii) Since both triangles are congruent all the corresponding parts will be equal.

So,

B=C


iv) Since both triangles are congruent all the corresponding parts will be equal.

So,

BD=CD .

NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles Exercise 7.1

1. Complete the following statements:

(a) Two line segments are congruent if ___________.
(b) Among two congruent angles, one has a measure of 70o ; the measure of the other angle is ___________.
(c) When we write A=B , we actually mean ___________.

Answer:

a) Two line segments are congruent if they are identical in shape and size and which is the case when the length of two line segments are equal.

b) 700 As the congruent things are a photocopy of each other.

c) When we write A=B , We mean that both the angles(A & B) are equal.

2. Give any two real-life examples for congruent shapes.

Answer:

Any two things that have identical shape and size are congruent like all the same kind of pens are congruent to one another. every same kind of bench in class are congruent to one another.all the similar football is congruent to one another.

3. If ΔABCΔFED under the correspondence ABCFED, write all the corresponding congruent parts of the triangles.

Answer:

Corresponding parts of the two congruent triangles ABCFED, are :

Sides:

ABandFE,

BCandED,

ACandFD.

Angles:

ABC=FED

BCA=EDF

CAB=DFE

4. If ΔDEFΔBCA, write the part(s) of ΔBCA that correspond to

(i)E (ii)EF (iii)F (iv)DF

Answer:

Given,

ΔDEFΔBCA,

The part of ΔBCA that correspond to

(i)E=C

(ii)EF=CA

(iii)F=A

(iv)DF=BA

NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles Exercise 7.2

1.(a) Which congruence criterion do you use in the following?

(a)GivenAC=DF

AB=DE

BC=EF

So,ΔABCΔDEF

Answer:

Since we are comparing all the sides of two triangles, The SSS (side, side, side) Congruent criterion is used.

1.(b) Which congruence criterion do you use in the following?

(b)Given:ZX=RP

RQ=ZY

PRQ=XZY

So,ΔPQRΔXYZ

Answer:

Since we are comparing two sides and one angle of the two triangles, the SAS (sie, angle, side) congruent criterion is used to prove them congruent.

1.(c) Which congruence criterion do you use in the following?

(c)Given:MLN=FGH

NML=GFH

ML=FG

So,ΔLMNΔGFH

Answer:

Since we are comparing two angles and one side, ASA(Angle, Side, Angle) congruency criterion is used to prove the congruency.


1.(d) Which congruence criterion do you use in the following?

(d)Given:EB=DB

AE=BC

A=C=90o

So,ΔABEΔCDB

Answer:

Since we are comparing two sides and one angle of the two triangles, the SSA (Side, Side, Angle) congruent criterion is used to prove the congruency.

2.(a) You want to show that ΔARTΔPEN,

(a) If you have to use SSS criterion, then you need to show

(i)AR= (ii)RT= (iii)AT=

Answer:

As we know that in the criterion of proving congruent, all three corresponding sides are equal to another. So to prove the congruency, we kneed to know the following things:

(i)AR=PE

(ii)RT=EN

(iii)AT=PN

2.(b) You want to show that ΔARTΔPEN,

(b) If it is given that T=N and you are to use SAS criterion, you need to have

(i)RT=and(ii)PN=

Answer:

As we know in SAS criterion the two sides and one angle are identical to their corresponding parts of another triangle. So to prove congruency we need to prove that,

(i)RT=ENand(ii)PN=AT

2.(c) You want to show that ΔARTΔPEN,

(c) If it is given that AT=PN and you are to use ASA criterion, you need to have

(i)?(ii)?

Answer:

Given, ΔARTΔPEN,

also, AT=PN

Now, As we know in the ASA criterion of proving congruency, the one and side two angles are equal to their corresponding parts. So,

(i)RAT=EPN

(ii)RTA=ENP


3. You have to show that AMPAMQ . In the following proof, supply the missing reasons.








Answer

Steps
Reasons
(i)PM=QM
Given in the question
(ii)PMA=QMA
Given in the question.
(iii)AM=AM
the side which is common in both triangle
(iv)AMPAMQ
By SAS Congruence Rule





4. In ABC , A=30o , B=40o and C=110o .

In PQR , P=30o , Q=40o and R=110o . A student says that ABCPQR by AAA congruence criterion. Is he justified? Why or why not?

Answer:

No, because it is not necessary that two triangles will be congruent if their all three corresponding angles are equal. in this case, the triangles might be zoomed copy of one another.

5. In the figure, the two triangles are congruent. The corresponding parts are marked. We can write ΔRAT ?


Answer:

Comparing from the figure.

RW,AO,andTN

By SAS Congruency criterion, we can say that

ΔRATΔWON ]

6. Complete the congruence statement:

Answer:

Comparing from the figure, we get,

BB,AA,andCT

So By SSS Congruency Rule,

ΔBCAΔBTA

Also,

Comparing from the figure, we get,

PR,TQ,andQS

So By SSS Congruency Rule,

ΔQRSΔTPQ .

7. In a squared sheet, draw two triangles of equal areas such that
(i) the triangles are congruent.
(ii) the triangles are not congruent.

What can you say about their perimeters?

Answer:

When two triangles are congruent, the corresponding parts are exactly identical so they have the same area and perimeter.

While the triangles are not congruent but have the same area, then the perimeter of both triangles are not equal.

8. Draw a rough sketch of two triangles such that they have five pairs of congruent parts but still the triangles are not congruent.

Answer:

Five pairs of congruent parts can be three pairs of sides and two pairs of angles. In that case, the SAS or ASA criterion would prove them to be congruent. Hence, such a figure is not possible.


9. If ΔABC and ΔPQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use?

Answer:

Given ΔABCΔPQR

One additional pair which is not given in the figure is BC=QR

We used the ASA Criterion as the two corresponding angles are given and we figured out the side by congruency.

10. Explain, why ΔABCΔFED

Answer:

Comparing both triangles, we have,

A=F

B=E=900

BC=ED

So By RHS congruency criterion,

ΔABCΔFED .

Congruence of Triangles Chapter 7 Maths Class 7-Topics

  • Congruence Of Plane Figures
  • Congruence Of Among Line Segments
  • Congruence Of Angles
  • Congruence Of Triangles
  • Criteria For Congruence Of Triangles
  • Congruence Among Right-Angled Triangles

NCERT Solutions for Class 7 Maths Chapter Wise

Chapter No.

Chapter Name

Chapter 1

Integers

Chapter 2

Fractions and Decimals

Chapter 3

Data Handling

Chapter 4

Simple Equations

Chapter 5

Lines and Angles

Chapter 6

The Triangle and its Properties

Chapter 7

Congruence of Triangles

Chapter 8

Chapter 9

Rational Numbers

Chapter 10

Practical Geometry

Chapter 11

Perimeter and Area

Chapter 12

Algebraic Expressions

Chapter 13

Exponents and Powers

Chapter 14

Symmetry

Chapter 15

NCERT Solutions for Class 7 Subject Wise

Important Points of NCERT Class 7 Maths Chapter 7 Congruence of Triangles

Questions discussed in the NCERT solutions for Class 7 Maths chapter 7 Congruence of Triangles are based on the following congruence criteria.

  • SSS Congruence of two triangles- Under a given correspondence, two triangles are congruent if the three sides of the one triangle are equal in measure to the three corresponding sides of the other triangle.
  • SAS Congruence of two triangles- Under a given correspondence, two triangles are congruent if two sides and the angle included between them in one of the triangles are equal in measures to the corresponding sides and the angle included between them of the other triangle.
  • ASA Congruence of two triangles- Under a given correspondence, two triangles are congruent if two angles and the side included between them in one of the triangles are equal in measures to the corresponding angles and the side included between them of the other triangle.
  • RHS Congruence of two right-angled triangles- Under a given correspondence, two right-angled triangles are congruent if the hypotenuse and a leg of one of the triangles are equal to the hypotenuse and the corresponding leg of the other triangle.

The same questions described in the NCERT solutions for Class 7 Maths chapter 7 Congruence of Triangles can be expected for exams.

Also Check NCERT Books and NCERT Syllabus here:

Frequently Asked Questions (FAQs)

1. Whether the class 7 maths chapter Congruence of Triangles is useful in higher studies

Yes, the chapter is important for higher studies in the field of maths and science and also in class 8, 9 and 10 maths also. Therefore practice congruence of triangles class 7 pdf which can be downloaded form the link given above in this article. 

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Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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