ncert-solutions-class-7-maths-ch-7-congruence-of-triangles

NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles

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:

$(i) ABC\leftrightarrow PQR$ and $(ii) ABC\leftrightarrow QRP$

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 :

$(i) ABC\leftrightarrow RPQ$

$(ii) BCA\leftrightarrow PQR$

$(iii) CAB\leftrightarrow PQR$

$(iv) CBA \leftrightarrow PQR$

NCERT solutions for class 7 chapter 7 congruence of triangles 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.

$\Delta ABC \cong\Delta PQR$

ii) Since,

ED = MN

DF = NL

FE = LM

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

$\Delta EDF\cong\Delta MNL$ .

iii) Since

AC = PR

BC = QR But

$AB\neq QR$

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.

$\Delta ADB\cong\Delta ADC$ .

2. In Fig 7.15, and D is the mid-point of $\overline{BC}.$ .

(i) State the three pairs of equal parts in $\Delta ADB$ and $\Delta ADC.$
(ii) Is $\Delta ADB\cong \Delta ADC ?$ Give reasons.
(iii) Is $\angle B = \angle C ?$ Why?

Answer:

Here in $\Delta ADB$ and $\Delta ADC.$

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,

$\Delta ADB\cong \Delta ADC$

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

$\angle B = \angle C$ .

3. In Fig 7.16, and . Which of the following statements is meaningfully written?

$(i) \Delta ABC\cong \Delta ABD$ $(ii) \Delta ABC\cong \Delta BAD$

Answer:

Given,

and

AB = AB ( common side )

So By SSS congruency rule,

$\Delta ABC\cong \Delta BAD$ .

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

1. ABC is an isosceles triangle with (Fig 7.17). Take a trace-copy of $\Delta ABC$ and also name it as $\Delta ABC$
(i) State the three pairs of equal parts in $\Delta ABC \; and \: \Delta ACB$ .
(ii) Is $\Delta ABC \cong \Delta ACB$ ? Why or why not?
(iii) Is $\angle B = \angle C$ ? Why or why not?

Answer:

Here, in $\Delta ABC \; and \: \Delta ACB$ .

i)the three pairs of equal parts in $\Delta ABC \; and \: \Delta ACB$ are

AB = AC

BC = CB

AC = AB

ii)

Hence By SSS Congruency rule, they both are congruent.

$\Delta ABC \cong \Delta ACB$

iii) Yes, $\angle B = \angle C$ because $\Delta ABC \; and \: \Delta ACB$ are congruent and by equating the corresponding parts of the triangles we get,

$\angle B = \angle C$ .

1. Which angle is included between the sides $\overline{DE}$ and $\overline{EF}$ of $\bigtriangleup DEF$ ?

Answer:

Since both the sides $\overline{DE}$ and $\overline{EF}$ intersects at E,

$\angle E$ is included between the sides $\overline{DE}$ and $\overline{EF}$ of $\bigtriangleup DEF$ .

2. By applying SAS congruence rule, you want to establish that $\bigtriangleup PQR\cong \bigtriangleup FED$ . It is given that and . 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 $\bigtriangleup PQR\cong \bigtriangleup FED$ we need,

And

$\angle P = \angle F$ .

Hence the extra information we need is $\angle P = \angle F$ .

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 $\Delta ABC$ and $\Delta DEF$

AB = DE

AC = DF

$\angle A \neq \angle D$

Hence, they are not congruent.

ii) In $\Delta ACB$ and $\Delta RPQ$

AC = RP = 2.5 cm

CB = PQ = 3 cm

$\angle C = \angle P = 35^0$

Hence by SAS congruency rule, they are congruent.

$\Delta ACB \cong\Delta RPQ$ .

iii) In $\Delta DFE$ and $\Delta PQR$

DF= PQ = 3.5 cm

FE= QR = 3 cm

$\angle F = \angle Q$

Hence, by SAS congruency rule, they are congruent.

$\Delta DFE\cong\Delta PQR$

iv) In $\Delta QPR$ and $\Delta SRP$

QP = SR = 3.5 cm

PR = RP (Common side)

$\angle QPR = \angle SRP$

Hence, by SAS congruency rule, they are congruent.

$\Delta QPR\cong\Delta SRP$ .

1. What is the side included between the angles and N of $\bigtriangleup M\! N\! P$ ?

Answer:

The side MN is the side which is included between the angles and N of $\bigtriangleup M\! N\! P$ .

2. You want to establish $\bigtriangleup DEF\cong \bigtriangleup MNP$ , using the ASA congruence rule. You are given that $\angle D= \angle M$ and $\angle F= \angle P$ . 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 $\bigtriangleup DEF\cong \bigtriangleup MNP$

$\angle D= \angle M$

$\angle F= \angle P$

And The side joining these angles is

$\overline {DF}= \overline {MP}$ .

So the information that is needed in order to prove congruency is $\overline {DF}= \overline {MP}$ .

4. In Fig 7.25, $\overline{AB}$ and $\overline{CD}$ bisect each other at .

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

(ii) Which of the following statements are true?

$(a) \bigtriangleup AOC\cong \bigtriangleup DOB$
$(b) \bigtriangleup AOC\cong \bigtriangleup BOD$

Answer:

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

CO = DO (given)

OA = OB (given )

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

ii) So by SAS congruency rule,

$\Delta COA \cong\Delta DOB$

that is

$\bigtriangleup AOC\cong \bigtriangleup BOD$

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 $\Delta ABC$ and $\Delta FED$

AB = FE = 3.5 cm

$\angle A = \angle F = 40 ^0$

$\angle B = \angle E = 60^0$

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

$\Delta ABC \cong \Delta FED$

ii) in $\Delta PQR$ and $\Delta FDE$

$\angle Q= \angle D= 90 ^0$

$\angle R= \angle E = 50^0$

But,

$\overline {EF}\neq\overline {RP}$

So, given triangles are not congruent.

iii) in $\Delta RPQ$ and $\Delta LMN$

RQ = LN = 6 cm

$\angle R = \angle L = 60 ^0$

$\angle Q= \angle N= 30^0$

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

$\Delta RPQ\cong \Delta LMN$ .

iv) in $\Delta ADB$ and $\Delta BCA$

AB = BA (common side)

$\angle CAB = \angle DBA= 30 ^0$

$\angle D= \angle C=180^0-30^0-30^0-45^0=75^0$

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

$\Delta ADB \cong \Delta BCA$

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.

$\bigtriangleup DEF$ $\bigtriangleup PQR$

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

Answer:

Given in $\bigtriangleup DEF$ and $\bigtriangleup PQR$ .

$\\\angle D=\angle Q= 60^{o}\\\angle F=\angle R= 80^{o}\\DF=QR= 5cm$

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

$\bigtriangleup DEF\cong\Delta QPR$ .

ii)

Given in $\bigtriangleup DEF$ and $\bigtriangleup PQR$ .

$\\\angle D=\angle Q= 60^{o}\\\angle F=\angle R= 80^{o}\\DF=QP= 6cm$

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 $\bigtriangleup DEF$ and $\bigtriangleup PQR$ .

$\\\angle D=\angle Q= 60^{o}\\\angle F=\angle R= 80^{o}\\DF=QP= 6cm$

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

(i) State the three pairs of equal parts in triangles and .
(ii) Is $\Delta BAC\cong \Delta DAC ?$ Give reasons.
(iii) Is Justify your answer.
(iv) Is $CD = CB \; ?$ Give reasons.

Answer:

Given in triangles and

$\angle DAC=\angle BAC$

$\angle DCA=\angle BCA$

$\overline {AC}=\overline {AC}$ ( common side)

ii)

So, By ASA congruency criterion,triangles and <img alt="DAC" height="12"

src="https://lh4.googleusercontent.com/zhQ_yBWAlSJqSfp2tsO5YE4_uVc3FTYADxv89NSEJ8VevtSJ_pprc5YbEEn8MuPLzhTtrX8KGAm-RIuE7NuaBFPyGWSGk32WNXm2KZ5Myo13pStnLsDWnbufiTt2oTBMKGEyFPQ" style="margin-left: 0px; margin-top: 0px;" width="43" /> are congruent.

$\Delta BAC\cong\Delta DAC$

iii)

Since $\Delta BAC\cong\Delta DAC$ , all corresponding parts will be equal. So

iv)

Since $\Delta BAC\cong\Delta DAC$ , 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 $\Delta PQR$ and $\Delta DEF$

$\angle Q = \angle E$

$PQ\neq DE$

Hence they are not congruent.

ii)

In $\Delta ACB$ and $\Delta BDA$

$\angle D = \angle C$

( same side )

So, by RHS congruency rule,

$\Delta ACB\cong\Delta BDA$

iii)

In $\Delta ABC$ and $\Delta ADC$

$\angle B = \angle D$

( same side )

So, by RHS congruency rule,

$\Delta ABC\cong\Delta ADC$

iv)

In $\Delta PSQ$ and $\Delta PSR$

$\angle PSQ = \angle PSR$

( same side )

So, by RHS congruency rule,

$\Delta PSQ\cong\Delta PSR$

2. It is to be established by RHS congruence rule that $\bigtriangleup ABC\cong \bigtriangleup RPQ$ . What additional information is needed, if it is given that

$\angle B= \angle P= 90^{\underline{0}}$ and

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

$\angle B= \angle P= 90^0$ ( Right angle )

( Side )

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

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

$\bigtriangleup ABC\cong \bigtriangleup RPQ$ .

3. In Fig 7.33, and are altitudes of $\bigtriangleup ABC$ such that .

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

Answer:

i) Given, in $\bigtriangleup CBD$ and $\bigtriangleup BCE$ .

$\angle CEB=\angle BDC=90^o$

$\overline{ BC} = \overline{ CB}$

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

$\bigtriangleup CBD\cong \bigtriangleup BCE$

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

So.

$\bigtriangleup CBD\cong \bigtriangleup BCE$ .

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

(i) State the three pairs of equal parts in $\bigtriangleup AD\! B$ and $\bigtriangleup ADC$ .
(ii) Is $\bigtriangleup ADB\cong \bigtriangleup ADC$ ? Why or why not?
(iii) Is $\angle B= \angle C$ ? Why or why not?
(iv) Is ? Why or why not?

Answer:

i) Given in $\bigtriangleup AD\! B$ and $\bigtriangleup ADC$ .

$\angle ADB = \angle ADC=90^0$

( Common side)

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

$\bigtriangleup ADB\cong \bigtriangleup ADC$

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

So,

$\angle B= \angle C$

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

So,

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 $70^{o}$ ; the measure of the other angle is ___________.
(c) When we write $\angle A = \angle B$ , 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) As the congruent things are a photocopy of each other.

c) When we write $\angle A = \angle B$ , 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 $\Delta ABC\cong \Delta FED$ under the correspondence $ABC\leftrightarrow FED,$ write all the corresponding congruent parts of the triangles.

Answer:

Corresponding parts of the two congruent triangles $ABC\leftrightarrow FED,$ are :

Sides:

$\overline {AB}\:\:and \:\:\overline {FE},$

$\overline {BC}\:\:and\:\:\overline {ED},$

$\overline {AC}\:\:and\:\:\overline {FD}.$

Angles:

$\angle {ABC}=\angle FED$

$\angle {BCA}=\angle EDF$

$\angle {CAB}=\angle DFE$

4. If $\Delta DEF \cong \Delta BCA,$ write the part(s) of $\Delta BCA$ that correspond to

$(i)\; \angle E$ $(ii) \; \overline{EF}$ $(iii) \; \angle F$ $(iv) \; \overline{DF}$

Answer:

Given,

$\Delta DEF \cong \Delta BCA,$

The part of $\Delta BCA$ that correspond to

$(i)\; \angle E=\angle C$

$(ii) \; \overline{EF}=\overline{CA}$

$(iii) \; \angle F=\angle A$

$(iv) \; \overline{DF}=\overline{BA}$

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?

$\\(a)\: Given\; AC=DF$

$So,\; \; \Delta ABC \cong \Delta DEF$

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?

$(b)\; Given:ZX=RP$

$\angle PRQ =\angle XZY$

$So, \Delta PQR\cong \Delta XYZ$

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?

$(c)\; Given:\angle MLN=\angle FGH$

$\angle NML=\angle GFH$

$So,\; \Delta LMN\cong \Delta GFH$

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?

$(d) Given:\; EB = DB$

$\angle A = \angle C = 90^{o}$

$So, \Delta ABE\cong \Delta CDB$

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 $\Delta ART\cong \Delta PEN,$

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

<img height="96"

src="https://lh5.googleusercontent.com/5yU6Q3cgsLY5D5x81iceWzX3qJtAXwkv5Irc0mEzuHmIDoGPT3S_0y128yyxkeqjUIsp60vwOvqq9V0spr3iG8lbXW2TrFBfOWvjHybtd25MiMmhrM5SDP_NcjPdp_0Rc5y1ovs" style="margin-left: 0px; margin-top: 0px;" width="229" />

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:

2.(b) You want to show that $\Delta ART\cong \Delta PEN,$

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

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,

2.(c) You want to show that $\Delta ART\cong \Delta PEN,$

$(i)?\; \; \; \; \; \; \; \; \; \; \; \; \; \; \;\; \; \; \; \; (ii)?$

Answer:

Given, $\Delta ART\cong \Delta PEN,$

also,

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

$(i)\:\angle{RAT}=\angle{EPN}$

$(ii)\:\angle{RTA}=\angle{ENP}$

3. You have to show that $\bigtriangleup AMP\cong \bigtriangleup AMQ$ . In the following proof, supply the missing reasons.

Steps	Reasons

$(ii)\angle PMA = \angle QMA$

$(iv)\bigtriangleup AMP\cong \bigtriangleup AMQ$

Answer

Steps	Reasons
	Given in the question
$(ii)\angle PMA = \angle QMA$	Given in the question.
	the side which is common in both triangle
$(iv)\bigtriangleup AMP\cong \bigtriangleup AMQ$	By SAS Congruence Rule

4. In $\bigtriangleup ABC$ , $\angle A= 30^{o}$ , $\angle B= 40^{o}$ and $\angle C= 110^{o}$ .

In $\bigtriangleup PQR$ , $\angle P=30^{o}$ , $\angle Q=40^{o}$ and $\angle R=110^{o}$ . A student says that $\bigtriangleup ABC\cong \bigtriangleup PQR$ 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 $\Delta RAT\cong$ ?

Answer:

Comparing from the figure.

$R\leftrightarrow W,A\leftrightarrow O,\:and\:\:T\leftrightarrow N$

By SAS Congruency criterion, we can say that

$\Delta RAT\cong \Delta WON$ ]

6. Complete the congruence statement:

Answer:

Comparing from the figure, we get,

$B\leftrightarrow B,A\leftrightarrow A,\:\:and\:\:C\leftrightarrow T$

So By SSS Congruency Rule,

$\Delta BCA\cong \Delta BTA$

Also,

Comparing from the figure, we get,

$P\leftrightarrow R,T\leftrightarrow Q,\:\:and\:\:Q\leftrightarrow S$

So By SSS Congruency Rule,

$\Delta QRS\cong \Delta TPQ$ .

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 $\Delta ABC$ and $\Delta PQR$ are to be congruent, name one additional pair of corresponding parts. What criterion did you use?

Answer:

Given $\Delta ABC \cong \Delta PQR$

One additional pair which is not given in the figure is $\overline {BC}=\overline {QR}$

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

10. Explain, why $\Delta ABC\cong \Delta FED$

Answer:

Comparing both triangles, we have,

$\angle A = \angle F$

$\angle B = \angle E=90^0$

$\overline {BC} = \overline {ED}$

So By RHS congruency criterion,

$\Delta ABC\cong \Delta FED$ .

Responsive Table

Chapter No.Chapter Name

Chapter 1	NCERT solutions for class 7 maths chapter 1 Integers
Chapter 2	NCERT solutions for class 7 maths chapter 2 Fractions and Decimals
Chapter 3	NCERT solutions for class 7 maths chapter 3 Data Handling
Chapter 4	NCERT solutions for class 7 maths chapter 4 Simple Equations
Chapter 5	NCERT solutions for class 7 maths chapter 5 Lines and Angles
Chapter 6	NCERT solutions for class 7 maths chapter 6 The Triangle and its Properties
Chapter 7	NCERT solutions for class 7 maths chapter 7 Congruence of Triangles
Chapter 8	NCERT solutions for class 7 maths chapter 8 comparing quantities
Chapter 9	NCERT solutions for class 7 maths chapter 9 Rational Numbers
Chapter 10	NCERT solutions for class 7 maths chapter 10 Practical Geometry
Chapter 11	NCERT solutions for class 7 maths chapter 11 Perimeter and Area
Chapter 12	NCERT solutions for class 7 maths chapter 12 Algebraic Expressions
Chapter 13	NCERT solutions for class 7 maths chapter 13 Exponents and Powers
Chapter 14	NCERT solutions for class 7 maths chapter 14 Symmetry
Chapter 15	NCERT solutions for class 7 maths chapter 15 visualising-solid-shapes

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NCERT solutions for class 7 maths chapter 7 congruence of triangles topic 7.5

NCERT solutions for class 7 chapter 7 congruence of triangles topic 7.6

NCERT solutions for class 7 maths chapter 7 congruence of triangles topic 7.7

NCERT solutions for class 7 maths chapter 7 congruence of triangles exercise 7.1

NCERT solutions for class 7 maths chapter 7 congruence of triangles exercise 7.2

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