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Congruence Of Triangles

Class 7th Mathematics CBSE Solution
Exercise 7.1
  1. Complete the following statements: (a) Two line segments are congruent if _____.…
  2. Give any two real-life examples for congruent shapes.
  3. If ∆ABC ∆FED under the correspondence ABC FED, write all the corresponding…
  4. If DEF congruent BCA, write the part(s) of BCA that correspond to: (i) angle e…
Exercise 7.2
  1. Which congruence criterion do you use in the following? (a) Given: AC=DF AB=DF…
  2. You want to show that ART is equivalent to PEN, (a) If you have to use SSS…
  3. You have to show that deltaaminus or plus congruent deltaamq . In the following…
  4. In delta ABC, A = 30o, B = 40o and C = 110o. In deltapqr , P = 30o, Q = 40o and…
  5. In the figure, the two triangles are congruent. The corresponding parts are…
  6. Complete the congruence statement: root 4 deltabca congruent ? p deltaqrs…
  7. In a squared sheet, draw two triangles of equal areas such that: (i) The…
  8. Draw a rough sketch of two triangles such that they have five pairs of congruent…
  9. If are to be congruent, name one additional pair of corresponding parts. What…
  10. Explain, why
Exercise 7.1
Question 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 of equal length


(b) Among two congruent angles, one has a measure of 70o; the measure of the other angle is 70o


(c) When we write ∠A = ∠B, it means m ∠A = m ∠B
Question 2.

Give any two real-life examples for congruent shapes.
Answer:

There could be infinite number of examples but two real-life examples for congruent shapes are as follows:


(i) Cricket balls of same brand.


(ii) Biscuits in the same packet
Question 3.

If ∆ABC ≅ ∆FED under the correspondence ABC FED, write all the corresponding congruent parts of the triangles.
Answer:

We know that,


If two triangles are congruent than corresponding parts of both the triangles are equal



Therefore,


Corresponding parts of triangle ABC and FED are as follows:


∠A ∠F


∠B ∠E


∠C ∠D


AB FE


CA DF
Question 4.

If ∆DEF ∆BCA, write the part(s) of ∆BCA that correspond to:

(i) (ii) (iii) (iv)


Answer:


(i) If triangle DEF and BCA are congruent than:


∠E corresponds to ∠C


(ii) If triangle DEF and BCA are congruent than:


EF corresponds to CA


(iii) If triangle DEF and BCA are congruent than:


∠F corresponds to ∠A


(iv) If triangle DEF and BCA are congruent than:


DF corresponds to BA
Exercise 7.2
Question 1.

Which congruence criterion do you use in the following?

(a) Given: AC=DF

AB=DF

BC=EF

∆ABC ∆DEF





(b) Given: ZX=R

RQ=ZY

∠PQR = ∠XYZ

∆PQR ∆XYZ





(c) ∠MLN = ∠FGH

∠NML = ∠GFH

ML=FG

∆LMN ∆GFH





(d) Given: EB=DB

AE=BC

∠A = ∠C=90o

∆ABC ∆CDB
Answer:

(a) In the above-given question, we have given that:


AB = DE


Also, BC = EF


And, AC = DF


Therefore, both the triangles are equal by SSS congruence criterion


As the sides of both the triangles are equal.


(b) In the above-given question, we have given that:


ZX = RP


Also,


RQ = ZY


And,


∠PQR = ∠XYZ


Therefore, both the triangles are equal by SAS congruence criterion


As


Two sides of both the triangles are equal. Also, ∠PQR and ∠XYZ are equal.


(c) In the above-given question, we have given that:


∠MLN = ∠FGH


Also,


∠NML = ∠GFH


And,


ML = FG


Therefore, both the triangles are equal by SAS congruence criterion


As two angles and the side included between these angles of ∠LMN are equal to two angles and the side included between these angles of ∠GFH.


(d) In the above-given question, we have given that:


EB = DB


Also,


AE = BC


And,


∠A = ∠C = 90o


Therefore, both the triangles are equal by RHS congruence criterion


As, two right triangles, one side, and hypotenuse of the above-given triangles are equal
Question 2.

You want to show that ∆ART ∆PEN,

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

(i) AR =

(ii) RT =

(iii) AT =

(iv) AT =



(b) If it is given that ∠T = ∠N and you are to use

SAS criterion, you need to have

(i) RT = and (ii) PN =



(c) If it is given that AT = PN and you are to use

ASA criterion, you need to have

(i) ? (ii) ?
Answer:

(a) In the given question, we have to show that:



Now,


If we have to use SSS criterion, then we need to show the following things:


(i) AR = PE


(ii) RT = EN


(iii) AT = PN


(b) In the above given question, we have:


If it is given that ∠T = ∠N and we have to use SAS criterion


Then, in that case we have to show the following things:


(i) RT = EN


(ii) PN = AT


(c) In the above question, we have:


If it is given that AT = PN and we have to use ASA criterion


Then, it that case we have to show the following things:


(i) ∠ATR = ∠PNE


(ii) ∠RAT = ∠EPN
Question 3.

You have to show that . In the following proof, supply the missing reasons.
Answer:

In the above question, we have to show that:



For this, we have to show the following things:
Question 4.

In ABC, ∠A = 30o, ∠B = 40o and ∠C = 110o. In, ∠P = 30o, ∠Q = 40o and ∠R = 110o. A student says that by AAA congruence criterion. Is he justified? Why or why not?
Answer:

According to the question, in ABC, ∠A = 30o, ∠B = 40o and ∠C = 110o


And, in , ∠P = 30o, ∠Q = 40o and ∠R = 110o


We have to justify that:


Is , by AAA congruence rule


As the given property shows that the two triangles have their all three respective angles equal.


But, no information explains any of the sides.


Thus, it can be concluded that AAA Property does not prove the triangle to be congruent.
Question 5.

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

From the above given figures,


It can be observed that:


∠RAT = ∠WON


∠ART = ∠OWN


AR = OW


∴ By ASA congruence criterion
Question 6.

Complete the congruence statement:

?

?
Answer:

It is given in the question that,

BC = BT, TA = CA and BA = BA (common)

Therefore,

Similarly,

TQ = QS, PT = RQ and PQ = RS

Therefore,

By SSS congruence rule
Question 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:

(i) In the first part, we have:



and have the same area


and are congruent to each other


And,


Both the triangles having equal perimeter


(ii) In the second part, we have:



Both the triangles have same height and base


Therefore,


Areas of both the triangles are equal


As,


These triangles are not congruent to each other


The perimeter of both the triangles will be different
Question 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:

We have to find two triangles such that there five parts are congruent to each other but yet the triangles are not congruent



In the figure shown above,
∠ A = ∠ D
∠ B = ∠ E
∠ C = ∠ F
AC = EF
BC = DE

Thus, we have five congruent parts and the sides AB and DF are not equal.
Hence, five parts are congruent and the triangles are not congruent.
Question 9.

If are to be congruent, name one additional pair of corresponding parts. What criterion did you use?
Answer:

From the above given questions, we have:


If triangle ABC and triangle PQR need to be congruent then the other additional pair of corresponding part is:


BC = QR


As, two angles of given triangles are already equal.


Therefore,


By ASA congruence rule, we have:
Question 10.

Explain, why
Answer:

From the given figure, it is clear:


∠ABC = ∠FED = 90° (i)


Also,


∠BAC = ∠EFD (ii)

Also,

From figures,

BC = ED (iii)

Therefore,


From (i), (ii) and (iii) and by ASA congruence rule,


FED


By AAS congruence criterion