**Congruence of Triangles**

**Exerci****se 7.2**

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

(a)

Given: AC = DF

AB = DE

BC = EF

So, DABC @ DDEF

Ans: DABC @ DDEF by SSS rule

(b)

Given: ZX = RP

RQ = ZY

ÐPRQ = ÐXZY

So, DPQR @ DXYZ

Ans: DPQR @ DXYZ by SAS rule

(c)

Given: ÐMLN = ÐFGH

ÐNML = ÐGFH

ML = FG

So, DLMN @ DGFH

Ans: DLMN @ DGFH by ASA rule

(d)

Given: EB = DB

AE = BC

ÐA = ÐC = 90°

So, DABE @ DCDB

Ans: DABE @ DCDB by RHS rule

2). You want to show that DART @ DPEN,

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

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

(b) If it is given that ÐT = ÐN 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) ?

Solution:

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

(i) AR @ PE

(ii) RT @ EN

(iii) AT @ PN

(b) If it is given that ÐT = ÐN and you are to use SAS criterion,

you need to have

(i) RT @ EN and

(ii) PN @ AT

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

you need to have

(i) ÐA @ ÐP

(ii) ÐT @ ÐN

3). You have to show that DAMP @ DAMQ. In the following proof, supply the missing reasons.

Steps Reasons

(i) PM @ QM (i) . . . . . . . . . . . . . . .

(ii) ÐPMA @ ÐQMA (ii) . . . . . . . . . . . . . . .

(iii) AM @ AM (iii) . . . . . . . . . . . . . . .

(iv) DAMP @ DAMQ (iv) . . . . . . . . . . . . . . .

Solution:

Steps Reasons

(i) PM @ QM (i) given

(ii) ÐPMA @ ÐQMA (ii) given

(iii) AM @ AM (iii) Common Side

(iv) DAMP @ DAMQ (iv) SAS Rule

4). In DABC, ÐA = 30° , ÐB = 40° and ÐC = 110° In DPQR, ÐP = 30° , ÐQ = 40° and ÐR = 110°. A student says that DABC @ DPQR by AAA congruence criterion. Is he justified? Why or why not?

Ans: No. The student is not justified because there is no AAA congruence criterion.

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

In DRAT and DWON

side AR @ side OW given

ÐA @ ÐO given

side AT @ side ON given

DRAT @ DWON by SAS Rule

We can also show that the two triangles are congruent by ASA Rule

II Method

In DRAT and DWON

ÐA @ ÐO given

side AT @ side ON given

ÐT @ ÐN given

DRAT @ DWON by ASA Rule

6). Complete the congruence statement:

(i).

In DBCA and DBTA

ÐABC @ÐABT given

side BC @ side ON given

ÐC @ ÐT given

DBCA @ DBTA by ASA Rule

(ii).

In DQRS and DTPQ

ÐR @ ÐP given

side QR @ side TP given

ÐRSQ @ ÐPQT given

DQRS @ DTPQ by ASA Rule

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?

Solution:

(i) the triangles are congruent.

In DABC and DPQR

side AB @ side PQ given

side AC @ side PR given

side BC @ side QR given

DABC @ DPQR by SSS Rule

AB + BC + AC = PQ + PR + QR

Perimeter of DABC = Perimeter of DPQR

(ii) the triangles are not congruent.

In DABC and DPQR

side AB ≠ side PQ given

side AC ≠ side PR given

side BC ≠ side QR given

DABC ≠ DPQR by SSS Rule

AB + BC + AC ≠ PQ + PR + QR

Perimeter of DABC ≠ Perimeter of DPQR

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

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

In DABC and DPQR

ÐB @ ÐQ given

ÐC @ ÐR given

The additional pair of corresponding parts so that the two triangles are congruent can be

side QR @ side TP

DABC @ DPRQ by ASA Rule

10). Explain, why DABC @ DFED.

In DABC and DFED

ÐA @ ÐF given (I)

ÐB @ ÐE given (II)

In DABC

By angle sum property of triangle

ÐA + ÐB+ ÐC = 180^{0}

ÐC = 180^{0} – (ÐA + ÐB) (III)

Similarly In DFED

ÐF + ÐE+ ÐD = 180^{0}

ÐD = 180^{0} – (ÐF + ÐE) (IV)

From I, II, III and IV, we have

ÐC = ÐÐD (V)

In DABC and DFED

ÐB @ ÐE given

side BC @ side ED given

ÐC @ ÐD by (V)

DABC @ DFED by ASA Rule

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