To prove two triangles congruent,

We use ASA Criteria –Angle Side Angle

 

In ASA Congruency Criteria,

  • 2 angles of both the triangles are equal
  • The side between these angles of both the triangles are equal.

For example

52.jpg

 

Here,

  2 angles and the side between them are equal.

 

So, they are congruent

 

Let’s take some examples

 

Are these triangles congruent?

53.jpg

In ∆ABC and ∆RPQ,

∠A = ∠R     ( Both are 30 °)

AB = RP     ( Both are 3 cm )

∠B  = ∠P    ( Both are 40 °)

∴ ∆ABC ≅ ∆RPQ    ( ASA Congruence Rule)

 

Here,

A R

B P

C Q

 

Are these triangles congruent?

54.jpg

In ∆ABC and ∆RPQ,

∠J = ∠N      (Both are 90 °)

JK = NM     (Both are 7 cm )

∠K  = ∠M    (Both are 30 °)

∴ ∆IJK ≅ ∆LNM    (ASA Congruence Rule)

 

Here,

J N

K M

I L

 

Are these triangles congruent?

55.jpg

In these two triangles,

  the side is not same

 

So, these two triangles are not congruent

  ∴ ∆QRP ≇ ∆DEF

 

Are these triangles congruent?

56.jpg

 

In ∆PRQ and ∆MLN,

∠R = ∠L        ( Both are 60 °)

RQ = LN       ( Both  are 6 cm )

∠Q  = ∠N      (Both are 30 °)

∴ ∆PQR ≅ ∆MLN     (ASA Congruence Rule)

 

Here,

R L

Q N

P M

 

 

Are these triangles congruent?

57.jpg

Here, ∠BAD = ∠BAC + ∠CAD

    = 30° + 45°

    = 75°         …(1)

 

And, ∠ABC = ∠ABD + ∠CBD

                   = 30° + 45°

    = 75°       …(2)

 

From (1) & (2)

  ∠BAD = ∠ABC = 75°

 

In ∆ABD and ∆ BAC,

    ∠BAD = ∠ABC       (Proved above )

  AB = BA         (Common )

    ∠DBA = ∠CAB     (Both are 30 °)

∴ ∆ABD ≅ ∆BAC      (ASA Congruency)

Here,

A B

B A

D C

 

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

∆DEF ∆DEF
(a) ∠D = 60°, ∠F = 80°, DF = 5 cm

∠Q = 60°, ∠R = 80°, QR = 5 cm

60.jpg

In ∆DEF and ∆QPR,

∠D = ∠Q      (Both are 60 °)

DF = QR      (Both are 5 cm )

∠F = ∠R       (Both are 80 °)

∴ ∆DEF ≅ ∆QPR     (ASA Congruence Rule)

 

Here,

D Q

F R

E P

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

∆DEF ∆DEF
(a) ∠D = 60°, ∠F = 80°, DF = 6 cm

∠Q = 60°, ∠R = 80°, QP = 6 cm

59.jpg

In ∆DEF and ∆QPR,

∠D = ∠Q      (Both are 60 °)

DF = QR      (Both are 5 cm )

∠F = ∠R      (Both are 80 °)

∴ ∆DEF ≅ ∆QPR    (ASA Congruence Rule)

 

Here,

D Q

F R

E P

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

∆DEF ∆DEF
(a) ∠E = 80°, ∠F = 30°, EF = 5 cm

∠P = 80°, PQ = 5 cm, ∠R = 30° cm

62.jpg

Since PQ is not the side between ∠P and ∠R

Thus, side between two angles is not equal

 

∴ The triangles are not congruent

So, ∆DEF ≇ ∆QPR

 

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

State the three pairs of equal parts in triangles BAC and DAC.

63.jpg

In ∆BAC and ∆DAC,

  ∠BAC = ∠DAC      (AZ bisects ∠DAB )

AC = AC                 (Common )

  ∠BCA = ∠DCA     (AZ bisects ∠DCB )

∴  ∆BAC ≅ ∆DAC      (ASA Congruence Rule)

 

 

  1. Chapter 7 Class 7 Congruence of Triangles
  2. Concept wise

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.